Subgroups — C₆₇×#22466 #30701717

Order 17152.

1384 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2897	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1907	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1883	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1792	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
13	#576×C₆₇	—	2	4	no	—
14	#576×C₆₇	C₄	1	8	yes	complemented
15	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
16	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
17	C₂×#135×C₆₇	—	2	1	no	—
18	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₆₇	—	2	1	no	—
20	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
23	#520×C₆₇	C₂²	1	2	yes	complemented
24	#443×C₆₇	—	2	4	no	—
25	#440×C₆₇	—	2	2	no	—
26	C₄×Q₁₆×C₆₇	—	2	1	no	—
27	C₄×D₈×C₆₇	—	2	1	no	—
28	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
32	#418×C₆₇	—	2	2	no	—
33	#418×C₆₇	C₂²	1	2	yes	complemented
34	#417×C₆₇	—	2	2	no	—
35	#417×C₆₇	C₂²	1	2	yes	—
36	#417×C₆₇	C₂²	1	2	yes	complemented
37	#416×C₆₇	—	2	2	no	—
38	#416×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#102×C₆₇	—	2	2	no	—
40	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
41	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
42	#405×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#390×C₆₇	C₂²	1	2	yes	complemented
45	#343×C₆₇	C₂²	1	2	yes	complemented
46	#341×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	#339×C₆₇	C₂²	1	1	yes	characteristic, complemented
48	H(5,𝔽₂)×C₆₇	—	2	4	no	—
49	H(5,𝔽₂)×C₆₇	—	2	4	no	—
50	H(5,𝔽₂)×C₆₇	—	4	2	no	—
51	H(5,𝔽₂)×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	C₂³	1	2	yes	—
55	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	2	no	—
58	C₂²×D₄×C₆₇	—	4	2	no	—
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
61	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
62	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	8	yes	complemented
63	#135×C₆₇	—	2	8	no	—
64	#135×C₆₇	C₂×C₄	1	4	yes	—
65	#135×C₆₇	D₄	1	4	yes	—
66	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
67	C₂×SD₁₆×C₆₇	—	2	2	no	—
68	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
69	C₂×D₈×C₆₇	—	2	2	no	—
70	C₂×D₈×C₆₇	—	2	8	no	—
71	C₂×D₈×C₆₇	—	2	1	no	—
72	C₂×D₈×C₆₇	—	2	1	no	—
73	C₂×D₈×C₆₇	—	2	2	no	—
74	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
75	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
76	#131×C₆₇	—	2	2	no	—
77	#131×C₆₇	—	2	4	no	—
78	#131×C₆₇	—	4	2	no	—
79	#131×C₆₇	C₂×C₄	1	4	yes	complemented
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	—	2	2	no	—
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	C₂²×C₈×C₆₇	—	2	2	no	—
87	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	4	2	no	—
92	#117×C₆₇	—	2	2	no	—
93	#117×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	#108×C₆₇	—	2	2	no	—
99	#107×C₆₇	—	2	1	no	—
100	#107×C₆₇	—	2	1	no	—
101	#104×C₆₇	—	2	4	no	—
102	#104×C₆₇	—	2	2	no	—
103	#103×C₆₇	—	2	2	no	—
104	#103×C₆₇	—	2	2	no	—
105	#103×C₆₇	D₄	1	1	yes	characteristic
106	#103×C₆₇	D₄	1	1	yes	characteristic
107	#102×C₆₇	—	2	2	no	—
108	#102×C₆₇	—	2	2	no	—
109	#102×C₆₇	—	2	2	no	—
110	#102×C₆₇	—	2	2	no	—
111	#102×C₆₇	—	2	2	no	—
112	#102×C₆₇	—	2	2	no	—
113	#102×C₆₇	D₄	1	1	yes	characteristic
114	#102×C₆₇	D₄	1	1	yes	characteristic
115	#100×C₆₇	—	4	2	no	—
116	#97×C₆₇	—	2	1	no	—
117	C₄×C₈×C₆₇	—	2	2	no	—
118	C₄×C₈×C₆₇	—	2	1	no	—
119	C₂⁴×C₆₇	—	2	2	no	—
120	C₂⁴×C₆₇	—	4	2	no	—
121	#41×C₆₇	—	2	1	no	—
122	#41×C₆₇	—	2	2	no	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	1	no	—
