Subgroups — C₆₇×#22453 #30701704

Order 17152.

1362 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#575×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#444×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2880	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1876	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1811	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1789	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	2	yes	—
13	#575×C₆₇	—	2	4	no	—
14	#575×C₆₇	—	2	4	no	—
15	C₂×#137×C₆₇	C₄	1	2	yes	complemented
16	C₂×#137×C₆₇	C₄	1	2	yes	complemented
17	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	2	yes	complemented
18	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	2	yes	complemented
19	C₂×#135×C₆₇	C₂²	1	2	yes	—
20	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
23	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
24	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#444×C₆₇	—	2	4	no	—
26	#443×C₆₇	—	2	4	no	—
27	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
33	#408×C₆₇	—	2	2	no	—
34	#408×C₆₇	—	2	2	no	—
35	#408×C₆₇	C₂²	1	2	yes	complemented
36	#408×C₆₇	C₂²	1	2	yes	complemented
37	#407×C₆₇	—	2	2	no	—
38	#407×C₆₇	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
40	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
44	#341×C₆₇	C₂²	1	2	yes	complemented
45	#336×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	#336×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	4	no	—
50	C₂×#41×C₆₇	—	2	4	no	—
51	C₂×#41×C₆₇	—	4	2	no	—
52	C₂×#41×C₆₇	—	4	2	no	—
53	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×#41×C₆₇	C₂³	1	2	yes	—
56	C₂×#41×C₆₇	C₂³	1	2	yes	—
57	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂²×D₄×C₆₇	—	2	2	no	—
59	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂³×C₄×C₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	#137×C₆₇	—	2	8	no	—
63	#137×C₆₇	—	2	8	no	—
64	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
65	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
66	#135×C₆₇	—	2	4	no	—
67	#135×C₆₇	—	2	4	no	—
68	#135×C₆₇	—	2	2	no	—
69	#135×C₆₇	—	2	2	no	—
70	#135×C₆₇	D₄	1	4	yes	—
71	#135×C₆₇	D₄	1	4	yes	—
72	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
76	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
77	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
78	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	#131×C₆₇	—	2	4	no	—
83	#131×C₆₇	—	2	4	no	—
84	#131×C₆₇	—	4	2	no	—
85	#131×C₆₇	—	4	2	no	—
86	C₂×M₁₆×C₆₇	—	2	2	no	—
87	C₂×M₁₆×C₆₇	—	2	2	no	—
88	C₂×M₁₆×C₆₇	—	2	2	no	—
89	C₂×M₁₆×C₆₇	—	2	2	no	—
90	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
91	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
92	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
93	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
94	C₂²×C₈×C₆₇	—	2	2	no	—
95	C₂²×C₈×C₆₇	—	2	2	no	—
96	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	#108×C₆₇	—	2	2	no	—
104	#108×C₆₇	—	2	2	no	—
105	#104×C₆₇	—	2	4	no	—
106	#104×C₆₇	—	2	4	no	—
107	#104×C₆₇	—	4	2	no	—
108	#104×C₆₇	—	4	2	no	—
109	C₂²⋊C₈×C₆₇	—	2	2	no	—
110	C₂²⋊C₈×C₆₇	—	2	2	no	—
111	C₂²⋊C₈×C₆₇	—	2	2	no	—
112	C₂²⋊C₈×C₆₇	—	2	2	no	—
113	C₂²⋊C₈×C₆₇	—	2	2	no	—
114	C₂²⋊C₈×C₆₇	—	2	2	no	—
115	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
116	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
117	#97×C₆₇	—	2	2	no	—
118	C₄×C₈×C₆₇	—	2	2	no	—
119	#95×C₆₇	—	2	1	no	—
120	C₂⁴×C₆₇	—	2	2	no	—
121	C₂⁴×C₆₇	#41	1	1	yes	characteristic
122	#41×C₆₇	—	2	2	no	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	4	no	—
125	#41×C₆₇	—	2	4	no	—
126	#41×C₆₇	—	2	2	no	—
127	#41×C₆₇	—	2	2	no	—
128	#41×C₆₇	—	2	2	no	—
129	#41×C₆₇	—	2	2	no	—
130	#41×C₆₇	—	4	4	no	—
131	#41×C₆₇	—	4	4	no	—
