Subgroups — C₆₇×#18767 #30698018

Order 17152.

1234 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1642	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1631	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	—	2	1	no	—
11	#550×C₆₇	—	2	2	no	—
12	#550×C₆₇	—	2	2	no	—
13	#550×C₆₇	—	2	2	no	—
14	#550×C₆₇	C₂²	1	4	yes	—
15	#533×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₇	—	2	1	no	—
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	—	2	2	no	—
19	C₂×#121×C₆₇	—	2	1	no	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	#492×C₆₇	C₂²	1	2	yes	complemented
24	#486×C₆₇	C₂²	1	2	yes	complemented
25	#483×C₆₇	—	2	1	no	—
26	#479×C₆₇	—	2	2	no	—
27	#477×C₆₇	C₂²	1	2	yes	complemented
28	#476×C₆₇	C₂²	1	2	yes	complemented
29	#475×C₆₇	C₂²	1	2	yes	complemented
30	#461×C₆₇	C₂²	1	2	yes	complemented
31	#460×C₆₇	—	2	2	no	—
32	#450×C₆₇	—	2	1	no	—
33	#450×C₆₇	—	2	2	no	—
34	C₄×SD₁₆×C₆₇	—	2	2	no	—
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	#428×C₆₇	—	2	1	no	—
38	#423×C₆₇	C₂²	1	1	yes	characteristic
39	#419×C₆₇	—	2	1	no	—
40	#331×C₆₇	C₂²	1	4	yes	complemented
41	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	complemented
42	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
43	C₂×#41×C₆₇	—	2	4	no	—
44	C₂×#41×C₆₇	—	2	4	no	—
45	C₂×#41×C₆₇	—	4	4	no	—
46	C₂×#41×C₆₇	—	4	4	no	—
47	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	D₄	1	2	yes	—
54	C₂³×C₄×C₆₇	—	2	2	no	—
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂×SD₁₆×C₆₇	—	2	2	no	—
57	C₂×SD₁₆×C₆₇	—	2	4	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
61	#127×C₆₇	—	2	4	no	—
62	#127×C₆₇	—	2	4	no	—
63	#127×C₆₇	—	2	1	no	—
64	#127×C₆₇	—	2	1	no	—
65	#127×C₆₇	D₄	1	2	yes	—
66	#127×C₆₇	C₂³	1	2	yes	—
67	#125×C₆₇	—	2	1	no	—
68	C₂²⋊Q₈×C₆₇	—	2	1	no	—
69	C₂²⋊Q₈×C₆₇	—	2	1	no	—
70	#121×C₆₇	—	2	4	no	—
71	#121×C₆₇	—	2	8	no	—
72	#121×C₆₇	—	2	8	no	—
73	#121×C₆₇	—	2	4	no	—
74	C₄×Q₈×C₆₇	—	2	1	no	—
75	C₄×Q₈×C₆₇	—	2	1	no	—
76	C₄×Q₈×C₆₇	—	2	1	no	—
77	C₄×Q₈×C₆₇	—	2	1	no	—
78	C₄×Q₈×C₆₇	D₄	1	2	yes	—
79	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
80	C₄×D₄×C₆₇	—	2	4	no	—
81	C₄×D₄×C₆₇	—	2	4	no	—
82	C₄×D₄×C₆₇	—	2	1	no	—
83	C₄×D₄×C₆₇	—	2	1	no	—
84	C₄×D₄×C₆₇	D₄	1	2	yes	—
85	C₄×D₄×C₆₇	C₂³	1	2	yes	—
86	#117×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	—	2	1	no	—
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	#107×C₆₇	—	2	2	no	—
95	#106×C₆₇	—	2	2	no	—
96	#106×C₆₇	—	2	2	no	—
97	C₄⋊C₈×C₆₇	—	2	1	no	—
98	C₄⋊C₈×C₆₇	—	2	1	no	—
99	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
100	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
101	#103×C₆₇	—	2	2	no	—
102	#103×C₆₇	—	2	2	no	—
103	#103×C₆₇	—	2	2	no	—
104	#102×C₆₇	—	2	4	no	—
105	#102×C₆₇	—	2	2	no	—
106	C₂²⋊C₈×C₆₇	—	2	1	no	—
107	C₂²⋊C₈×C₆₇	—	2	1	no	—
108	C₄×C₈×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	C₂³	1	2	yes	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	1	no	—
112	#41×C₆₇	—	4	8	no	—
113	#41×C₆₇	—	4	16	no	—
114	#41×C₆₇	—	4	8	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	1	no	—
119	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	complemented
120	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	8	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	4	2	no	—
130	C₂×D₄×C₆₇	—	4	4	no	—
131	C₂×D₄×C₆₇	—	4	8	no	—
132	C₂×D₄×C₆₇	—	4	2	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	8	2	no	—