125	#41×C₆₇	—	2	8	no	—
126	#41×C₆₇	—	2	8	no	—
127	#41×C₆₇	—	4	4	no	—
128	#41×C₆₇	—	4	4	no	—
129	#41×C₆₇	C₂²×C₄	1	4	yes	—
130	#41×C₆₇	C₂²×C₄	1	2	yes	—
131	#41×C₆₇	C₂×D₄	1	4	yes	—
132	#41×C₆₇	#41	1	2	yes	—
133	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
134	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	1	no	—
137	C₂×D₄×C₆₇	—	2	8	no	—
138	C₂×D₄×C₆₇	—	2	8	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	4	4	no	—
145	C₂×D₄×C₆₇	—	4	4	no	—
146	C₂×D₄×C₆₇	—	4	2	no	—
147	C₂×D₄×C₆₇	—	4	2	no	—
148	C₂×D₄×C₆₇	—	4	8	no	—
149	C₂×D₄×C₆₇	—	4	8	no	—
150	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
151	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	4	2	no	—
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	Q₁₆×C₆₇	C₂²⋊C₄	1	4	yes	complemented
161	SD₁₆×C₆₇	—	2	8	no	—
162	SD₁₆×C₆₇	C₂²⋊C₄	1	8	yes	complemented
163	D₈×C₆₇	—	2	8	no	—
164	D₈×C₆₇	—	2	4	no	—
165	D₈×C₆₇	—	2	4	no	—
166	D₈×C₆₇	C₂²⋊C₄	1	4	yes	complemented
167	M₁₆×C₆₇	—	2	1	no	—
168	M₁₆×C₆₇	—	2	4	no	—
169	M₁₆×C₆₇	—	2	2	no	—
170	M₁₆×C₆₇	—	2	1	no	—
171	M₁₆×C₆₇	—	2	1	no	—
172	M₁₆×C₆₇	—	2	2	no	—
173	M₁₆×C₆₇	—	4	4	no	—
174	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	1	no	—
178	C₂×C₈×C₆₇	—	2	4	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	1	no	—
181	C₂×C₈×C₆₇	—	4	4	no	—
182	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
183	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	4	2	no	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₆₇	—	2	1	no	—
194	C₂²⋊C₄×C₆₇	—	4	2	no	—
195	C₂²⋊C₄×C₆₇	—	8	4	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	—	2	1	no	—
199	C₄²×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	—	2	1	no	—
202	C₂³×C₆₇	—	2	4	no	—
203	C₂³×C₆₇	—	4	8	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	—	4	2	no	—
206	C₂³×C₆₇	—	8	2	no	—
207	C₂³×C₆₇	—	8	2	no	—
208	C₂³×C₆₇	—	16	2	no	—
209	C₂³×C₆₇	—	16	2	no	—
210	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
211	Q₈×C₆₇	—	2	1	no	—
212	Q₈×C₆₇	—	2	2	no	—
213	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
214	Q₈×C₆₇	C₄×D₄	1	2	yes	—
215	D₄×C₆₇	—	2	2	no	—
216	D₄×C₆₇	—	2	4	no	—
217	D₄×C₆₇	—	2	2	no	—
218	D₄×C₆₇	—	2	4	no	—
219	D₄×C₆₇	—	2	4	no	—
220	D₄×C₆₇	—	2	4	no	—
221	D₄×C₆₇	—	2	1	no	—
222	D₄×C₆₇	—	2	2	no	—
223	D₄×C₆₇	—	4	8	no	—
224	D₄×C₆₇	—	4	8	no	—
225	D₄×C₆₇	—	4	2	no	—
226	D₄×C₆₇	—	4	2	no	—
227	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
228	D₄×C₆₇	C₄×D₄	1	2	yes	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	4	8	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	1	no	—
241	C₂×C₄×C₆₇	—	4	2	no	—
242	C₂×C₄×C₆₇	—	8	2	no	—
243	C₂×C₄×C₆₇	—	16	2	no	—
244	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
250	C₈×C₆₇	—	2	2	no	—
251	C₈×C₆₇	—	2	1	no	—
252	C₈×C₆₇	—	2	1	no	—
253	C₈×C₆₇	—	4	2	no	—
254	C₈×C₆₇	—	4	1	no	—
255	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
256	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂²×C₆₇	—	2	1	no	—
258	C₂²×C₆₇	—	2	1	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	4	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	1	no	—
263	C₂²×C₆₇	—	4	2	no	—
264	C₂²×C₆₇	—	4	1	no	—
265	C₂²×C₆₇	—	4	4	no	—
266	C₂²×C₆₇	—	8	4	no	—
267	C₂²×C₆₇	—	8	2	no	—
268	C₂²×C₆₇	—	8	2	no	—
269	C₂²×C₆₇	—	16	2	no	—
270	C₂²×C₆₇	—	16	8	no	—
271	C₂²×C₆₇	#518	1	1	yes	characteristic
272	C₂²×C₆₇	D₄²	1	1	yes	characteristic
273	C₂²×C₆₇	#546	1	1	yes	characteristic
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	4	no	—
276	C₄×C₆₇	—	4	2	no	—
