Subgroups — C₆₇×#18862 #30698113

Order 17152.

1098 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3497	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3228	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3113	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1651	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1649	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1642	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1641	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1557	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	—	2	1	no	—
13	#550×C₆₇	—	2	1	no	—
14	#550×C₆₇	C₂²	1	2	yes	—
15	D₄×Q₈×C₆₇	—	2	1	no	—
16	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	#548×C₆₇	—	2	2	no	—
18	#548×C₆₇	—	2	2	no	—
19	#537×C₆₇	—	2	2	no	—
20	#537×C₆₇	—	2	1	no	—
21	#533×C₆₇	C₂²	1	1	yes	characteristic
22	#532×C₆₇	—	2	2	no	—
23	#532×C₆₇	C₂²	1	1	yes	characteristic
24	#532×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	#500×C₆₇	C₂²	1	1	yes	characteristic
27	#498×C₆₇	C₂²	1	1	yes	characteristic
28	#486×C₆₇	C₂²	1	1	yes	characteristic
29	#486×C₆₇	C₂²	1	1	yes	characteristic
30	#483×C₆₇	—	2	1	no	—
31	#482×C₆₇	—	2	2	no	—
32	#478×C₆₇	—	2	2	no	—
33	#477×C₆₇	—	2	1	no	—
34	#475×C₆₇	C₂²	1	2	yes	—
35	#474×C₆₇	C₂²	1	2	yes	—
36	#468×C₆₇	—	2	2	no	—
37	#462×C₆₇	C₂²	1	2	yes	—
38	#461×C₆₇	C₂²	1	2	yes	—
39	#460×C₆₇	—	2	1	no	—
40	#451×C₆₇	—	2	1	no	—
41	#441×C₆₇	—	2	2	no	—
42	#432×C₆₇	C₂²	1	1	yes	characteristic
43	#431×C₆₇	C₂²	1	1	yes	characteristic
44	#423×C₆₇	C₂²	1	1	yes	characteristic
45	#332×C₆₇	C₂²	1	2	yes	—
46	#331×C₆₇	C₂²	1	2	yes	—
47	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
48	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
49	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
50	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
51	2₋¹⁺⁴×C₆₇	—	4	8	no	—
52	C₂×#41×C₆₇	—	2	4	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	—	2	2	no	—
55	C₂×#41×C₆₇	—	2	1	no	—
56	C₂×#41×C₆₇	—	2	1	no	—
57	C₂²×Q₈×C₆₇	—	2	2	no	—
58	C₂²×Q₈×C₆₇	—	2	2	no	—
59	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
60	C₂×Q₁₆×C₆₇	—	2	2	no	—
61	C₂×SD₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	1	no	—
63	C₄⋊Q₈×C₆₇	—	2	2	no	—
64	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
68	#127×C₆₇	—	2	2	no	—
69	#127×C₆₇	D₄	1	1	yes	characteristic
70	#127×C₆₇	C₂³	1	1	yes	characteristic
71	#125×C₆₇	—	2	1	no	—
72	#124×C₆₇	—	2	4	no	—
73	#124×C₆₇	—	2	1	no	—
74	#124×C₆₇	—	2	1	no	—
75	C₂²⋊Q₈×C₆₇	—	2	2	no	—
76	C₂²⋊Q₈×C₆₇	—	2	4	no	—
77	C₂²⋊Q₈×C₆₇	—	2	2	no	—
78	C₂²⋊Q₈×C₆₇	—	2	4	no	—
79	C₂²⋊Q₈×C₆₇	—	2	1	no	—
80	C₂²⋊Q₈×C₆₇	—	2	1	no	—
81	#121×C₆₇	—	2	4	no	—
82	#121×C₆₇	—	2	2	no	—
83	#121×C₆₇	—	2	4	no	—
84	#121×C₆₇	—	2	4	no	—
85	#121×C₆₇	—	2	1	no	—
86	#121×C₆₇	—	2	1	no	—
87	#121×C₆₇	—	2	2	no	—
88	#121×C₆₇	D₄	1	2	yes	—
89	#121×C₆₇	D₄	1	2	yes	—
90	C₄×Q₈×C₆₇	—	2	1	no	—
91	C₄×Q₈×C₆₇	—	2	1	no	—
92	C₄×Q₈×C₆₇	—	2	1	no	—
93	C₄×Q₈×C₆₇	—	2	1	no	—
94	C₄×Q₈×C₆₇	D₄	1	2	yes	—
95	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
96	C₄×D₄×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	2	no	—
98	C₄×D₄×C₆₇	—	2	2	no	—
99	C₄×D₄×C₆₇	—	2	1	no	—
100	C₄×D₄×C₆₇	—	2	1	no	—
101	C₄×D₄×C₆₇	D₄	1	2	yes	—
102	C₄×D₄×C₆₇	C₂³	1	2	yes	—
103	#117×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	—	2	1	no	—
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	#107×C₆₇	—	2	2	no	—
109	#107×C₆₇	—	2	2	no	—
110	#106×C₆₇	—	2	2	no	—
111	#106×C₆₇	—	2	2	no	—
112	C₄⋊C₈×C₆₇	—	2	1	no	—
113	C₄⋊C₈×C₆₇	—	2	1	no	—
