Subgroups — C₆₇×#22462 #30701713

Order 17152.

1320 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3584	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2901	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1906	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1881	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1794	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	#585×C₆₇	C₂²	1	2	yes	—
13	#578×C₆₇	—	2	1	no	—
14	#578×C₆₇	C₄	1	2	yes	complemented
15	#577×C₆₇	—	2	2	no	—
16	#577×C₆₇	C₄	1	4	yes	complemented
17	#576×C₆₇	—	2	1	no	—
18	#576×C₆₇	C₄	1	2	yes	complemented
19	#575×C₆₇	—	2	2	no	—
20	#575×C₆₇	C₂²	1	2	yes	—
21	C₂×#135×C₆₇	—	2	2	no	—
22	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
24	#568×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×#41×C₆₇	C₂²	1	2	yes	complemented
27	#444×C₆₇	—	2	2	no	—
28	#443×C₆₇	—	2	2	no	—
29	C₄×Q₁₆×C₆₇	—	2	1	no	—
30	C₄×SD₁₆×C₆₇	—	2	2	no	—
31	C₄×D₈×C₆₇	—	2	1	no	—
32	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#427×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#421×C₆₇	—	2	4	no	—
35	#421×C₆₇	C₂²	1	2	yes	—
36	#421×C₆₇	C₂²	1	2	yes	complemented
37	#421×C₆₇	C₂²	1	2	yes	complemented
38	C₂×#104×C₆₇	—	2	4	no	—
39	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
41	#416×C₆₇	C₂²	1	1	yes	characteristic
42	#416×C₆₇	C₂²	1	2	yes	complemented
43	#410×C₆₇	C₂²	1	2	yes	complemented
44	#408×C₆₇	C₂²	1	2	yes	complemented
45	#405×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	#344×C₆₇	C₂²	1	2	yes	complemented
48	#341×C₆₇	C₂²	1	1	yes	characteristic, complemented
49	#339×C₆₇	C₂²	1	1	yes	characteristic, complemented
50	2₋¹⁺⁴×C₆₇	—	2	2	no	—
51	2₋¹⁺⁴×C₆₇	—	2	2	no	—
52	2₋¹⁺⁴×C₆₇	C₂×C₄	1	4	yes	complemented
53	H(5,𝔽₂)×C₆₇	—	2	2	no	—
54	H(5,𝔽₂)×C₆₇	—	2	2	no	—
55	H(5,𝔽₂)×C₆₇	—	4	2	no	—
56	H(5,𝔽₂)×C₆₇	C₂×C₄	1	4	yes	complemented
57	C₂×#41×C₆₇	—	2	4	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	—	4	2	no	—
60	C₂×#41×C₆₇	C₂³	1	2	yes	—
61	C₂×#41×C₆₇	C₂³	1	2	yes	—
62	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
63	#137×C₆₇	—	2	2	no	—
64	#137×C₆₇	—	2	4	no	—
65	#137×C₆₇	—	2	4	no	—
66	#137×C₆₇	C₂×C₄	1	4	yes	complemented
67	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
68	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
69	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
70	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	4	yes	complemented
71	#135×C₆₇	—	2	4	no	—
72	#135×C₆₇	—	2	2	no	—
73	#135×C₆₇	—	2	4	no	—
74	#135×C₆₇	—	2	2	no	—
75	#135×C₆₇	—	2	1	no	—
76	#135×C₆₇	—	2	1	no	—
77	#135×C₆₇	—	2	1	no	—
78	#135×C₆₇	—	2	1	no	—
79	#135×C₆₇	C₂×C₄	1	2	yes	—
80	#135×C₆₇	C₂×C₄	1	2	yes	—
81	#135×C₆₇	D₄	1	2	yes	—
82	#135×C₆₇	D₄	1	2	yes	—
83	C₂×Q₁₆×C₆₇	—	2	2	no	—
84	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
85	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
86	C₂×SD₁₆×C₆₇	—	2	4	no	—
87	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
88	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
89	C₂×D₈×C₆₇	—	2	2	no	—
90	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
92	#131×C₆₇	—	2	1	no	—
