Subgroups — C₆₇×#18894 #30698145

Order 17152.

1022 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1627	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1622	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1542	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₆₇	—	2	1	no	—
11	#554×C₆₇	—	2	1	no	—
12	#554×C₆₇	—	2	2	no	—
13	#554×C₆₇	—	2	2	no	—
14	#554×C₆₇	C₂²	1	2	yes	complemented
15	D₄×Q₈×C₆₇	—	2	1	no	—
16	D₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
17	#544×C₆₇	—	2	1	no	—
18	#543×C₆₇	—	2	2	no	—
19	#533×C₆₇	C₂²	1	1	yes	characteristic
20	#533×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
23	#518×C₆₇	—	2	1	no	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	#492×C₆₇	—	2	1	no	—
26	#492×C₆₇	C₂²	1	2	yes	complemented
27	#486×C₆₇	—	2	1	no	—
28	#486×C₆₇	C₂²	1	2	yes	complemented
29	#460×C₆₇	—	2	2	no	—
30	#460×C₆₇	C₂²	1	4	yes	complemented
31	#459×C₆₇	—	2	2	no	—
32	#459×C₆₇	C₂²	1	4	yes	—
33	#450×C₆₇	—	2	2	no	—
34	#450×C₆₇	—	2	2	no	—
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#102×C₆₇	—	2	1	no	—
39	#333×C₆₇	C₂²	1	2	yes	complemented
40	#332×C₆₇	C₂²	1	2	yes	complemented
41	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	complemented
42	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
43	C₂²×Q₈×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	—	2	1	no	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×D₈×C₆₇	—	2	4	no	—
52	C₂×D₈×C₆₇	—	4	4	no	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	#127×C₆₇	—	2	1	no	—
61	#127×C₆₇	—	2	1	no	—
62	#127×C₆₇	—	2	1	no	—
63	#127×C₆₇	C₂³	1	2	yes	—
64	#125×C₆₇	—	2	2	no	—
65	#125×C₆₇	—	2	2	no	—
66	#125×C₆₇	—	2	1	no	—
67	#125×C₆₇	—	2	1	no	—
68	C₂²⋊Q₈×C₆₇	—	2	4	no	—
69	C₂²⋊Q₈×C₆₇	—	2	4	no	—
70	C₂²⋊Q₈×C₆₇	—	2	4	no	—
71	C₂²⋊Q₈×C₆₇	—	2	4	no	—
72	C₂²⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	C₂²⋊Q₈×C₆₇	—	2	1	no	—
75	C₂²⋊Q₈×C₆₇	—	2	1	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	4	no	—
78	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
79	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
80	C₄×Q₈×C₆₇	—	2	1	no	—
81	C₄×Q₈×C₆₇	—	2	1	no	—
82	C₄×Q₈×C₆₇	D₄	1	2	yes	—
83	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	—	2	1	no	—
89	C₄×D₄×C₆₇	—	2	1	no	—
90	C₄×D₄×C₆₇	D₄	1	2	yes	—
91	C₄×D₄×C₆₇	C₂³	1	2	yes	—
92	#117×C₆₇	—	2	1	no	—
93	#117×C₆₇	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₆₇	—	2	1	no	—
101	C₄⋊C₈×C₆₇	—	2	1	no	—
102	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
104	#102×C₆₇	—	2	4	no	—
105	#102×C₆₇	—	2	4	no	—
106	#102×C₆₇	—	2	4	no	—
107	C₂²⋊C₈×C₆₇	—	2	1	no	—
108	C₂²⋊C₈×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	—	2	1	no	—
110	C₄×C₈×C₆₇	C₂³	1	2	yes	—
111	C₂⁴×C₆₇	—	2	1	no	—
112	C₂⁴×C₆₇	—	4	1	no	—
113	C₂×Q₈×C₆₇	—	2	4	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	8	4	no	—
117	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	4	1	no	—
125	C₂×D₄×C₆₇	—	4	1	no	—
126	C₂×D₄×C₆₇	—	4	1	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	—	4	2	no	—
129	C₂×D₄×C₆₇	—	8	2	no	—
130	C₂×D₄×C₆₇	—	8	2	no	—
131	C₂×D₄×C₆₇	SD₁₆	1	4	yes	complemented