135	C₂×D₄×C₆₇	—	8	4	no	—
136	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	8	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	2	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	4	8	no	—
145	C₂²×C₄×C₆₇	—	4	2	no	—
146	C₂²×C₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	SD₁₆×C₆₇	—	4	4	no	—
151	SD₁₆×C₆₇	—	8	4	no	—
152	M₁₆×C₆₇	—	4	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	1	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	1	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	4	1	no	—
165	C₄⋊C₄×C₆₇	—	4	1	no	—
166	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²⋊C₄×C₆₇	—	2	8	no	—
169	C₂²⋊C₄×C₆₇	—	2	2	no	—
170	C₂²⋊C₄×C₆₇	—	2	2	no	—
171	C₂²⋊C₄×C₆₇	—	4	1	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₄²×C₆₇	—	4	1	no	—
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	4	2	no	—
182	C₂³×C₆₇	—	4	4	no	—
183	C₂³×C₆₇	—	4	4	no	—
184	C₂³×C₆₇	—	4	2	no	—
185	C₂³×C₆₇	—	4	4	no	—
186	C₂³×C₆₇	—	4	2	no	—
187	C₂³×C₆₇	#120	1	1	yes	characteristic
188	Q₈×C₆₇	—	2	2	no	—
189	Q₈×C₆₇	—	2	4	no	—
190	Q₈×C₆₇	—	4	2	no	—
191	Q₈×C₆₇	—	4	2	no	—
192	Q₈×C₆₇	C₂×SD₁₆	1	4	yes	—
193	D₄×C₆₇	—	2	2	no	—
194	D₄×C₆₇	—	2	2	no	—
195	D₄×C₆₇	—	4	4	no	—
196	D₄×C₆₇	—	4	2	no	—
197	D₄×C₆₇	—	4	4	no	—
198	D₄×C₆₇	—	4	16	no	—
199	D₄×C₆₇	—	4	2	no	—
200	D₄×C₆₇	—	4	2	no	—
201	D₄×C₆₇	—	4	4	no	—
202	D₄×C₆₇	—	4	4	no	—
203	D₄×C₆₇	—	8	2	no	—
204	D₄×C₆₇	—	8	8	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	4	16	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	4	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
220	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
221	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
223	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
224	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₈×C₆₇	—	4	2	no	—
228	C₈×C₆₇	—	4	2	no	—
229	C₈×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	2	1	no	—
231	C₂²×C₆₇	—	2	1	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	2	no	—
234	C₂²×C₆₇	—	4	2	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	8	4	no	—
239	C₂²×C₆₇	—	8	8	no	—
240	C₂²×C₆₇	#450	1	2	yes	—
241	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
242	C₄×C₆₇	—	2	4	no	—
243	C₄×C₆₇	—	2	1	no	—
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	—	4	2	no	—
248	C₄×C₆₇	—	8	1	no	—
249	C₄×C₆₇	—	8	1	no	—
250	C₄×C₆₇	#450	1	2	yes	—
251	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
252	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₄×C₆₇	#575	1	2	yes	—
254	#18767	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₆₇	—	2	2	no	—
256	C₂×C₆₇	—	8	2	no	—
257	C₂×C₆₇	—	8	4	no	—
258	C₂×C₆₇	C₂×#450	1	1	yes	characteristic
259	C₂×C₆₇	#3005	1	1	yes	characteristic
260	C₂×C₆₇	#3019	1	1	yes	characteristic
261	C₂×#550	C₂×C₆₇	1	1	yes	characteristic
262	#3208	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#3101	C₂×C₆₇	1	1	yes	characteristic, complemented
264	#3078	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1642	C₂×C₆₇	1	4	yes	complemented
266	#1631	C₂×C₆₇	1	4	yes	complemented
267	#1545	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#1486	C₂×C₆₇	1	1	yes	characteristic, complemented
269	#1479	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₆₇	#18767	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×#41	—	2	1	no	—
272	#550	—	2	2	no	—