277	C₄×C₆₇	—	4	1	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	—	8	2	no	—
280	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
281	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
282	C₄×C₆₇	D₄²	1	1	yes	characteristic
283	C₄×C₆₇	#546	1	1	yes	characteristic
284	#22466	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
285	C₂×C₆₇	—	2	1	no	—
286	C₂×C₆₇	—	2	1	no	—
287	C₂×C₆₇	—	4	4	no	—
288	C₂×C₆₇	—	4	2	no	—
289	C₂×C₆₇	—	8	4	no	—
290	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
291	C₂×C₆₇	#3290	1	1	yes	characteristic
292	C₂×C₆₇	#3292	1	1	yes	characteristic
293	C₂×#576	C₂×C₆₇	1	1	yes	characteristic
294	C₂×#443	C₂×C₆₇	1	1	yes	characteristic, complemented
295	#2951	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#2897	C₂×C₆₇	1	2	yes	complemented
297	#1978	C₂×C₆₇	1	1	yes	characteristic, complemented
298	#1970	C₂×C₆₇	1	1	yes	characteristic, complemented
299	#1907	C₂×C₆₇	1	2	yes	complemented
300	#1883	C₂×C₆₇	1	2	yes	complemented
301	#1816	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1792	C₂×C₆₇	1	2	yes	complemented
303	#1764	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₆₇	#22466	1	1	yes	characteristic, direct product factor, Sylow
305	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
306	#576	—	2	4	no	—
307	#576	C₄×C₆₇	1	8	yes	complemented
308	C₂×AGL(1,ℤ/8)	—	2	2	no	—
309	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
310	C₂×#135	—	2	1	no	—
311	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×D₈	—	2	1	no	—
313	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
316	#520	C₂²×C₆₇	1	2	yes	complemented
317	#443	—	2	4	no	—
318	#440	—	2	2	no	—
319	C₄×Q₁₆	—	2	1	no	—
320	C₄×D₈	—	2	1	no	—
321	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic, complemented
322	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
323	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
325	#418	—	2	2	no	—
326	#418	C₂²×C₆₇	1	2	yes	complemented
327	#417	—	2	2	no	—
328	#417	C₂²×C₆₇	1	2	yes	—
329	#417	C₂²×C₆₇	1	2	yes	complemented
330	#416	—	2	2	no	—
331	#416	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#102	—	2	2	no	—
333	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
334	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
335	#405	C₂²×C₆₇	1	1	yes	characteristic, complemented
336	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
337	#390	C₂²×C₆₇	1	2	yes	complemented
338	#343	C₂²×C₆₇	1	2	yes	complemented
339	#341	C₂²×C₆₇	1	1	yes	characteristic, complemented
340	#339	C₂²×C₆₇	1	1	yes	characteristic, complemented
341	H(5,𝔽₂)	—	2	4	no	—
342	H(5,𝔽₂)	—	2	4	no	—
343	H(5,𝔽₂)	—	4	2	no	—
344	H(5,𝔽₂)	C₂×C₄×C₆₇	1	8	yes	complemented
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	C₂³×C₆₇	1	2	yes	—
348	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	—	4	2	no	—
352	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
353	AGL(1,ℤ/8)	—	2	4	no	—
354	AGL(1,ℤ/8)	—	2	16	no	—
355	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	8	yes	complemented
356	#135	—	2	8	no	—
357	#135	C₂×C₄×C₆₇	1	4	yes	—
358	#135	D₄×C₆₇	1	4	yes	—
359	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
360	C₂×SD₁₆	—	2	2	no	—
361	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
362	C₂×D₈	—	2	2	no	—
363	C₂×D₈	—	2	8	no	—
364	C₂×D₈	—	2	1	no	—
365	C₂×D₈	—	2	1	no	—
366	C₂×D₈	—	2	2	no	—
367	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
368	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
369	#131	—	2	2	no	—
370	#131	—	2	4	no	—
371	#131	—	4	2	no	—