93	#131×C₆₇	—	2	1	no	—
94	#131×C₆₇	—	2	4	no	—
95	#131×C₆₇	—	4	2	no	—
96	#131×C₆₇	C₂×C₄	1	2	yes	complemented
97	#131×C₆₇	C₂×C₄	1	2	yes	complemented
98	C₂×M₁₆×C₆₇	—	2	1	no	—
99	C₂×M₁₆×C₆₇	—	2	4	no	—
100	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
101	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₆₇	—	2	1	no	—
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₄×Q₈×C₆₇	—	2	2	no	—
107	C₄×D₄×C₆₇	—	2	2	no	—
108	C₄×D₄×C₆₇	—	4	2	no	—
109	#117×C₆₇	—	2	2	no	—
110	#117×C₆₇	C₂³	1	2	yes	—
111	#117×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	—	2	2	no	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	#108×C₆₇	—	2	1	no	—
115	#108×C₆₇	—	2	1	no	—
116	#107×C₆₇	—	2	1	no	—
117	#106×C₆₇	—	2	1	no	—
118	#104×C₆₇	—	2	4	no	—
119	#104×C₆₇	—	2	4	no	—
120	#104×C₆₇	—	2	4	no	—
121	#104×C₆₇	—	2	2	no	—
122	#104×C₆₇	—	2	2	no	—
123	#104×C₆₇	D₄	1	2	yes	—
124	#104×C₆₇	D₄	1	2	yes	—
125	#103×C₆₇	—	2	2	no	—
126	#103×C₆₇	—	2	1	no	—
127	#102×C₆₇	—	2	2	no	—
128	#102×C₆₇	—	2	1	no	—
129	#99×C₆₇	—	4	2	no	—
130	C₂²⋊C₈×C₆₇	—	4	2	no	—
131	#97×C₆₇	—	2	1	no	—
132	C₄×C₈×C₆₇	—	2	1	no	—
133	C₄×C₈×C₆₇	—	2	1	no	—
134	C₄×C₈×C₆₇	—	2	1	no	—
135	#41×C₆₇	—	2	1	no	—
136	#41×C₆₇	—	2	2	no	—
137	#41×C₆₇	—	2	2	no	—
138	#41×C₆₇	—	2	1	no	—
139	#41×C₆₇	—	2	4	no	—
140	#41×C₆₇	—	2	4	no	—
141	#41×C₆₇	—	2	4	no	—
142	#41×C₆₇	—	2	4	no	—
143	#41×C₆₇	—	2	2	no	—
144	#41×C₆₇	—	2	2	no	—
145	#41×C₆₇	—	2	2	no	—
146	#41×C₆₇	—	2	2	no	—
147	#41×C₆₇	—	4	4	no	—
148	#41×C₆₇	—	4	4	no	—
149	#41×C₆₇	—	4	4	no	—
150	#41×C₆₇	—	4	4	no	—
151	#41×C₆₇	—	4	2	no	—
152	#41×C₆₇	—	4	2	no	—
153	#41×C₆₇	C₂²×C₄	1	4	yes	—
154	#41×C₆₇	C₂²×C₄	1	2	yes	—
155	#41×C₆₇	C₂×D₄	1	4	yes	—
156	#41×C₆₇	C₂×D₄	1	2	yes	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	4	no	—
159	C₂×Q₈×C₆₇	—	4	2	no	—
160	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
161	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
162	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	—	2	1	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	4	no	—
167	C₂×D₄×C₆₇	—	2	2	no	—
168	C₂×D₄×C₆₇	—	4	4	no	—
169	C₂×D₄×C₆₇	—	4	2	no	—
170	C₂×D₄×C₆₇	—	4	4	no	—
171	C₂×D₄×C₆₇	—	4	4	no	—
172	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
173	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	1	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	4	4	no	—
180	C₂²×C₄×C₆₇	—	4	2	no	—
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₆₇	—	2	4	no	—
183	Q₁₆×C₆₇	—	2	1	no	—
184	Q₁₆×C₆₇	—	2	1	no	—
185	Q₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
186	Q₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
187	SD₁₆×C₆₇	—	2	4	no	—
188	SD₁₆×C₆₇	—	2	4	no	—
189	SD₁₆×C₆₇	—	2	2	no	—
190	SD₁₆×C₆₇	—	2	2	no	—
191	SD₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
192	SD₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
193	D₈×C₆₇	—	2	4	no	—
194	D₈×C₆₇	—	2	1	no	—
195	D₈×C₆₇	—	2	1	no	—
196	D₈×C₆₇	C₂²⋊C₄	1	2	yes	—
197	D₈×C₆₇	C₂²⋊C₄	1	2	yes	—
198	M₁₆×C₆₇	—	2	1	no	—
199	M₁₆×C₆₇	—	2	1	no	—
200	M₁₆×C₆₇	—	2	2	no	—
201	M₁₆×C₆₇	—	2	2	no	—
202	M₁₆×C₆₇	—	2	2	no	—
203	M₁₆×C₆₇	—	2	2	no	—
204	M₁₆×C₆₇	—	4	4	no	—