132	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	8	2	no	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	SD₁₆×C₆₇	—	8	4	no	—
143	SD₁₆×C₆₇	—	8	4	no	—
144	D₈×C₆₇	—	4	8	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	4	2	no	—
149	C₄⋊C₄×C₆₇	—	2	4	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	1	no	—
160	C₄⋊C₄×C₆₇	—	2	1	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₆₇	—	2	4	no	—
167	C₂²⋊C₄×C₆₇	—	2	4	no	—
168	C₂²⋊C₄×C₆₇	—	2	2	no	—
169	C₂²⋊C₄×C₆₇	—	2	2	no	—
170	C₂²⋊C₄×C₆₇	—	2	2	no	—
171	C₂²⋊C₄×C₆₇	—	2	2	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	8	1	no	—
182	C₂³×C₆₇	—	8	1	no	—
183	C₂³×C₆₇	—	8	2	no	—
184	C₂³×C₆₇	—	8	2	no	—
185	C₂³×C₆₇	—	8	2	no	—
186	C₂³×C₆₇	#120	1	1	yes	characteristic
187	Q₈×C₆₇	—	2	4	no	—
188	Q₈×C₆₇	—	4	2	no	—
189	Q₈×C₆₇	—	4	2	no	—
190	Q₈×C₆₇	—	8	1	no	—
191	Q₈×C₆₇	—	8	4	no	—
192	D₄×C₆₇	—	2	2	no	—
193	D₄×C₆₇	—	2	4	no	—
194	D₄×C₆₇	—	4	4	no	—
195	D₄×C₆₇	—	4	2	no	—
196	D₄×C₆₇	—	4	2	no	—
197	D₄×C₆₇	—	4	4	no	—
198	D₄×C₆₇	—	8	2	no	—
199	D₄×C₆₇	—	8	2	no	—
200	D₄×C₆₇	—	8	2	no	—
201	D₄×C₆₇	C₂×SD₁₆	1	4	yes	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	4	no	—
210	C₂×C₄×C₆₇	—	8	2	no	—
211	C₂×C₄×C₆₇	—	8	1	no	—
212	C₂×C₄×C₆₇	—	8	4	no	—
213	C₂×C₄×C₆₇	—	8	4	no	—
214	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
215	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
216	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
218	C₂×C₄×C₆₇	#137	1	2	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₈×C₆₇	—	4	2	no	—
223	C₈×C₆₇	—	4	2	no	—
224	C₈×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	4	2	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	8	4	no	—
231	C₂²×C₆₇	—	8	2	no	—
232	C₂²×C₆₇	—	8	2	no	—
233	C₂²×C₆₇	—	8	2	no	—
234	C₂²×C₆₇	—	16	4	no	—
235	C₂²×C₆₇	#448	1	2	yes	—
236	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	2	1	no	—
239	C₄×C₆₇	—	2	1	no	—
240	C₄×C₆₇	—	4	2	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	—	4	2	no	—
245	C₄×C₆₇	#448	1	2	yes	—
246	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
247	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
248	C₄×C₆₇	#576	1	2	yes	—
249	#18894	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂×C₆₇	—	2	2	no	—
251	C₂×C₆₇	—	4	4	no	—
252	C₂×C₆₇	—	16	2	no	—
253	C₂×C₆₇	C₂×#448	1	1	yes	characteristic
254	C₂×C₆₇	#3008	1	1	yes	characteristic
255	C₂×C₆₇	#3021	1	1	yes	characteristic
256	#3501	C₂×C₆₇	1	1	yes	characteristic, complemented
257	#3208	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#3208	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
260	#1627	C₂×C₆₇	1	4	yes	complemented
261	#1622	C₂×C₆₇	1	4	yes	complemented
262	#1542	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1481	C₂×C₆₇	1	1	yes	characteristic, complemented
264	#1478	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₆₇	#18894	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²×D₈	—	2	1	no	—
267	#554	—	2	1	no	—
268	#554	—	2	2	no	—
269	#554	—	2	2	no	—
270	#554	C₂²×C₆₇	1	2	yes	complemented
271	D₄×Q₈	—	2	1	no	—
272	D₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
273	#544	—	2	1	no	—
274	#543	—	2	2	no	—