132	#41×C₆₇	—	4	2	no	—
133	#41×C₆₇	—	4	2	no	—
134	#41×C₆₇	—	4	4	no	—
135	#41×C₆₇	—	4	4	no	—
136	#41×C₆₇	C₂×D₄	1	4	yes	—
137	#41×C₆₇	C₂×D₄	1	4	yes	—
138	C₂×Q₈×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	—	2	4	no	—
140	C₂×Q₈×C₆₇	—	4	2	no	—
141	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	4	no	—
145	C₂×D₄×C₆₇	—	2	2	no	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	2	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	4	2	no	—
150	C₂×D₄×C₆₇	—	4	4	no	—
151	C₂×D₄×C₆₇	—	4	4	no	—
152	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
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	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	4	no	—
167	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₆₇	—	2	4	no	—
172	Q₁₆×C₆₇	—	2	2	no	—
173	Q₁₆×C₆₇	C₂²⋊C₄	1	4	yes	complemented
174	SD₁₆×C₆₇	—	2	4	no	—
175	SD₁₆×C₆₇	—	2	4	no	—
176	SD₁₆×C₆₇	—	2	4	no	—
177	SD₁₆×C₆₇	C₂²⋊C₄	1	8	yes	complemented
178	D₈×C₆₇	—	2	4	no	—
179	D₈×C₆₇	—	2	2	no	—
180	D₈×C₆₇	C₂²⋊C₄	1	4	yes	complemented
181	M₁₆×C₆₇	—	2	4	no	—
182	M₁₆×C₆₇	—	2	4	no	—
183	M₁₆×C₆₇	—	2	2	no	—
184	M₁₆×C₆₇	—	2	2	no	—
185	M₁₆×C₆₇	—	2	2	no	—
186	M₁₆×C₆₇	—	2	2	no	—
187	M₁₆×C₆₇	—	4	4	no	—
188	M₁₆×C₆₇	—	4	4	no	—
189	C₂×C₈×C₆₇	—	2	1	no	—
190	C₂×C₈×C₆₇	—	2	1	no	—
191	C₂×C₈×C₆₇	—	2	1	no	—
192	C₂×C₈×C₆₇	—	2	1	no	—
193	C₂×C₈×C₆₇	—	2	1	no	—
194	C₂×C₈×C₆₇	—	2	1	no	—
195	C₂×C₈×C₆₇	—	2	2	no	—
196	C₂×C₈×C₆₇	—	2	2	no	—
197	C₂×C₈×C₆₇	—	4	4	no	—
198	C₂×C₈×C₆₇	—	4	4	no	—
199	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
200	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
201	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
202	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
204	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
205	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
207	C₂²⋊C₄×C₆₇	—	4	2	no	—
208	C₄²×C₆₇	—	2	1	no	—
209	C₄²×C₆₇	—	2	1	no	—
210	C₄²×C₆₇	—	2	1	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	1	no	—
214	C₂³×C₆₇	—	2	1	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	4	1	no	—
217	C₂³×C₆₇	—	4	1	no	—
218	C₂³×C₆₇	—	4	4	no	—
219	C₂³×C₆₇	—	4	4	no	—
220	C₂³×C₆₇	—	8	2	no	—
221	C₂³×C₆₇	C₄×D₄	1	2	yes	—
222	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
223	Q₈×C₆₇	—	2	2	no	—
224	Q₈×C₆₇	—	2	2	no	—
225	Q₈×C₆₇	—	2	2	no	—
226	Q₈×C₆₇	—	4	4	no	—
227	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	2	no	—
231	D₄×C₆₇	—	2	2	no	—
232	D₄×C₆₇	—	2	2	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	4	4	no	—
235	D₄×C₆₇	—	4	4	no	—
236	D₄×C₆₇	—	4	2	no	—
237	D₄×C₆₇	—	4	2	no	—
238	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	—	4	4	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	2	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	1	no	—
259	C₂×C₄×C₆₇	—	4	1	no	—
260	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
267	C₈×C₆₇	—	2	2	no	—
268	C₈×C₆₇	—	2	1	no	—
269	C₈×C₆₇	—	2	2	no	—
270	C₈×C₆₇	—	2	1	no	—
271	C₈×C₆₇	—	4	2	no	—
272	C₈×C₆₇	—	4	2	no	—
273	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
274	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
275	C₂²×C₆₇	—	2	1	no	—
276	C₂²×C₆₇	—	2	1	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	4	no	—
279	C₂²×C₆₇	—	2	2	no	—
280	C₂²×C₆₇	—	2	2	no	—
281	C₂²×C₆₇	—	4	2	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	2	no	—
284	C₂²×C₆₇	—	4	2	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	8	1	no	—
287	C₂²×C₆₇	—	8	1	no	—
288	C₂²×C₆₇	—	8	8	no	—
289	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
290	C₂²×C₆₇	#546	1	1	yes	characteristic
291	C₂²×C₆₇	#546	1	1	yes	characteristic
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	4	no	—
294	C₄×C₆₇	—	4	2	no	—
295	C₄×C₆₇	—	4	2	no	—
296	C₄×C₆₇	—	8	1	no	—
297	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