205	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
206	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
207	C₂×C₈×C₆₇	—	2	1	no	—
208	C₂×C₈×C₆₇	—	2	1	no	—
209	C₂×C₈×C₆₇	—	2	2	no	—
210	C₂×C₈×C₆₇	—	2	2	no	—
211	C₂×C₈×C₆₇	—	2	1	no	—
212	C₂×C₈×C₆₇	—	2	1	no	—
213	C₂×C₈×C₆₇	—	2	1	no	—
214	C₂×C₈×C₆₇	—	2	2	no	—
215	C₂×C₈×C₆₇	—	4	4	no	—
216	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
217	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
218	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
219	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
220	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
221	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
222	C₄⋊C₄×C₆₇	—	2	2	no	—
223	C₄⋊C₄×C₆₇	—	2	1	no	—
224	C₄⋊C₄×C₆₇	—	4	2	no	—
225	C₂²⋊C₄×C₆₇	—	2	2	no	—
226	C₂²⋊C₄×C₆₇	—	2	1	no	—
227	C₂²⋊C₄×C₆₇	—	4	2	no	—
228	C₄²×C₆₇	—	2	1	no	—
229	C₄²×C₆₇	—	2	2	no	—
230	C₄²×C₆₇	—	2	1	no	—
231	C₄²×C₆₇	—	4	2	no	—
232	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
233	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
234	C₂³×C₆₇	—	2	2	no	—
235	C₂³×C₆₇	—	2	1	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	4	4	no	—
238	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
239	Q₈×C₆₇	—	2	1	no	—
240	Q₈×C₆₇	—	2	2	no	—
241	Q₈×C₆₇	—	2	2	no	—
242	Q₈×C₆₇	—	2	2	no	—
243	Q₈×C₆₇	—	4	4	no	—
244	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
245	Q₈×C₆₇	C₄×D₄	1	2	yes	—
246	D₄×C₆₇	—	2	2	no	—
247	D₄×C₆₇	—	2	4	no	—
248	D₄×C₆₇	—	2	2	no	—
249	D₄×C₆₇	—	2	4	no	—
250	D₄×C₆₇	—	2	2	no	—
251	D₄×C₆₇	—	2	2	no	—
252	D₄×C₆₇	—	2	1	no	—
253	D₄×C₆₇	—	2	2	no	—
254	D₄×C₆₇	—	4	4	no	—
255	D₄×C₆₇	—	4	4	no	—
256	D₄×C₆₇	—	4	4	no	—
257	D₄×C₆₇	—	4	2	no	—
258	D₄×C₆₇	—	4	2	no	—
259	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
260	D₄×C₆₇	C₄×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	1	no	—
270	C₂×C₄×C₆₇	—	4	4	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	4	4	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	2	no	—
275	C₂×C₄×C₆₇	—	4	2	no	—
276	C₂×C₄×C₆₇	—	4	2	no	—
277	C₂×C₄×C₆₇	—	4	1	no	—
278	C₂×C₄×C₆₇	—	8	4	no	—
279	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
280	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
285	C₈×C₆₇	—	2	2	no	—
286	C₈×C₆₇	—	2	2	no	—
287	C₈×C₆₇	—	4	2	no	—
288	C₈×C₆₇	—	4	1	no	—
289	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
290	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
291	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
292	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	2	4	no	—
295	C₂²×C₆₇	—	4	1	no	—
296	C₂²×C₆₇	—	4	2	no	—
297	C₂²×C₆₇	—	8	2	no	—
298	C₂²×C₆₇	—	8	1	no	—
299	C₂²×C₆₇	—	8	2	no	—
300	C₂²×C₆₇	—	16	4	no	—
301	C₂²×C₆₇	#518	1	1	yes	characteristic
302	C₂²×C₆₇	D₄²	1	1	yes	characteristic
303	C₂²×C₆₇	#546	1	1	yes	characteristic
304	C₄×C₆₇	—	2	2	no	—
305	C₄×C₆₇	—	2	2	no	—
306	C₄×C₆₇	—	2	4	no	—
307	C₄×C₆₇	—	4	1	no	—
308	C₄×C₆₇	—	4	2	no	—
309	C₄×C₆₇	—	8	1	no	—
310	C₄×C₆₇	—	8	2	no	—
311	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
312	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