275	#533	C₂²×C₆₇	1	1	yes	characteristic
276	#533	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×C₂²⋊Q₈	—	2	2	no	—
279	#518	—	2	1	no	—
280	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
281	#492	—	2	1	no	—
282	#492	C₂²×C₆₇	1	2	yes	complemented
283	#486	—	2	1	no	—
284	#486	C₂²×C₆₇	1	2	yes	complemented
285	#460	—	2	2	no	—
286	#460	C₂²×C₆₇	1	4	yes	complemented
287	#459	—	2	2	no	—
288	#459	C₂²×C₆₇	1	4	yes	—
289	#450	—	2	2	no	—
290	#450	—	2	2	no	—
291	#431	C₂²×C₆₇	1	1	yes	characteristic
292	#431	C₂²×C₆₇	1	1	yes	characteristic
293	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#102	—	2	1	no	—
295	#333	C₂²×C₆₇	1	2	yes	complemented
296	#332	C₂²×C₆₇	1	2	yes	complemented
297	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	complemented
298	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
299	C₂²×Q₈	—	2	2	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
304	C₂³×C₄	—	2	1	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	C₂×D₈	—	2	4	no	—
308	C₂×D₈	—	4	4	no	—
309	C₂²×C₈	—	2	1	no	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
313	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
314	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
315	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
316	#127	—	2	1	no	—
317	#127	—	2	1	no	—
318	#127	—	2	1	no	—
319	#127	C₂³×C₆₇	1	2	yes	—
320	#125	—	2	2	no	—
321	#125	—	2	2	no	—
322	#125	—	2	1	no	—
323	#125	—	2	1	no	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	2	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	1	no	—
331	C₂²⋊Q₈	—	2	1	no	—
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
335	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
336	C₄×Q₈	—	2	1	no	—
337	C₄×Q₈	—	2	1	no	—
338	C₄×Q₈	D₄×C₆₇	1	2	yes	—
339	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	—	2	1	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	D₄×C₆₇	1	2	yes	—
347	C₄×D₄	C₂³×C₆₇	1	2	yes	—
348	#117	—	2	1	no	—
349	#117	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	4	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₂²⋊C₄	—	2	4	no	—
354	C₂×C₂²⋊C₄	—	2	1	no	—
355	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
359	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
360	#102	—	2	4	no	—
361	#102	—	2	4	no	—
362	#102	—	2	4	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	C₂²⋊C₈	—	2	1	no	—
365	C₄×C₈	—	2	1	no	—
366	C₄×C₈	C₂³×C₆₇	1	2	yes	—
367	C₂⁴	—	2	1	no	—
368	C₂⁴	—	4	1	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	8	4	no	—
373	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
374	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	1	no	—
381	C₂×D₄	—	4	1	no	—
382	C₂×D₄	—	4	1	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	8	2	no	—
386	C₂×D₄	—	8	2	no	—
387	C₂×D₄	SD₁₆×C₆₇	1	4	yes	complemented
388	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	8	2	no	—
395	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
398	SD₁₆	—	8	4	no	—
399	SD₁₆	—	8	4	no	—
400	D₈	—	4	8	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	4	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