273	#550	—	2	2	no	—
274	#550	—	2	2	no	—
275	#550	C₂²×C₆₇	1	4	yes	—
276	#533	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×#127	—	2	1	no	—
278	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×#121	—	2	2	no	—
280	C₂×#121	—	2	1	no	—
281	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
282	C₂×C₄×D₄	—	2	1	no	—
283	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
284	#492	C₂²×C₆₇	1	2	yes	complemented
285	#486	C₂²×C₆₇	1	2	yes	complemented
286	#483	—	2	1	no	—
287	#479	—	2	2	no	—
288	#477	C₂²×C₆₇	1	2	yes	complemented
289	#476	C₂²×C₆₇	1	2	yes	complemented
290	#475	C₂²×C₆₇	1	2	yes	complemented
291	#461	C₂²×C₆₇	1	2	yes	complemented
292	#460	—	2	2	no	—
293	#450	—	2	1	no	—
294	#450	—	2	2	no	—
295	C₄×SD₁₆	—	2	2	no	—
296	#431	C₂²×C₆₇	1	1	yes	characteristic
297	#431	C₂²×C₆₇	1	1	yes	characteristic
298	#428	—	2	1	no	—
299	#423	C₂²×C₆₇	1	1	yes	characteristic
300	#419	—	2	1	no	—
301	#331	C₂²×C₆₇	1	4	yes	complemented
302	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	complemented
303	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
304	C₂×#41	—	2	4	no	—
305	C₂×#41	—	2	4	no	—
306	C₂×#41	—	4	4	no	—
307	C₂×#41	—	4	4	no	—
308	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
309	C₂²×D₄	—	2	2	no	—
310	C₂²×D₄	—	2	2	no	—
311	C₂²×D₄	—	2	1	no	—
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
314	C₂²×D₄	D₄×C₆₇	1	2	yes	—
315	C₂³×C₄	—	2	2	no	—
316	C₂³×C₄	—	2	1	no	—
317	C₂×SD₁₆	—	2	2	no	—
318	C₂×SD₁₆	—	2	4	no	—
319	C₂×M₁₆	—	2	1	no	—
320	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
321	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
322	#127	—	2	4	no	—
323	#127	—	2	4	no	—
324	#127	—	2	1	no	—
325	#127	—	2	1	no	—
326	#127	D₄×C₆₇	1	2	yes	—
327	#127	C₂³×C₆₇	1	2	yes	—
328	#125	—	2	1	no	—
329	C₂²⋊Q₈	—	2	1	no	—
330	C₂²⋊Q₈	—	2	1	no	—
331	#121	—	2	4	no	—
332	#121	—	2	8	no	—
333	#121	—	2	8	no	—
334	#121	—	2	4	no	—
335	C₄×Q₈	—	2	1	no	—
336	C₄×Q₈	—	2	1	no	—
337	C₄×Q₈	—	2	1	no	—
338	C₄×Q₈	—	2	1	no	—
339	C₄×Q₈	D₄×C₆₇	1	2	yes	—
340	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	—	2	1	no	—
344	C₄×D₄	—	2	1	no	—
345	C₄×D₄	D₄×C₆₇	1	2	yes	—
346	C₄×D₄	C₂³×C₆₇	1	2	yes	—
347	#117	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
355	#107	—	2	2	no	—
356	#106	—	2	2	no	—
357	#106	—	2	2	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	—	2	1	no	—
360	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
361	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
362	#103	—	2	2	no	—
363	#103	—	2	2	no	—
364	#103	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	4	no	—
367	C₂²⋊C₈	—	2	1	no	—
368	C₂²⋊C₈	—	2	1	no	—
369	C₄×C₈	—	2	1	no	—
370	C₄×C₈	C₂³×C₆₇	1	2	yes	—
371	C₂⁴	—	2	2	no	—
372	C₂⁴	—	2	1	no	—
373	#41	—	4	16	no	—
374	#41	—	4	8	no	—
375	#41	—	4	8	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	SD₁₆×C₆₇	1	4	yes	complemented
381	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	4	8	no	—
392	C₂×D₄	—	4	4	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	—	8	2	no	—
396	C₂×D₄	—	8	4	no	—
397	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
398	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	8	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	4	8	no	—
406	C₂²×C₄	—	4	2	no	—
407	C₂²×C₄	—	4	2	no	—
408	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
411	SD₁₆	—	4	4	no	—
412	SD₁₆	—	8	4	no	—