313	C₄×C₆₇	D₄²	1	1	yes	characteristic
314	C₄×C₆₇	#546	1	1	yes	characteristic
315	#22462	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂×C₆₇	—	2	1	no	—
317	C₂×C₆₇	—	2	1	no	—
318	C₂×C₆₇	—	2	1	no	—
319	C₂×C₆₇	—	4	4	no	—
320	C₂×C₆₇	—	4	2	no	—
321	C₂×C₆₇	—	8	2	no	—
322	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
323	#3584	C₂×C₆₇	1	1	yes	characteristic
324	#2957	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
326	#2901	C₂×C₆₇	1	2	yes	complemented
327	#1976	C₂×C₆₇	1	1	yes	characteristic, complemented
328	#1974	C₂×C₆₇	1	1	yes	characteristic, complemented
329	#1906	C₂×C₆₇	1	2	yes	complemented
330	#1881	C₂×C₆₇	1	2	yes	complemented
331	#1814	C₂×C₆₇	1	1	yes	characteristic, complemented
332	#1794	C₂×C₆₇	1	2	yes	complemented
333	#1764	C₂×C₆₇	1	1	yes	characteristic, complemented
334	C₆₇	#22462	1	1	yes	characteristic, direct product factor, Sylow
335	#585	C₂²×C₆₇	1	2	yes	—
336	#578	—	2	1	no	—
337	#578	C₄×C₆₇	1	2	yes	complemented
338	#577	—	2	2	no	—
339	#577	C₄×C₆₇	1	4	yes	complemented
340	#576	—	2	1	no	—
341	#576	C₄×C₆₇	1	2	yes	complemented
342	#575	—	2	2	no	—
343	#575	C₂²×C₆₇	1	2	yes	—
344	C₂×#135	—	2	2	no	—
345	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
346	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
347	#568	C₂²×C₆₇	1	2	yes	complemented
348	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
349	C₄×#41	C₂²×C₆₇	1	2	yes	complemented
350	#444	—	2	2	no	—
351	#443	—	2	2	no	—
352	C₄×Q₁₆	—	2	1	no	—
353	C₄×SD₁₆	—	2	2	no	—
354	C₄×D₈	—	2	1	no	—
355	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic, complemented
356	#427	C₂²×C₆₇	1	1	yes	characteristic, complemented
357	#421	—	2	4	no	—
358	#421	C₂²×C₆₇	1	2	yes	—
359	#421	C₂²×C₆₇	1	2	yes	complemented
360	#421	C₂²×C₆₇	1	2	yes	complemented
361	C₂×#104	—	2	4	no	—
362	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
363	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
364	#416	C₂²×C₆₇	1	1	yes	characteristic
365	#416	C₂²×C₆₇	1	2	yes	complemented
366	#410	C₂²×C₆₇	1	2	yes	complemented
367	#408	C₂²×C₆₇	1	2	yes	complemented
368	#405	C₂²×C₆₇	1	1	yes	characteristic, complemented
369	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
370	#344	C₂²×C₆₇	1	2	yes	complemented
371	#341	C₂²×C₆₇	1	1	yes	characteristic, complemented
372	#339	C₂²×C₆₇	1	1	yes	characteristic, complemented
373	2₋¹⁺⁴	—	2	2	no	—
374	2₋¹⁺⁴	—	2	2	no	—
375	2₋¹⁺⁴	C₂×C₄×C₆₇	1	4	yes	complemented
376	H(5,𝔽₂)	—	2	2	no	—
377	H(5,𝔽₂)	—	2	2	no	—
378	H(5,𝔽₂)	—	4	2	no	—
379	H(5,𝔽₂)	C₂×C₄×C₆₇	1	4	yes	complemented
380	C₂×#41	—	2	4	no	—
381	C₂×#41	—	2	4	no	—
382	C₂×#41	—	4	2	no	—
383	C₂×#41	C₂³×C₆₇	1	2	yes	—
384	C₂×#41	C₂³×C₆₇	1	2	yes	—
385	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
386	#137	—	2	2	no	—
387	#137	—	2	4	no	—
388	#137	—	2	4	no	—
389	#137	C₂×C₄×C₆₇	1	4	yes	complemented
390	AGL(1,ℤ/8)	—	2	2	no	—
391	AGL(1,ℤ/8)	—	2	4	no	—
392	AGL(1,ℤ/8)	—	2	4	no	—
393	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	4	yes	complemented
394	#135	—	2	4	no	—
395	#135	—	2	2	no	—
396	#135	—	2	4	no	—
397	#135	—	2	2	no	—
398	#135	—	2	1	no	—
399	#135	—	2	1	no	—
400	#135	—	2	1	no	—