372	#131	C₂×C₄×C₆₇	1	4	yes	complemented
373	C₂×M₁₆	—	2	1	no	—
374	C₂×M₁₆	—	2	2	no	—
375	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
376	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
377	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
378	C₂²×C₈	—	2	1	no	—
379	C₂²×C₈	—	2	2	no	—
380	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
382	C₄×Q₈	—	2	2	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	4	2	no	—
385	#117	—	2	2	no	—
386	#117	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
389	C₂×C₂²⋊C₄	—	4	2	no	—
390	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
391	#108	—	2	2	no	—
392	#107	—	2	1	no	—
393	#107	—	2	1	no	—
394	#104	—	2	4	no	—
395	#104	—	2	2	no	—
396	#103	—	2	2	no	—
397	#103	—	2	2	no	—
398	#103	D₄×C₆₇	1	1	yes	characteristic
399	#103	D₄×C₆₇	1	1	yes	characteristic
400	#102	—	2	2	no	—
401	#102	—	2	2	no	—
402	#102	—	2	2	no	—
403	#102	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	D₄×C₆₇	1	1	yes	characteristic
407	#102	D₄×C₆₇	1	1	yes	characteristic
408	#100	—	4	2	no	—
409	#97	—	2	1	no	—
410	C₄×C₈	—	2	2	no	—
411	C₄×C₈	—	2	1	no	—
412	C₂⁴	—	2	2	no	—
413	C₂⁴	—	4	2	no	—
414	#41	—	2	1	no	—
415	#41	—	2	2	no	—
416	#41	—	2	2	no	—
417	#41	—	2	1	no	—
418	#41	—	2	8	no	—
419	#41	—	2	8	no	—
420	#41	—	4	4	no	—
421	#41	—	4	4	no	—
422	#41	C₂²×C₄×C₆₇	1	4	yes	—
423	#41	C₂²×C₄×C₆₇	1	2	yes	—
424	#41	C₂×D₄×C₆₇	1	4	yes	—
425	#41	#41×C₆₇	1	2	yes	—
426	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
427	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	8	no	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	4	4	no	—
438	C₂×D₄	—	4	4	no	—
439	C₂×D₄	—	4	2	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	8	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
444	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
445	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
453	Q₁₆	C₂²⋊C₄×C₆₇	1	4	yes	complemented
454	SD₁₆	—	2	8	no	—
455	SD₁₆	C₂²⋊C₄×C₆₇	1	8	yes	complemented
456	D₈	—	2	8	no	—
457	D₈	—	2	4	no	—
458	D₈	—	2	4	no	—
459	D₈	C₂²⋊C₄×C₆₇	1	4	yes	complemented
460	M₁₆	—	2	1	no	—
461	M₁₆	—	2	4	no	—
462	M₁₆	—	2	2	no	—
463	M₁₆	—	2	1	no	—
464	M₁₆	—	2	1	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	4	4	no	—
467	M₁₆	C₂²×C₄×C₆₇	1	2	yes	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	4	4	no	—
475	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
476	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
480	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
485	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	4	2	no	—
488	C₂²⋊C₄	—	8	4	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	4	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	4	no	—
496	C₂³	—	4	8	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	—	8	2	no	—
501	C₂³	—	16	2	no	—
502	C₂³	—	16	2	no	—
503	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
504	Q₈	—	2	1	no	—
505	Q₈	—	2	2	no	—
506	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
507	Q₈	C₄×D₄×C₆₇	1	2	yes	—
508	D₄	—	2	2	no	—
509	D₄	—	2	4	no	—
510	D₄	—	2	2	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	1	no	—
515	D₄	—	2	2	no	—
516	D₄	—	4	8	no	—
517	D₄	—	4	8	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
521	D₄	C₄×D₄×C₆₇	1	2	yes	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	8	2	no	—
536	C₂×C₄	—	16	2	no	—
537	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
539	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
541	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
