Subgroups — C₆₇×#33012 #30712263

Order 17152.

1334 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#480×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#449×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2624	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1862	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1824	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
16	#520×C₆₇	—	2	4	no	—
17	#520×C₆₇	—	2	4	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	—	2	2	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#117×C₆₇	—	2	1	no	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	#480×C₆₇	—	2	4	no	—
26	#449×C₆₇	—	2	4	no	—
27	C₂×#107×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#399×C₆₇	—	2	1	no	—
36	#397×C₆₇	—	2	1	no	—
37	#394×C₆₇	—	2	2	no	—
38	#390×C₆₇	C₂²	1	2	yes	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂×#41×C₆₇	—	2	4	no	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	D₄	1	4	yes	—
49	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	—	2	2	no	—
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×SD₁₆×C₆₇	—	2	2	no	—
59	C₂×SD₁₆×C₆₇	—	2	4	no	—
60	C₂×D₈×C₆₇	—	2	2	no	—
61	C₂×D₈×C₆₇	—	2	4	no	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	#121×C₆₇	—	2	2	no	—
66	#121×C₆₇	—	2	2	no	—
67	C₄×Q₈×C₆₇	—	2	4	no	—
68	C₄×Q₈×C₆₇	—	2	4	no	—
69	C₄×D₄×C₆₇	—	2	16	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	—	2	4	no	—
72	#117×C₆₇	—	2	8	no	—
73	#117×C₆₇	—	2	2	no	—
74	#117×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
92	C₂×C₄²×C₆₇	—	2	2	no	—
93	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
94	#107×C₆₇	—	2	4	no	—
95	#103×C₆₇	—	2	2	no	—
96	#103×C₆₇	—	2	4	no	—
97	#102×C₆₇	—	2	2	no	—
98	#102×C₆₇	—	2	4	no	—
99	#102×C₆₇	—	2	4	no	—
100	C₂²⋊C₈×C₆₇	—	2	2	no	—
101	C₂²⋊C₈×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	—	2	1	no	—
105	C₂⁴×C₆₇	—	2	1	no	—
106	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
107	#41×C₆₇	—	2	16	no	—
108	#41×C₆₇	—	2	8	no	—
109	#41×C₆₇	—	2	8	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×Q₈×C₆₇	#41	1	4	yes	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	4	no	—
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	—	2	1	no	—
118	C₂×D₄×C₆₇	—	2	1	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	4	2	no	—
123	C₂×D₄×C₆₇	—	8	2	no	—
124	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₆₇	#41	1	4	yes	—
126	C₂²×C₄×C₆₇	—	2	8	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	8	2	no	—
137	C₂²×C₄×C₆₇	—	8	2	no	—
138	C₂²×C₄×C₆₇	—	8	2	no	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	SD₁₆×C₆₇	—	4	4	no	—
146	D₈×C₆₇	—	4	4	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	1	no	—
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	8	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	2	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	2	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	8	2	no	—
164	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	4	no	—
167	C₂²⋊C₄×C₆₇	—	4	2	no	—
168	C₂²⋊C₄×C₆₇	—	8	2	no	—
169	C₂²⋊C₄×C₆₇	—	8	2	no	—
170	C₂²⋊C₄×C₆₇	—	8	1	no	—
171	C₄²×C₆₇	—	2	8	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	1	no	—
179	C₂³×C₆₇	—	2	1	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	—	4	1	no	—
183	C₂³×C₆₇	—	4	1	no	—
184	C₂³×C₆₇	—	4	2	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	1	no	—
187	C₂³×C₆₇	—	4	1	no	—
188	C₂³×C₆₇	#120	1	2	yes	—
189	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₆₇	—	2	2	no	—
191	Q₈×C₆₇	—	2	4	no	—
192	Q₈×C₆₇	#123	1	4	yes	—
193	D₄×C₆₇	—	2	16	no	—
194	D₄×C₆₇	—	2	2	no	—
195	D₄×C₆₇	—	2	4	no	—
196	D₄×C₆₇	—	4	1	no	—
197	D₄×C₆₇	—	4	1	no	—
198	D₄×C₆₇	—	4	2	no	—
199	D₄×C₆₇	—	8	4	no	—
200	D₄×C₆₇	#123	1	4	yes	—
201	C₂×C₄×C₆₇	—	2	16	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	8	4	no	—
223	C₂×C₄×C₆₇	—	8	8	no	—
