Subgroups — C₆₇×#32909 #30712160

Order 17152.

1462 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#464×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#451×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#448×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2850	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1874	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1872	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1805	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
19	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
20	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₆₇	—	2	1	no	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#464×C₆₇	—	2	8	no	—
28	#451×C₆₇	—	2	4	no	—
29	#448×C₆₇	—	2	4	no	—
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#103×C₆₇	C₂²	1	2	yes	—
33	C₂×#103×C₆₇	C₂²	1	2	yes	—
34	C₂×#102×C₆₇	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#391×C₆₇	C₂²	1	1	yes	characteristic
38	#390×C₆₇	C₂²	1	1	yes	characteristic
39	#388×C₆₇	—	2	1	no	—
40	#388×C₆₇	—	2	1	no	—
41	#386×C₆₇	—	2	1	no	—
42	#386×C₆₇	—	2	1	no	—
43	#382×C₆₇	C₂²	1	1	yes	characteristic
44	#340×C₆₇	C₂²	1	2	yes	—
45	#340×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₆₇	—	2	4	no	—
48	C₂²×Q₈×C₆₇	—	2	4	no	—
49	C₂²×Q₈×C₆₇	—	2	1	no	—
50	C₂²×Q₈×C₆₇	—	2	2	no	—
51	C₂²×Q₈×C₆₇	D₄	1	4	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×Q₁₆×C₆₇	—	2	4	no	—
62	C₂×Q₁₆×C₆₇	—	2	4	no	—
63	C₂×Q₁₆×C₆₇	—	2	4	no	—
64	C₂×SD₁₆×C₆₇	—	2	4	no	—
65	C₂×SD₁₆×C₆₇	—	2	4	no	—
66	C₂×SD₁₆×C₆₇	—	2	4	no	—
67	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	#124×C₆₇	—	2	4	no	—
70	C₂²⋊Q₈×C₆₇	—	2	2	no	—
71	C₂²⋊Q₈×C₆₇	—	2	2	no	—
72	#121×C₆₇	—	2	2	no	—
73	#121×C₆₇	—	2	2	no	—
74	C₄×Q₈×C₆₇	—	2	8	no	—
75	C₄×Q₈×C₆₇	—	2	8	no	—
76	C₄×Q₈×C₆₇	—	2	4	no	—
77	C₄×Q₈×C₆₇	—	2	4	no	—
78	C₄×Q₈×C₆₇	—	2	4	no	—
79	C₄×Q₈×C₆₇	D₄	1	8	yes	complemented
80	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	—	2	8	no	—
96	C₂×C₄²×C₆₇	—	2	2	no	—
97	C₂×C₄²×C₆₇	—	2	2	no	—
98	C₂×C₄²×C₆₇	—	2	4	no	—
99	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₆₇	—	2	4	no	—
104	#103×C₆₇	—	2	4	no	—
105	#103×C₆₇	—	2	4	no	—
106	#103×C₆₇	—	2	4	no	—
107	#102×C₆₇	—	2	4	no	—
108	C₂²⋊C₈×C₆₇	—	2	2	no	—
109	C₂²⋊C₈×C₆₇	—	2	2	no	—
110	#95×C₆₇	—	2	1	no	—
111	#95×C₆₇	—	2	1	no	—
112	#95×C₆₇	—	2	1	no	—
113	#95×C₆₇	—	2	1	no	—
114	C₂⁴×C₆₇	—	2	1	no	—
115	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	16	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	16	no	—
120	C₂×Q₈×C₆₇	—	2	4	no	—
121	C₂×Q₈×C₆₇	—	2	2	no	—
122	C₂×Q₈×C₆₇	—	2	2	no	—
123	C₂×Q₈×C₆₇	—	2	2	no	—
124	C₂×Q₈×C₆₇	—	2	2	no	—
125	C₂×Q₈×C₆₇	—	2	1	no	—
126	C₂×Q₈×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	2	no	—
128	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
129	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
130	C₂×Q₈×C₆₇	#41	1	4	yes	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	1	no	—
133	C₂×D₄×C₆₇	—	2	1	no	—
134	C₂×D₄×C₆₇	—	2	2	no	—
135	C₂×D₄×C₆₇	—	4	2	no	—
136	C₂×D₄×C₆₇	—	4	2	no	—
137	C₂×D₄×C₆₇	—	4	2	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	2	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	4	1	no	—
155	C₂²×C₄×C₆₇	—	4	2	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	#41	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	Q₁₆×C₆₇	—	4	8	no	—
166	SD₁₆×C₆₇	—	4	8	no	—
167	C₂×C₈×C₆₇	—	2	2	no	—
168	C₂×C₈×C₆₇	—	2	2	no	—
169	C₂×C₈×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	8	no	—
174	C₄⋊C₄×C₆₇	—	2	4	no	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	2	2	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	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	4	2	no	—
182	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₂²⋊C₄×C₆₇	—	4	2	no	—