420	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
421	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
434	C₂³	—	2	2	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	8	1	no	—
438	C₂³	—	8	1	no	—
439	C₂³	—	8	2	no	—
440	C₂³	—	8	2	no	—
441	C₂³	—	8	2	no	—
442	C₂³	#120×C₆₇	1	1	yes	characteristic
443	Q₈	—	2	4	no	—
444	Q₈	—	4	2	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	8	1	no	—
447	Q₈	—	8	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	2	2	no	—
450	D₄	—	4	4	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	2	no	—
453	D₄	—	4	4	no	—
454	D₄	—	8	2	no	—
455	D₄	—	8	2	no	—
456	D₄	—	8	2	no	—
457	D₄	C₂×SD₁₆×C₆₇	1	4	yes	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	8	2	no	—
467	C₂×C₄	—	8	1	no	—
468	C₂×C₄	—	8	4	no	—
469	C₂×C₄	—	8	4	no	—
470	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
471	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
472	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
474	C₂×C₄	#137×C₆₇	1	2	yes	—
475	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₈	—	4	2	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	8	4	no	—
487	C₂²	—	8	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	16	4	no	—
491	C₂²	#448×C₆₇	1	2	yes	—
492	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	2	1	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	#448×C₆₇	1	2	yes	—
502	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
503	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
504	C₄	#576×C₆₇	1	2	yes	—
505	C₂	—	2	2	no	—
506	C₂	—	4	4	no	—
507	C₂	—	16	2	no	—
508	C₂	C₂×#448×C₆₇	1	1	yes	characteristic
509	C₂	C₆₇×#3008	1	1	yes	characteristic
510	C₂	C₆₇×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#459×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
1. C₆₇×#3501	C₂	semidirect
2. C₆₇×#3208	C₂	semidirect
3. C₆₇×#3208	C₂	semidirect
5. C₆₇×#1627	C₂	semidirect
6. C₆₇×#1622	C₂	semidirect
7. C₆₇×#1542	C₂	semidirect
8. C₆₇×#1481	C₂	semidirect
9. C₆₇×#1478	C₂	semidirect
14. #554×C₆₇	C₂²	semidirect
16. D₄×Q₈×C₆₇	C₂²	semidirect
26. #492×C₆₇	C₂²	semidirect
28. #486×C₆₇	C₂²	semidirect
30. #460×C₆₇	C₂²	semidirect
39. #333×C₆₇	C₂²	semidirect
40. #332×C₆₇	C₂²	semidirect
41. Q₈⋊C₈×C₆₇	C₂²	semidirect
42. D₄⋊C₈×C₆₇	C₂²	semidirect
131. C₂×D₄×C₆₇	SD₁₆	semidirect
249. #18894	C₆₇	direct (N × H)
256. #3501	C₂×C₆₇	semidirect
257. #3208	C₂×C₆₇	semidirect
258. #3208	C₂×C₆₇	semidirect
260. #1627	C₂×C₆₇	semidirect
261. #1622	C₂×C₆₇	semidirect
262. #1542	C₂×C₆₇	semidirect
263. #1481	C₂×C₆₇	semidirect
264. #1478	C₂×C₆₇	semidirect
265. C₆₇	#18894	direct (N × H)
270. #554	C₂²×C₆₇	semidirect
272. D₄×Q₈	C₂²×C₆₇	semidirect
282. #492	C₂²×C₆₇	semidirect
284. #486	C₂²×C₆₇	semidirect
286. #460	C₂²×C₆₇	semidirect
295. #333	C₂²×C₆₇	semidirect
296. #332	C₂²×C₆₇	semidirect
297. Q₈⋊C₈	C₂²×C₆₇	semidirect
298. D₄⋊C₈	C₂²×C₆₇	semidirect
387. C₂×D₄	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3501	2
C₆₇×#3208	2
C₂×#459×C₆₇	2
C₆₇×#1627	2
C₆₇×#1622	2
C₆₇×#1542	2
C₆₇×#1481	2
C₆₇×#1478	2
#18894	67

Subgroups — C67×#18894 #30698145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18894 #30698145