114	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
115	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
116	#103×C₆₇	—	2	2	no	—
117	#103×C₆₇	—	2	2	no	—
118	#103×C₆₇	—	2	2	no	—
119	#102×C₆₇	—	2	2	no	—
120	#102×C₆₇	—	2	2	no	—
121	#102×C₆₇	—	2	2	no	—
122	C₂²⋊C₈×C₆₇	—	2	1	no	—
123	C₂²⋊C₈×C₆₇	—	2	1	no	—
124	#97×C₆₇	—	2	1	no	—
125	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
127	#41×C₆₇	—	4	16	no	—
128	#41×C₆₇	—	4	4	no	—
129	#41×C₆₇	—	4	4	no	—
130	#41×C₆₇	—	4	8	no	—
131	#41×C₆₇	—	4	4	no	—
132	#41×C₆₇	—	4	4	no	—
133	C₂×Q₈×C₆₇	—	2	4	no	—
134	C₂×Q₈×C₆₇	—	2	4	no	—
135	C₂×Q₈×C₆₇	—	2	1	no	—
136	C₂×Q₈×C₆₇	—	2	1	no	—
137	C₂×Q₈×C₆₇	—	2	1	no	—
138	C₂×Q₈×C₆₇	—	2	1	no	—
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	4	8	no	—
141	C₂×Q₈×C₆₇	—	4	4	no	—
142	C₂×Q₈×C₆₇	—	4	4	no	—
143	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
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	4	no	—
148	C₂×D₄×C₆₇	—	2	1	no	—
149	C₂×D₄×C₆₇	—	2	1	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	Q₁₆×C₆₇	—	4	4	no	—
164	SD₁₆×C₆₇	—	8	2	no	—
165	M₁₆×C₆₇	—	8	1	no	—
166	C₂×C₈×C₆₇	—	2	1	no	—
167	C₂×C₈×C₆₇	—	2	1	no	—
168	C₂×C₈×C₆₇	—	2	2	no	—
169	C₂×C₈×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	4	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	—	2	1	no	—
177	C₄⋊C₄×C₆₇	—	2	1	no	—
178	C₄⋊C₄×C₆₇	—	2	1	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	4	1	no	—
181	C₄⋊C₄×C₆₇	—	4	1	no	—
182	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₂²⋊C₄×C₆₇	—	2	4	no	—
186	C₂²⋊C₄×C₆₇	—	2	4	no	—
187	C₂²⋊C₄×C₆₇	—	2	2	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	2	no	—
190	C₂²⋊C₄×C₆₇	—	2	2	no	—
191	C₂²⋊C₄×C₆₇	—	4	1	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	—	4	1	no	—
195	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	#120	1	1	yes	characteristic
202	Q₈×C₆₇	—	2	4	no	—
203	Q₈×C₆₇	—	2	2	no	—
204	Q₈×C₆₇	—	4	8	no	—
205	Q₈×C₆₇	—	4	2	no	—
206	Q₈×C₆₇	—	4	2	no	—
207	Q₈×C₆₇	—	4	2	no	—
208	Q₈×C₆₇	—	4	1	no	—
209	Q₈×C₆₇	—	4	1	no	—
210	Q₈×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
211	D₄×C₆₇	—	2	2	no	—
212	D₄×C₆₇	—	2	2	no	—
213	D₄×C₆₇	—	4	2	no	—
214	D₄×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	4	2	no	—
216	D₄×C₆₇	—	4	4	no	—
217	D₄×C₆₇	—	4	4	no	—
218	D₄×C₆₇	—	4	2	no	—
219	D₄×C₆₇	—	4	1	no	—
220	D₄×C₆₇	—	4	1	no	—
221	D₄×C₆₇	—	8	1	no	—
222	D₄×C₆₇	—	8	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	8	no	—
233	C₂×C₄×C₆₇	—	4	8	no	—
234	C₂×C₄×C₆₇	—	4	4	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	8	1	no	—
240	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
241	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
242	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
246	C₈×C₆₇	—	4	1	no	—
247	C₈×C₆₇	—	4	1	no	—
248	C₈×C₆₇	—	4	2	no	—
249	C₈×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	2	1	no	—
252	C₂²×C₆₇	—	2	1	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	—	4	2	no	—
255	C₂²×C₆₇	—	4	2	no	—
256	C₂²×C₆₇	—	4	2	no	—
257	C₂²×C₆₇	—	4	2	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	2	1	no	—
264	C₄×C₆₇	—	2	1	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	4	2	no	—
267	C₄×C₆₇	—	4	2	no	—
268	C₄×C₆₇	—	8	1	no	—
269	C₄×C₆₇	—	8	1	no	—
270	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
271	C₄×C₆₇	#577	1	1	yes	characteristic
272	C₄×C₆₇	#578	1	1	yes	characteristic
273	#18862	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂×C₆₇	—	4	1	no	—
275	C₂×C₆₇	—	8	2	no	—
276	C₂×C₆₇	—	8	2	no	—
277	C₂×C₆₇	#3003	1	1	yes	characteristic