542	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
543	C₈	—	2	2	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	4	2	no	—
547	C₈	—	4	1	no	—
548	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
549	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
550	C₂²	—	2	1	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	8	4	no	—
560	C₂²	—	8	2	no	—
561	C₂²	—	8	2	no	—
562	C₂²	—	16	2	no	—
563	C₂²	—	16	8	no	—
564	C₂²	#518×C₆₇	1	1	yes	characteristic
565	C₂²	D₄²×C₆₇	1	1	yes	characteristic
566	C₂²	#546×C₆₇	1	1	yes	characteristic
567	C₄	—	2	2	no	—
568	C₄	—	2	4	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	8	2	no	—
573	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
574	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
575	C₄	D₄²×C₆₇	1	1	yes	characteristic
576	C₄	#546×C₆₇	1	1	yes	characteristic
577	C₂	—	2	1	no	—
578	C₂	—	2	1	no	—
579	C₂	—	4	2	no	—
580	C₂	—	4	4	no	—
581	C₂	—	8	4	no	—
582	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
583	C₂	C₆₇×#3290	1	1	yes	characteristic
584	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#576×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₆₇	C₂	semidirect
3. C₆₇×#2951	C₂	semidirect
4. C₆₇×#2897	C₂	semidirect
5. C₆₇×#1978	C₂	semidirect
6. C₆₇×#1970	C₂	semidirect
7. C₆₇×#1907	C₂	semidirect
8. C₆₇×#1883	C₂	semidirect
9. C₆₇×#1816	C₂	semidirect
10. C₆₇×#1792	C₂	semidirect
11. C₆₇×#1764	C₂	semidirect
14. #576×C₆₇	C₄	semidirect
22. C₂×#131×C₆₇	C₂²	semidirect
23. #520×C₆₇	C₂²	semidirect
28. C₂×#108×C₆₇	C₂²	semidirect
29. C₂×#107×C₆₇	C₂²	semidirect
31. C₂×#104×C₆₇	C₂²	semidirect
33. #418×C₆₇	C₂²	semidirect
36. #417×C₆₇	C₂²	semidirect
40. C₂×#102×C₆₇	C₂²	semidirect
41. C₂×#100×C₆₇	C₂²	semidirect
42. #405×C₆₇	C₂²	semidirect
43. C₂×C₄×C₈×C₆₇	C₂²	semidirect
44. #390×C₆₇	C₂²	semidirect
45. #343×C₆₇	C₂²	semidirect
46. #341×C₆₇	C₂²	semidirect
47. #339×C₆₇	C₂²	semidirect
51. H(5,𝔽₂)×C₆₇	C₂×C₄	semidirect
62. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
74. C₂×D₈×C₆₇	C₂×C₄	semidirect
79. #131×C₆₇	C₂×C₄	semidirect
160. Q₁₆×C₆₇	C₂²⋊C₄	semidirect
162. SD₁₆×C₆₇	C₂²⋊C₄	semidirect
166. D₈×C₆₇	C₂²⋊C₄	semidirect
284. #22466	C₆₇	direct (N × H)
294. C₂×#443	C₂×C₆₇	semidirect
295. #2951	C₂×C₆₇	semidirect
296. #2897	C₂×C₆₇	semidirect
297. #1978	C₂×C₆₇	semidirect
298. #1970	C₂×C₆₇	semidirect
299. #1907	C₂×C₆₇	semidirect
300. #1883	C₂×C₆₇	semidirect
301. #1816	C₂×C₆₇	semidirect
302. #1792	C₂×C₆₇	semidirect
303. #1764	C₂×C₆₇	semidirect
304. C₆₇	#22466	direct (N × H)
307. #576	C₄×C₆₇	semidirect
315. C₂×#131	C₂²×C₆₇	semidirect
316. #520	C₂²×C₆₇	semidirect
321. C₂×#108	C₂²×C₆₇	semidirect
322. C₂×#107	C₂²×C₆₇	semidirect
324. C₂×#104	C₂²×C₆₇	semidirect
326. #418	C₂²×C₆₇	semidirect
329. #417	C₂²×C₆₇	semidirect
333. C₂×#102	C₂²×C₆₇	semidirect
334. C₂×#100	C₂²×C₆₇	semidirect
335. #405	C₂²×C₆₇	semidirect
336. C₂×C₄×C₈	C₂²×C₆₇	semidirect
337. #390	C₂²×C₆₇	semidirect
338. #343	C₂²×C₆₇	semidirect
339. #341	C₂²×C₆₇	semidirect
340. #339	C₂²×C₆₇	semidirect
344. H(5,𝔽₂)	C₂×C₄×C₆₇	semidirect
355. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
367. C₂×D₈	C₂×C₄×C₆₇	semidirect
372. #131	C₂×C₄×C₆₇	semidirect
453. Q₁₆	C₂²⋊C₄×C₆₇	semidirect
455. SD₁₆	C₂²⋊C₄×C₆₇	semidirect
459. D₈	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2951	2
C₆₇×#2897	2
C₆₇×#1978	2
C₆₇×#1970	2
C₆₇×#1907	2
C₆₇×#1883	2
C₆₇×#1816	2
C₆₇×#1792	2
C₆₇×#1764	2
#22466	67

Subgroups — C67×#22466 #30701717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22466 #30701717