401	#135	—	2	1	no	—
402	#135	C₂×C₄×C₆₇	1	2	yes	—
403	#135	C₂×C₄×C₆₇	1	2	yes	—
404	#135	D₄×C₆₇	1	2	yes	—
405	#135	D₄×C₆₇	1	2	yes	—
406	C₂×Q₁₆	—	2	2	no	—
407	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
408	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
409	C₂×SD₁₆	—	2	4	no	—
410	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	complemented
411	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
412	C₂×D₈	—	2	2	no	—
413	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
414	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
415	#131	—	2	1	no	—
416	#131	—	2	1	no	—
417	#131	—	2	4	no	—
418	#131	—	4	2	no	—
419	#131	C₂×C₄×C₆₇	1	2	yes	complemented
420	#131	C₂×C₄×C₆₇	1	2	yes	complemented
421	C₂×M₁₆	—	2	1	no	—
422	C₂×M₁₆	—	2	4	no	—
423	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
424	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
425	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
426	C₂²×C₈	—	2	1	no	—
427	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
428	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
429	C₄×Q₈	—	2	2	no	—
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	—	4	2	no	—
432	#117	—	2	2	no	—
433	#117	C₂³×C₆₇	1	2	yes	—
434	#117	C₂³×C₆₇	1	1	yes	characteristic
435	C₂×C₄²	—	2	2	no	—
436	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
437	#108	—	2	1	no	—
438	#108	—	2	1	no	—
439	#107	—	2	1	no	—
440	#106	—	2	1	no	—
441	#104	—	2	4	no	—
442	#104	—	2	4	no	—
443	#104	—	2	4	no	—
444	#104	—	2	2	no	—
445	#104	—	2	2	no	—
446	#104	D₄×C₆₇	1	2	yes	—
447	#104	D₄×C₆₇	1	2	yes	—
448	#103	—	2	2	no	—
449	#103	—	2	1	no	—
450	#102	—	2	2	no	—
451	#102	—	2	1	no	—
452	#99	—	4	2	no	—
453	C₂²⋊C₈	—	4	2	no	—
454	#97	—	2	1	no	—
455	C₄×C₈	—	2	1	no	—
456	C₄×C₈	—	2	1	no	—
457	C₄×C₈	—	2	1	no	—
458	#41	—	2	1	no	—
459	#41	—	2	2	no	—
460	#41	—	2	2	no	—
461	#41	—	2	1	no	—
462	#41	—	2	4	no	—
463	#41	—	2	4	no	—
464	#41	—	2	4	no	—
465	#41	—	2	4	no	—
466	#41	—	2	2	no	—
467	#41	—	2	2	no	—
468	#41	—	2	2	no	—
469	#41	—	2	2	no	—
470	#41	—	4	4	no	—
471	#41	—	4	4	no	—
472	#41	—	4	4	no	—
473	#41	—	4	4	no	—
474	#41	—	4	2	no	—
475	#41	—	4	2	no	—
476	#41	C₂²×C₄×C₆₇	1	4	yes	—
477	#41	C₂²×C₄×C₆₇	1	2	yes	—
478	#41	C₂×D₄×C₆₇	1	4	yes	—
479	#41	C₂×D₄×C₆₇	1	2	yes	—
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	4	no	—
482	C₂×Q₈	—	4	2	no	—
483	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
484	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
485	C₂×Q₈	#41×C₆₇	1	1	yes	characteristic
486	C₂×D₄	—	2	2	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	2	2	no	—
491	C₂×D₄	—	4	4	no	—
492	C₂×D₄	—	4	2	no	—
493	C₂×D₄	—	4	4	no	—
494	C₂×D₄	—	4	4	no	—
495	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
496	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
497	C₂×D₄	#41×C₆₇	1	1	yes	characteristic
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	4	4	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
505	Q₁₆	—	2	4	no	—
506	Q₁₆	—	2	1	no	—
507	Q₁₆	—	2	1	no	—
508	Q₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
509	Q₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