298	C₄×C₆₇	#518	1	1	yes	characteristic
299	C₄×C₆₇	D₄²	1	1	yes	characteristic
300	C₄×C₆₇	D₄²	1	1	yes	characteristic
301	#22453	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂×C₆₇	—	2	1	no	—
303	C₂×C₆₇	—	2	1	no	—
304	C₂×C₆₇	—	4	2	no	—
305	C₂×C₆₇	—	4	4	no	—
306	C₂×C₆₇	—	8	2	no	—
307	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₆₇	#2955	1	1	yes	characteristic
309	C₂×C₆₇	#2956	1	1	yes	characteristic
310	C₂×#575	C₂×C₆₇	1	1	yes	characteristic
311	C₂×#444	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₂×#443	C₂×C₆₇	1	1	yes	characteristic, complemented
313	#2880	C₂×C₆₇	1	2	yes	complemented
314	#1975	C₂×C₆₇	1	1	yes	characteristic, complemented
315	#1974	C₂×C₆₇	1	1	yes	characteristic, complemented
316	#1876	C₂×C₆₇	1	2	yes	complemented
317	#1876	C₂×C₆₇	1	2	yes	complemented
318	#1811	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#1789	C₂×C₆₇	1	2	yes	complemented
320	#1753	C₂×C₆₇	1	1	yes	characteristic, complemented
321	C₆₇	#22453	1	1	yes	characteristic, direct product factor, Sylow
322	C₂²×#41	C₂²×C₆₇	1	2	yes	—
323	#575	—	2	4	no	—
324	#575	—	2	4	no	—
325	C₂×#137	C₄×C₆₇	1	2	yes	complemented
326	C₂×#137	C₄×C₆₇	1	2	yes	complemented
327	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	2	yes	complemented
328	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	2	yes	complemented
329	C₂×#135	C₂²×C₆₇	1	2	yes	—
330	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
333	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
334	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
335	#444	—	2	4	no	—
336	#443	—	2	4	no	—
337	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic, complemented
338	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic, complemented
339	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
340	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
341	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
342	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
343	#408	—	2	2	no	—
344	#408	—	2	2	no	—
345	#408	C₂²×C₆₇	1	2	yes	complemented
346	#408	C₂²×C₆₇	1	2	yes	complemented
347	#407	—	2	2	no	—
348	#407	C₂²×C₆₇	1	2	yes	—
349	C₂×C₂²⋊C₈	—	2	2	no	—
350	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic, complemented
352	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
353	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
354	#341	C₂²×C₆₇	1	2	yes	complemented
355	#336	C₂²×C₆₇	1	1	yes	characteristic, complemented
356	#336	C₂²×C₆₇	1	1	yes	characteristic, complemented
357	C₂×#41	—	2	2	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	4	no	—
360	C₂×#41	—	2	4	no	—
361	C₂×#41	—	4	2	no	—
362	C₂×#41	—	4	2	no	—
363	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
364	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
365	C₂×#41	C₂³×C₆₇	1	2	yes	—
366	C₂×#41	C₂³×C₆₇	1	2	yes	—
367	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
370	C₂³×C₄	—	2	2	no	—
371	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
372	#137	—	2	8	no	—
373	#137	—	2	8	no	—
374	AGL(1,ℤ/8)	—	2	8	no	—
375	AGL(1,ℤ/8)	—	2	8	no	—
376	#135	—	2	4	no	—
377	#135	—	2	4	no	—
378	#135	—	2	2	no	—
379	#135	—	2	2	no	—
380	#135	D₄×C₆₇	1	4	yes	—
381	#135	D₄×C₆₇	1	4	yes	—
382	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
383	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
384	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
385	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
386	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
387	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
388	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
389	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
390	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