224	C₂×C₄×C₆₇	#120	1	2	yes	—
225	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
228	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
229	C₈×C₆₇	—	4	1	no	—
230	C₈×C₆₇	—	4	1	no	—
231	C₈×C₆₇	—	4	2	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	2	4	no	—
234	C₂²×C₆₇	—	2	2	no	—
235	C₂²×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	4	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	8	1	no	—
243	C₂²×C₆₇	—	8	2	no	—
244	C₂²×C₆₇	—	8	2	no	—
245	C₂²×C₆₇	—	8	1	no	—
246	C₂²×C₆₇	—	8	1	no	—
247	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
248	C₂²×C₆₇	#525	1	2	yes	—
249	C₂²×C₆₇	#576	1	2	yes	—
250	C₂²×C₆₇	#577	1	1	yes	characteristic
251	C₂²×C₆₇	#578	1	1	yes	characteristic
252	C₄×C₆₇	—	2	2	no	—
253	C₄×C₆₇	—	2	4	no	—
254	C₄×C₆₇	—	4	2	no	—
255	C₄×C₆₇	—	4	4	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	—	8	2	no	—
258	C₄×C₆₇	—	8	1	no	—
259	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
260	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
261	C₄×C₆₇	#525	1	2	yes	—
262	#33012	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂×C₆₇	—	4	2	no	—
264	C₂×C₆₇	—	4	4	no	—
265	C₂×C₆₇	—	16	1	no	—
266	C₂×C₆₇	#2849	1	1	yes	characteristic
267	C₂×C₆₇	#3008	1	1	yes	characteristic
268	C₂×C₆₇	#3013	1	1	yes	characteristic
269	C₂×C₆₇	#3119	1	2	yes	—
270	C₂×C₆₇	#3124	1	2	yes	—
271	C₂×#520	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#480	C₂×C₆₇	1	1	yes	characteristic, complemented
273	C₂×#449	C₂×C₆₇	1	1	yes	characteristic
274	#2624	C₂×C₆₇	1	1	yes	characteristic, complemented
275	#1880	C₂×C₆₇	1	2	yes	complemented
276	#1879	C₂×C₆₇	1	2	yes	complemented
277	#1863	C₂×C₆₇	1	2	yes	complemented
278	#1862	C₂×C₆₇	1	2	yes	complemented
279	#1824	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#1818	C₂×C₆₇	1	1	yes	characteristic, complemented
281	#1788	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₆₇	#33012	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
284	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
285	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
287	#520	—	2	4	no	—
288	#520	—	2	4	no	—
289	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
290	C₂×C₄×D₄	—	2	2	no	—
291	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
292	C₂×#117	—	2	1	no	—
293	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
294	C₂²×C₄⋊C₄	—	2	1	no	—
295	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
296	#480	—	2	4	no	—
297	#449	—	2	4	no	—
298	C₂×#107	C₂²×C₆₇	1	2	yes	—
299	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×#103	C₂²×C₆₇	1	2	yes	—
301	C₂×#102	C₂²×C₆₇	1	2	yes	—
302	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#102	C₂²×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
306	#399	—	2	1	no	—
307	#397	—	2	1	no	—
308	#394	—	2	2	no	—
309	#390	C₂²×C₆₇	1	2	yes	—
310	#389	C₂²×C₆₇	1	2	yes	—
311	#380	C₂²×C₆₇	1	1	yes	characteristic
312	#340	C₂²×C₆₇	1	2	yes	—
313	#340	C₂²×C₆₇	1	2	yes	—
314	#336	C₂²×C₆₇	1	2	yes	—
315	#336	C₂²×C₆₇	1	2	yes	—
316	C₂×#41	—	2	4	no	—
317	C₂×#41	—	2	4	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	D₄×C₆₇	1	4	yes	—
320	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
324	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
325	C₂³×C₄	—	2	1	no	—
326	C₂³×C₄	—	2	1	no	—
327	C₂³×C₄	—	2	2	no	—
328	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
329	C₂×SD₁₆	—	2	2	no	—
330	C₂×SD₁₆	—	2	4	no	—
331	C₂×D₈	—	2	2	no	—
332	C₂×D₈	—	2	4	no	—
333	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
336	#121	—	2	2	no	—
337	#121	—	2	2	no	—
338	C₄×Q₈	—	2	4	no	—
339	C₄×Q₈	—	2	4	no	—
340	C₄×D₄	—	2	16	no	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	—	2	4	no	—
343	#117	—	2	8	no	—
344	#117	—	2	2	no	—
345	#117	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	8	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
363	C₂×C₄²	—	2	2	no	—