278	C₂×C₆₇	#3013	1	1	yes	characteristic
279	C₂×C₆₇	#3022	1	1	yes	characteristic
280	#3497	C₂×C₆₇	1	1	yes	characteristic
281	#3254	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#3228	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#3113	C₂×C₆₇	1	1	yes	characteristic, complemented
284	#1651	C₂×C₆₇	1	2	yes	complemented
285	#1649	C₂×C₆₇	1	2	yes	complemented
286	#1642	C₂×C₆₇	1	2	yes	complemented
287	#1641	C₂×C₆₇	1	2	yes	complemented
288	#1557	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#1496	C₂×C₆₇	1	1	yes	characteristic, complemented
290	#1487	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₆₇	#18862	1	1	yes	characteristic, direct product factor, Sylow
292	C₂×2₋¹⁺⁴	—	2	1	no	—
293	#550	—	2	1	no	—
294	#550	C₂²×C₆₇	1	2	yes	—
295	D₄×Q₈	—	2	1	no	—
296	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
297	#548	—	2	2	no	—
298	#548	—	2	2	no	—
299	#537	—	2	2	no	—
300	#537	—	2	1	no	—
301	#533	C₂²×C₆₇	1	1	yes	characteristic
302	#532	—	2	2	no	—
303	#532	C₂²×C₆₇	1	1	yes	characteristic
304	#532	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
306	#500	C₂²×C₆₇	1	1	yes	characteristic
307	#498	C₂²×C₆₇	1	1	yes	characteristic
308	#486	C₂²×C₆₇	1	1	yes	characteristic
309	#486	C₂²×C₆₇	1	1	yes	characteristic
310	#483	—	2	1	no	—
311	#482	—	2	2	no	—
312	#478	—	2	2	no	—
313	#477	—	2	1	no	—
314	#475	C₂²×C₆₇	1	2	yes	—
315	#474	C₂²×C₆₇	1	2	yes	—
316	#468	—	2	2	no	—
317	#462	C₂²×C₆₇	1	2	yes	—
318	#461	C₂²×C₆₇	1	2	yes	—
319	#460	—	2	1	no	—
320	#451	—	2	1	no	—
321	#441	—	2	2	no	—
322	#432	C₂²×C₆₇	1	1	yes	characteristic
323	#431	C₂²×C₆₇	1	1	yes	characteristic
324	#423	C₂²×C₆₇	1	1	yes	characteristic
325	#332	C₂²×C₆₇	1	2	yes	—
326	#331	C₂²×C₆₇	1	2	yes	—
327	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
328	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
329	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
330	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
331	2₋¹⁺⁴	—	4	8	no	—
332	C₂×#41	—	2	4	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	1	no	—
336	C₂×#41	—	2	1	no	—
337	C₂²×Q₈	—	2	2	no	—
338	C₂²×Q₈	—	2	2	no	—
339	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
340	C₂×Q₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₄⋊Q₈	—	2	2	no	—
344	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
345	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
346	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
348	#127	—	2	2	no	—
349	#127	D₄×C₆₇	1	1	yes	characteristic
350	#127	C₂³×C₆₇	1	1	yes	characteristic
351	#125	—	2	1	no	—
352	#124	—	2	4	no	—
353	#124	—	2	1	no	—
354	#124	—	2	1	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	#121	—	2	4	no	—
362	#121	—	2	2	no	—
363	#121	—	2	4	no	—
364	#121	—	2	4	no	—
365	#121	—	2	1	no	—
366	#121	—	2	1	no	—
367	#121	—	2	2	no	—
368	#121	D₄×C₆₇	1	2	yes	—
369	#121	D₄×C₆₇	1	2	yes	—
370	C₄×Q₈	—	2	1	no	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	D₄×C₆₇	1	2	yes	—
375	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
376	C₄×D₄	—	2	4	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	D₄×C₆₇	1	2	yes	—
382	C₄×D₄	C₂³×C₆₇	1	2	yes	—
383	#117	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
388	#107	—	2	2	no	—
389	#107	—	2	2	no	—
390	#106	—	2	2	no	—
391	#106	—	2	2	no	—
392	C₄⋊C₈	—	2	1	no	—
393	C₄⋊C₈	—	2	1	no	—
394	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
395	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
396	#103	—	2	2	no	—
397	#103	—	2	2	no	—
398	#103	—	2	2	no	—