391	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
392	#131	—	2	4	no	—
393	#131	—	2	4	no	—
394	#131	—	4	2	no	—
395	#131	—	4	2	no	—
396	C₂×M₁₆	—	2	2	no	—
397	C₂×M₁₆	—	2	2	no	—
398	C₂×M₁₆	—	2	2	no	—
399	C₂×M₁₆	—	2	2	no	—
400	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
401	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
402	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
403	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
404	C₂²×C₈	—	2	2	no	—
405	C₂²×C₈	—	2	2	no	—
406	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
408	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
409	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
412	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
413	#108	—	2	2	no	—
414	#108	—	2	2	no	—
415	#104	—	2	4	no	—
416	#104	—	2	4	no	—
417	#104	—	4	2	no	—
418	#104	—	4	2	no	—
419	C₂²⋊C₈	—	2	2	no	—
420	C₂²⋊C₈	—	2	2	no	—
421	C₂²⋊C₈	—	2	2	no	—
422	C₂²⋊C₈	—	2	2	no	—
423	C₂²⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
426	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
427	#97	—	2	2	no	—
428	C₄×C₈	—	2	2	no	—
429	#95	—	2	1	no	—
430	C₂⁴	—	2	2	no	—
431	C₂⁴	#41×C₆₇	1	1	yes	characteristic
432	#41	—	2	2	no	—
433	#41	—	2	2	no	—
434	#41	—	2	4	no	—
435	#41	—	2	4	no	—
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	2	2	no	—
439	#41	—	2	2	no	—
440	#41	—	4	4	no	—
441	#41	—	4	4	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	4	4	no	—
445	#41	—	4	4	no	—
446	#41	C₂×D₄×C₆₇	1	4	yes	—
447	#41	C₂×D₄×C₆₇	1	4	yes	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	4	no	—
450	C₂×Q₈	—	4	2	no	—
451	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
452	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	4	2	no	—
460	C₂×D₄	—	4	4	no	—
461	C₂×D₄	—	4	4	no	—
462	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
463	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
479	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
481	Q₁₆	—	2	4	no	—
482	Q₁₆	—	2	2	no	—
483	Q₁₆	C₂²⋊C₄×C₆₇	1	4	yes	complemented
484	SD₁₆	—	2	4	no	—
485	SD₁₆	—	2	4	no	—
486	SD₁₆	—	2	4	no	—
487	SD₁₆	C₂²⋊C₄×C₆₇	1	8	yes	complemented
488	D₈	—	2	4	no	—
489	D₈	—	2	2	no	—
490	D₈	C₂²⋊C₄×C₆₇	1	4	yes	complemented
491	M₁₆	—	2	4	no	—
492	M₁₆	—	2	4	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	—	2	2	no	—
495	M₁₆	—	2	2	no	—
496	M₁₆	—	2	2	no	—
497	M₁₆	—	4	4	no	—
498	M₁₆	—	4	4	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	4	4	no	—
508	C₂×C₈	—	4	4	no	—
509	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
510	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
511	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
512	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
513	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
514	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
515	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
516	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
517	C₂²⋊C₄	—	4	2	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	1	no	—
524	C₂³	—	2	1	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	4	1	no	—
527	C₂³	—	4	1	no	—
528	C₂³	—	4	4	no	—
529	C₂³	—	4	4	no	—
530	C₂³	—	8	2	no	—
531	C₂³	C₄×D₄×C₆₇	1	2	yes	—
532	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
533	Q₈	—	2	2	no	—
534	Q₈	—	2	2	no	—
535	Q₈	—	2	2	no	—
536	Q₈	—	4	4	no	—
537	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
538	D₄	—	2	4	no	—
539	D₄	—	2	4	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	2	no	—
542	D₄	—	2	2	no	—
543	D₄	—	4	4	no	—
544	D₄	—	4	4	no	—
545	D₄	—	4	4	no	—
546	D₄	—	4	2	no	—
547	D₄	—	4	2	no	—
548	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	4	4	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