510	SD₁₆	—	2	4	no	—
511	SD₁₆	—	2	4	no	—
512	SD₁₆	—	2	2	no	—
513	SD₁₆	—	2	2	no	—
514	SD₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
515	SD₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
516	D₈	—	2	4	no	—
517	D₈	—	2	1	no	—
518	D₈	—	2	1	no	—
519	D₈	C₂²⋊C₄×C₆₇	1	2	yes	—
520	D₈	C₂²⋊C₄×C₆₇	1	2	yes	—
521	M₁₆	—	2	1	no	—
522	M₁₆	—	2	1	no	—
523	M₁₆	—	2	2	no	—
524	M₁₆	—	2	2	no	—
525	M₁₆	—	2	2	no	—
526	M₁₆	—	2	2	no	—
527	M₁₆	—	4	4	no	—
528	M₁₆	C₂²×C₄×C₆₇	1	2	yes	—
529	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	2	no	—
533	C₂×C₈	—	2	2	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	2	no	—
538	C₂×C₈	—	4	4	no	—
539	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
540	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
541	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
542	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
543	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
544	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	4	2	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	2	no	—
554	C₄²	—	4	2	no	—
555	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
556	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
557	C₂³	—	2	2	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	2	2	no	—
560	C₂³	—	4	4	no	—
561	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
562	Q₈	—	2	1	no	—
563	Q₈	—	2	2	no	—
564	Q₈	—	2	2	no	—
565	Q₈	—	2	2	no	—
566	Q₈	—	4	4	no	—
567	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
568	Q₈	C₄×D₄×C₆₇	1	2	yes	—
569	D₄	—	2	2	no	—
570	D₄	—	2	4	no	—
571	D₄	—	2	2	no	—
572	D₄	—	2	4	no	—
573	D₄	—	2	2	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	1	no	—
576	D₄	—	2	2	no	—
577	D₄	—	4	4	no	—
578	D₄	—	4	4	no	—
579	D₄	—	4	4	no	—
580	D₄	—	4	2	no	—
581	D₄	—	4	2	no	—
582	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
583	D₄	C₄×D₄×C₆₇	1	2	yes	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	4	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	8	4	no	—
602	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
603	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
604	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
606	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
607	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
608	C₈	—	2	2	no	—
609	C₈	—	2	2	no	—
610	C₈	—	4	2	no	—
611	C₈	—	4	1	no	—
612	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
613	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
614	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
615	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
616	C₂²	—	2	2	no	—
617	C₂²	—	2	4	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	8	2	no	—
621	C₂²	—	8	1	no	—
622	C₂²	—	8	2	no	—
623	C₂²	—	16	4	no	—
624	C₂²	#518×C₆₇	1	1	yes	characteristic
625	C₂²	D₄²×C₆₇	1	1	yes	characteristic
626	C₂²	#546×C₆₇	1	1	yes	characteristic
627	C₄	—	2	2	no	—
628	C₄	—	2	2	no	—
629	C₄	—	2	4	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	2	no	—
632	C₄	—	8	2	no	—
633	C₄	—	8	1	no	—
634	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