413	M₁₆	—	4	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	4	1	no	—
426	C₄⋊C₄	—	4	1	no	—
427	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
428	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	4	1	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	4	1	no	—
436	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
440	C₂³	—	2	4	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	4	4	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	4	no	—
447	C₂³	—	4	2	no	—
448	C₂³	#120×C₆₇	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	C₂×SD₁₆×C₆₇	1	4	yes	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	4	4	no	—
457	D₄	—	4	16	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	4	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	4	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	8	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	16	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
481	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
482	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
483	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
484	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
488	C₈	—	4	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	2	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	8	8	no	—
500	C₂²	—	8	4	no	—
501	C₂²	#450×C₆₇	1	2	yes	—
502	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	4	no	—
506	C₄	—	2	1	no	—
507	C₄	—	4	2	no	—
508	C₄	—	4	2	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	1	no	—
511	C₄	#450×C₆₇	1	2	yes	—
512	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
513	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
514	C₄	#575×C₆₇	1	2	yes	—
515	C₂	—	2	2	no	—
516	C₂	—	8	4	no	—
517	C₂	—	8	2	no	—
518	C₂	C₂×#450×C₆₇	1	1	yes	characteristic
519	C₂	C₆₇×#3005	1	1	yes	characteristic
520	C₂	C₆₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#550×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3208	C₂	semidirect
3. C₆₇×#3101	C₂	semidirect
4. C₆₇×#3078	C₂	semidirect
5. C₆₇×#1642	C₂	semidirect
6. C₆₇×#1631	C₂	semidirect
7. C₆₇×#1545	C₂	semidirect
8. C₆₇×#1486	C₂	semidirect
9. C₆₇×#1479	C₂	semidirect
23. #492×C₆₇	C₂²	semidirect
24. #486×C₆₇	C₂²	semidirect
27. #477×C₆₇	C₂²	semidirect
28. #476×C₆₇	C₂²	semidirect
29. #475×C₆₇	C₂²	semidirect
30. #461×C₆₇	C₂²	semidirect
40. #331×C₆₇	C₂²	semidirect
41. Q₈⋊C₈×C₆₇	C₂²	semidirect
42. D₄⋊C₈×C₆₇	C₂²	semidirect
119. C₂×Q₈×C₆₇	SD₁₆	semidirect
254. #18767	C₆₇	direct (N × H)
262. #3208	C₂×C₆₇	semidirect
263. #3101	C₂×C₆₇	semidirect
264. #3078	C₂×C₆₇	semidirect
265. #1642	C₂×C₆₇	semidirect
266. #1631	C₂×C₆₇	semidirect
267. #1545	C₂×C₆₇	semidirect
268. #1486	C₂×C₆₇	semidirect
269. #1479	C₂×C₆₇	semidirect
270. C₆₇	#18767	direct (N × H)
284. #492	C₂²×C₆₇	semidirect
285. #486	C₂²×C₆₇	semidirect
288. #477	C₂²×C₆₇	semidirect
289. #476	C₂²×C₆₇	semidirect
290. #475	C₂²×C₆₇	semidirect
291. #461	C₂²×C₆₇	semidirect
301. #331	C₂²×C₆₇	semidirect
302. Q₈⋊C₈	C₂²×C₆₇	semidirect
303. D₄⋊C₈	C₂²×C₆₇	semidirect
380. C₂×Q₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#550×C₆₇	2
C₆₇×#3208	2
C₆₇×#3101	2
C₆₇×#3078	2
C₆₇×#1642	2
C₆₇×#1631	2
C₆₇×#1545	2
C₆₇×#1486	2
C₆₇×#1479	2
#18767	67

Subgroups — C67×#18767 #30698018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18767 #30698018