571	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
573	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
574	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
575	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
576	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
577	C₈	—	2	2	no	—
578	C₈	—	2	1	no	—
579	C₈	—	2	2	no	—
580	C₈	—	2	1	no	—
581	C₈	—	4	2	no	—
582	C₈	—	4	2	no	—
583	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
584	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	8	1	no	—
597	C₂²	—	8	1	no	—
598	C₂²	—	8	8	no	—
599	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
600	C₂²	#546×C₆₇	1	1	yes	characteristic
601	C₂²	#546×C₆₇	1	1	yes	characteristic
602	C₄	—	2	2	no	—
603	C₄	—	2	4	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	8	1	no	—
607	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
608	C₄	#518×C₆₇	1	1	yes	characteristic
609	C₄	D₄²×C₆₇	1	1	yes	characteristic
610	C₄	D₄²×C₆₇	1	1	yes	characteristic
611	C₂	—	2	1	no	—
612	C₂	—	2	1	no	—
613	C₂	—	4	2	no	—
614	C₂	—	4	4	no	—
615	C₂	—	8	2	no	—
616	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
617	C₂	C₆₇×#2955	1	1	yes	characteristic
618	C₂	C₆₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#575×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₂²⋊C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#444×C₆₇	C₂	semidirect
3. C₂×#443×C₆₇	C₂	semidirect
4. C₆₇×#2880	C₂	semidirect
5. C₆₇×#1975	C₂	semidirect
6. C₆₇×#1974	C₂	semidirect
7. C₆₇×#1876	C₂	semidirect
8. C₆₇×#1876	C₂	semidirect
9. C₆₇×#1811	C₂	semidirect
10. C₆₇×#1789	C₂	semidirect
11. C₆₇×#1753	C₂	semidirect
15. C₂×#137×C₆₇	C₄	semidirect
16. C₂×#137×C₆₇	C₄	semidirect
17. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
18. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
22. C₂×#131×C₆₇	C₂²	semidirect
23. C₂×#131×C₆₇	C₂²	semidirect
27. C₂×#108×C₆₇	C₂²	semidirect
28. C₂×#108×C₆₇	C₂²	semidirect
29. C₂×#104×C₆₇	C₂²	semidirect
30. C₂×#104×C₆₇	C₂²	semidirect
31. C₂×#104×C₆₇	C₂²	semidirect
32. C₂×#104×C₆₇	C₂²	semidirect
35. #408×C₆₇	C₂²	semidirect
36. #408×C₆₇	C₂²	semidirect
41. C₂×#97×C₆₇	C₂²	semidirect
42. C₂×C₄×C₈×C₆₇	C₂²	semidirect
44. #341×C₆₇	C₂²	semidirect
45. #336×C₆₇	C₂²	semidirect
46. #336×C₆₇	C₂²	semidirect
173. Q₁₆×C₆₇	C₂²⋊C₄	semidirect
177. SD₁₆×C₆₇	C₂²⋊C₄	semidirect
180. D₈×C₆₇	C₂²⋊C₄	semidirect
301. #22453	C₆₇	direct (N × H)
311. C₂×#444	C₂×C₆₇	semidirect
312. C₂×#443	C₂×C₆₇	semidirect
313. #2880	C₂×C₆₇	semidirect
314. #1975	C₂×C₆₇	semidirect
315. #1974	C₂×C₆₇	semidirect
316. #1876	C₂×C₆₇	semidirect
317. #1876	C₂×C₆₇	semidirect
318. #1811	C₂×C₆₇	semidirect
319. #1789	C₂×C₆₇	semidirect
320. #1753	C₂×C₆₇	semidirect
321. C₆₇	#22453	direct (N × H)
325. C₂×#137	C₄×C₆₇	semidirect
326. C₂×#137	C₄×C₆₇	semidirect
327. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
328. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
332. C₂×#131	C₂²×C₆₇	semidirect
333. C₂×#131	C₂²×C₆₇	semidirect
337. C₂×#108	C₂²×C₆₇	semidirect
338. C₂×#108	C₂²×C₆₇	semidirect
339. C₂×#104	C₂²×C₆₇	semidirect
340. C₂×#104	C₂²×C₆₇	semidirect
341. C₂×#104	C₂²×C₆₇	semidirect
342. C₂×#104	C₂²×C₆₇	semidirect
345. #408	C₂²×C₆₇	semidirect
346. #408	C₂²×C₆₇	semidirect
351. C₂×#97	C₂²×C₆₇	semidirect
352. C₂×C₄×C₈	C₂²×C₆₇	semidirect
354. #341	C₂²×C₆₇	semidirect
355. #336	C₂²×C₆₇	semidirect
356. #336	C₂²×C₆₇	semidirect
483. Q₁₆	C₂²⋊C₄×C₆₇	semidirect
487. SD₁₆	C₂²⋊C₄×C₆₇	semidirect
490. 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×#575×C₆₇	2
C₂×#444×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2880	2
C₆₇×#1975	2
C₆₇×#1974	2
C₆₇×#1876	2
C₆₇×#1811	2
C₆₇×#1789	2
C₆₇×#1753	2
#22453	67

Subgroups — C67×#22453 #30701704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22453 #30701704