364	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
365	#107	—	2	4	no	—
366	#103	—	2	2	no	—
367	#103	—	2	4	no	—
368	#102	—	2	2	no	—
369	#102	—	2	4	no	—
370	#102	—	2	4	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	#95	—	2	1	no	—
374	#95	—	2	2	no	—
375	C₂⁴	—	2	1	no	—
376	C₂⁴	—	2	1	no	—
377	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
378	#41	—	2	16	no	—
379	#41	—	2	8	no	—
380	#41	—	2	8	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
384	C₂×Q₈	#41×C₆₇	1	4	yes	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	8	2	no	—
395	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
396	C₂×D₄	#41×C₆₇	1	4	yes	—
397	C₂²×C₄	—	2	8	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	8	2	no	—
408	C₂²×C₄	—	8	2	no	—
409	C₂²×C₄	—	8	2	no	—
410	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
416	SD₁₆	—	4	4	no	—
417	D₈	—	4	4	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₄⋊C₄	—	2	8	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	8	2	no	—
435	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	4	2	no	—
439	C₂²⋊C₄	—	8	2	no	—
440	C₂²⋊C₄	—	8	2	no	—
441	C₂²⋊C₄	—	8	1	no	—
442	C₄²	—	2	8	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	4	2	no	—
452	C₂³	—	4	1	no	—
453	C₂³	—	4	1	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	4	1	no	—
459	C₂³	#120×C₆₇	1	2	yes	—
460	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
461	Q₈	—	2	2	no	—
462	Q₈	—	2	4	no	—
463	Q₈	#123×C₆₇	1	4	yes	—
464	D₄	—	2	16	no	—
465	D₄	—	2	2	no	—
466	D₄	—	2	4	no	—
467	D₄	—	4	1	no	—
468	D₄	—	4	1	no	—
469	D₄	—	4	2	no	—
470	D₄	—	8	4	no	—
471	D₄	#123×C₆₇	1	4	yes	—
472	C₂×C₄	—	2	16	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	8	4	no	—
494	C₂×C₄	—	8	8	no	—
495	C₂×C₄	#120×C₆₇	1	2	yes	—
496	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
498	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
499	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
500	C₈	—	4	1	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	8	1	no	—
514	C₂²	—	8	1	no	—
515	C₂²	—	8	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	—	8	1	no	—
518	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
519	C₂²	#525×C₆₇	1	2	yes	—
520	C₂²	#576×C₆₇	1	2	yes	—
521	C₂²	#577×C₆₇	1	1	yes	characteristic
522	C₂²	#578×C₆₇	1	1	yes	characteristic
523	C₄	—	2	2	no	—
524	C₄	—	2	4	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	4	no	—
528	C₄	—	8	2	no	—
529	C₄	—	8	1	no	—
530	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
531	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
532	C₄	#525×C₆₇	1	2	yes	—
533	C₂	—	4	2	no	—
534	C₂	—	4	4	no	—
535	C₂	—	16	1	no	—
536	C₂	C₆₇×#2849	1	1	yes	characteristic
537	C₂	C₆₇×#3008	1	1	yes	characteristic
538	C₂	C₆₇×#3013	1	1	yes	characteristic
539	C₂	C₆₇×#3119	1	2	yes	—
540	C₂	C₆₇×#3124	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#449×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#520×C₆₇	C₂	semidirect
2. C₂×#480×C₆₇	C₂	semidirect
4. C₆₇×#2624	C₂	semidirect
5. C₆₇×#1880	C₂	semidirect
6. C₆₇×#1879	C₂	semidirect
7. C₆₇×#1863	C₂	semidirect
8. C₆₇×#1862	C₂	semidirect
9. C₆₇×#1824	C₂	semidirect
10. C₆₇×#1818	C₂	semidirect
11. C₆₇×#1788	C₂	semidirect
262. #33012	C₆₇	direct (N × H)
271. C₂×#520	C₂×C₆₇	semidirect
272. C₂×#480	C₂×C₆₇	semidirect
274. #2624	C₂×C₆₇	semidirect
275. #1880	C₂×C₆₇	semidirect
276. #1879	C₂×C₆₇	semidirect
277. #1863	C₂×C₆₇	semidirect
278. #1862	C₂×C₆₇	semidirect
279. #1824	C₂×C₆₇	semidirect
280. #1818	C₂×C₆₇	semidirect
281. #1788	C₂×C₆₇	semidirect
282. C₆₇	#33012	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#520×C₆₇	2
C₂×#480×C₆₇	2
C₂×#449×C₆₇	2
C₆₇×#2624	2
C₆₇×#1880	2
C₆₇×#1879	2
C₆₇×#1863	2
C₆₇×#1862	2
C₆₇×#1824	2
C₆₇×#1818	2
C₆₇×#1788	2
#33012	67

Subgroups — C67×#33012 #30712263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33012 #30712263