399	#102	—	2	2	no	—
400	#102	—	2	2	no	—
401	#102	—	2	2	no	—
402	C₂²⋊C₈	—	2	1	no	—
403	C₂²⋊C₈	—	2	1	no	—
404	#97	—	2	1	no	—
405	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
407	#41	—	4	16	no	—
408	#41	—	4	4	no	—
409	#41	—	4	4	no	—
410	#41	—	4	8	no	—
411	#41	—	4	4	no	—
412	#41	—	4	4	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	4	8	no	—
421	C₂×Q₈	—	4	4	no	—
422	C₂×Q₈	—	4	4	no	—
423	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
433	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
443	Q₁₆	—	4	4	no	—
444	SD₁₆	—	8	2	no	—
445	M₁₆	—	8	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	—	4	1	no	—
462	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
463	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
464	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	2	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	4	1	no	—
472	C₄²	—	2	2	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	4	1	no	—
475	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
476	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
478	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	#120×C₆₇	1	1	yes	characteristic
482	Q₈	—	2	4	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	4	8	no	—
485	Q₈	—	4	2	no	—
486	Q₈	—	4	2	no	—
487	Q₈	—	4	2	no	—
488	Q₈	—	4	1	no	—
489	Q₈	—	4	1	no	—
490	Q₈	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
491	D₄	—	2	2	no	—
492	D₄	—	2	2	no	—
493	D₄	—	4	2	no	—
494	D₄	—	4	8	no	—
495	D₄	—	4	2	no	—
496	D₄	—	4	4	no	—
497	D₄	—	4	4	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	1	no	—
500	D₄	—	4	1	no	—
501	D₄	—	8	1	no	—
502	D₄	—	8	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	8	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	8	1	no	—
520	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
521	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
522	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
525	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
526	C₈	—	4	1	no	—
527	C₈	—	4	1	no	—
528	C₈	—	4	2	no	—
529	C₈	—	4	2	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	1	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	8	1	no	—
549	C₄	—	8	1	no	—
550	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
551	C₄	#577×C₆₇	1	1	yes	characteristic
552	C₄	#578×C₆₇	1	1	yes	characteristic
553	C₂	—	4	1	no	—
554	C₂	—	8	2	no	—
555	C₂	—	8	2	no	—
556	C₂	C₆₇×#3003	1	1	yes	characteristic
557	C₂	C₆₇×#3013	1	1	yes	characteristic
558	C₂	C₆₇×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3497	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×#3254	C₂	semidirect
3. C₆₇×#3228	C₂	semidirect
4. C₆₇×#3113	C₂	semidirect
5. C₆₇×#1651	C₂	semidirect
6. C₆₇×#1649	C₂	semidirect
7. C₆₇×#1642	C₂	semidirect
8. C₆₇×#1641	C₂	semidirect
9. C₆₇×#1557	C₂	semidirect
10. C₆₇×#1496	C₂	semidirect
11. C₆₇×#1487	C₂	semidirect
273. #18862	C₆₇	direct (N × H)
281. #3254	C₂×C₆₇	semidirect
282. #3228	C₂×C₆₇	semidirect
283. #3113	C₂×C₆₇	semidirect
284. #1651	C₂×C₆₇	semidirect
285. #1649	C₂×C₆₇	semidirect
286. #1642	C₂×C₆₇	semidirect
287. #1641	C₂×C₆₇	semidirect
288. #1557	C₂×C₆₇	semidirect
289. #1496	C₂×C₆₇	semidirect
290. #1487	C₂×C₆₇	semidirect
291. C₆₇	#18862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3497	2
C₆₇×#3254	2
C₆₇×#3228	2
C₆₇×#3113	2
C₆₇×#1651	2
C₆₇×#1649	2
C₆₇×#1642	2
C₆₇×#1641	2
C₆₇×#1557	2
C₆₇×#1496	2
C₆₇×#1487	2
#18862	67

Subgroups — C67×#18862 #30698113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18862 #30698113