635	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
636	C₄	D₄²×C₆₇	1	1	yes	characteristic
637	C₄	#546×C₆₇	1	1	yes	characteristic
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	2	1	no	—
641	C₂	—	4	2	no	—
642	C₂	—	4	4	no	—
643	C₂	—	8	2	no	—
644	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂×C₆₇	D₄×C₂²⋊C₄
prime radical	C₆₇×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2957	C₂	semidirect
3. C₄×#135×C₆₇	C₂	semidirect
4. C₆₇×#2901	C₂	semidirect
5. C₆₇×#1976	C₂	semidirect
6. C₆₇×#1974	C₂	semidirect
7. C₆₇×#1906	C₂	semidirect
8. C₆₇×#1881	C₂	semidirect
9. C₆₇×#1814	C₂	semidirect
10. C₆₇×#1794	C₂	semidirect
11. C₆₇×#1764	C₂	semidirect
14. #578×C₆₇	C₄	semidirect
16. #577×C₆₇	C₄	semidirect
18. #576×C₆₇	C₄	semidirect
24. #568×C₆₇	C₂²	semidirect
26. C₄×#41×C₆₇	C₂²	semidirect
32. C₂×#108×C₆₇	C₂²	semidirect
33. #427×C₆₇	C₂²	semidirect
36. #421×C₆₇	C₂²	semidirect
37. #421×C₆₇	C₂²	semidirect
40. C₂×#104×C₆₇	C₂²	semidirect
42. #416×C₆₇	C₂²	semidirect
43. #410×C₆₇	C₂²	semidirect
44. #408×C₆₇	C₂²	semidirect
45. #405×C₆₇	C₂²	semidirect
46. C₂×C₄×C₈×C₆₇	C₂²	semidirect
47. #344×C₆₇	C₂²	semidirect
48. #341×C₆₇	C₂²	semidirect
49. #339×C₆₇	C₂²	semidirect
52. 2₋¹⁺⁴×C₆₇	C₂×C₄	semidirect
56. H(5,𝔽₂)×C₆₇	C₂×C₄	semidirect
66. #137×C₆₇	C₂×C₄	semidirect
70. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
84. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
87. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
90. C₂×D₈×C₆₇	C₂×C₄	semidirect
96. #131×C₆₇	C₂×C₄	semidirect
97. #131×C₆₇	C₂×C₄	semidirect
315. #22462	C₆₇	direct (N × H)
324. #2957	C₂×C₆₇	semidirect
325. C₄×#135	C₂×C₆₇	semidirect
326. #2901	C₂×C₆₇	semidirect
327. #1976	C₂×C₆₇	semidirect
328. #1974	C₂×C₆₇	semidirect
329. #1906	C₂×C₆₇	semidirect
330. #1881	C₂×C₆₇	semidirect
331. #1814	C₂×C₆₇	semidirect
332. #1794	C₂×C₆₇	semidirect
333. #1764	C₂×C₆₇	semidirect
334. C₆₇	#22462	direct (N × H)
337. #578	C₄×C₆₇	semidirect
339. #577	C₄×C₆₇	semidirect
341. #576	C₄×C₆₇	semidirect
347. #568	C₂²×C₆₇	semidirect
349. C₄×#41	C₂²×C₆₇	semidirect
355. C₂×#108	C₂²×C₆₇	semidirect
356. #427	C₂²×C₆₇	semidirect
359. #421	C₂²×C₆₇	semidirect
360. #421	C₂²×C₆₇	semidirect
363. C₂×#104	C₂²×C₆₇	semidirect
365. #416	C₂²×C₆₇	semidirect
366. #410	C₂²×C₆₇	semidirect
367. #408	C₂²×C₆₇	semidirect
368. #405	C₂²×C₆₇	semidirect
369. C₂×C₄×C₈	C₂²×C₆₇	semidirect
370. #344	C₂²×C₆₇	semidirect
371. #341	C₂²×C₆₇	semidirect
372. #339	C₂²×C₆₇	semidirect
375. 2₋¹⁺⁴	C₂×C₄×C₆₇	semidirect
379. H(5,𝔽₂)	C₂×C₄×C₆₇	semidirect
389. #137	C₂×C₄×C₆₇	semidirect
393. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
407. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
410. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
413. C₂×D₈	C₂×C₄×C₆₇	semidirect
419. #131	C₂×C₄×C₆₇	semidirect
420. #131	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3584	2
C₆₇×#2957	2
C₄×#135×C₆₇	2
C₆₇×#2901	2
C₆₇×#1976	2
C₆₇×#1974	2
C₆₇×#1906	2
C₆₇×#1881	2
C₆₇×#1814	2
C₆₇×#1794	2
C₆₇×#1764	2
#22462	67

Subgroups — C67×#22462 #30701713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22462 #30701713