184	C₂²⋊C₄×C₆₇	—	4	2	no	—
185	C₂²⋊C₄×C₆₇	—	4	2	no	—
186	C₂²⋊C₄×C₆₇	—	4	2	no	—
187	C₂²⋊C₄×C₆₇	—	4	2	no	—
188	C₂²⋊C₄×C₆₇	—	4	2	no	—
189	C₄²×C₆₇	—	2	8	no	—
190	C₄²×C₆₇	—	2	8	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	C₂×D₄	1	4	yes	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	1	no	—
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	1	no	—
203	C₂³×C₆₇	—	4	1	no	—
204	C₂³×C₆₇	—	4	1	no	—
205	C₂³×C₆₇	—	4	1	no	—
206	C₂³×C₆₇	#121	1	2	yes	—
207	C₂³×C₆₇	#135	1	2	yes	—
208	C₂³×C₆₇	#137	1	2	yes	—
209	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₆₇	—	2	16	no	—
211	Q₈×C₆₇	—	2	4	no	—
212	Q₈×C₆₇	—	2	4	no	—
213	Q₈×C₆₇	—	2	4	no	—
214	Q₈×C₆₇	—	4	2	no	—
215	Q₈×C₆₇	—	4	2	no	—
216	Q₈×C₆₇	#121	1	8	yes	—
217	D₄×C₆₇	—	4	2	no	—
218	D₄×C₆₇	—	4	2	no	—
219	D₄×C₆₇	—	8	4	no	—
220	C₂×C₄×C₆₇	—	2	8	no	—
221	C₂×C₄×C₆₇	—	2	16	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	4	4	no	—
240	C₂×C₄×C₆₇	—	4	8	no	—
241	C₂×C₄×C₆₇	—	4	2	no	—
242	C₂×C₄×C₆₇	—	4	2	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	1	no	—
249	C₂×C₄×C₆₇	—	4	1	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	#120	1	2	yes	—
253	C₂×C₄×C₆₇	#120	1	2	yes	—
254	C₂×C₄×C₆₇	#121	1	2	yes	—
255	C₂×C₄×C₆₇	#137	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
259	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
260	C₈×C₆₇	—	4	2	no	—
261	C₈×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	8	1	no	—
269	C₂²×C₆₇	—	8	1	no	—
270	C₂²×C₆₇	—	8	1	no	—
271	C₂²×C₆₇	—	8	1	no	—
272	C₂²×C₆₇	—	8	1	no	—
273	C₂²×C₆₇	—	8	1	no	—
274	C₂²×C₆₇	—	8	1	no	—
275	C₂²×C₆₇	#464	1	4	yes	complemented
276	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
278	C₂²×C₆₇	#525	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
281	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
282	C₂²×C₆₇	#575	1	1	yes	characteristic
283	C₄×C₆₇	—	2	8	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	2	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	4	1	no	—
289	C₄×C₆₇	—	4	4	no	—
290	C₄×C₆₇	—	8	1	no	—
291	C₄×C₆₇	—	8	1	no	—
292	C₄×C₆₇	—	8	1	no	—
293	C₄×C₆₇	C₂×#121	1	2	yes	—
294	C₄×C₆₇	#525	1	2	yes	—
295	#32909	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	4	no	—
297	C₂×C₆₇	—	16	1	no	—
298	C₂×C₆₇	#2847	1	1	yes	characteristic
299	C₂×C₆₇	#3002	1	1	yes	characteristic
300	C₂×C₆₇	#3004	1	1	yes	characteristic
301	C₂×C₆₇	C₂×#464	1	1	yes	characteristic
302	C₂×C₆₇	#3037	1	1	yes	characteristic
303	C₂×C₆₇	#3108	1	1	yes	characteristic
304	C₂×C₆₇	#3117	1	1	yes	characteristic
305	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₂×#464	C₂×C₆₇	1	2	yes	complemented
307	C₂×#451	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₂×#448	C₂×C₆₇	1	1	yes	characteristic
309	#2850	C₂×C₆₇	1	1	yes	characteristic
310	#1880	C₂×C₆₇	1	2	yes	complemented
311	#1878	C₂×C₆₇	1	2	yes	complemented
312	#1874	C₂×C₆₇	1	1	yes	characteristic
313	#1872	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#1805	C₂×C₆₇	1	2	yes	complemented
315	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₆₇	#32909	1	1	yes	characteristic, direct product factor, Sylow
317	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
319	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
320	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×C₄×Q₈	—	2	8	no	—
324	C₂×C₄×Q₈	—	2	1	no	—
325	C₂×C₄×Q₈	—	2	1	no	—
326	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
327	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
328	C₂²×C₄⋊C₄	—	2	1	no	—
329	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
330	C₂²×C₄²	—	2	1	no	—
331	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
332	#464	—	2	8	no	—
333	#451	—	2	4	no	—
334	#448	—	2	4	no	—
335	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
336	C₂×#103	C₂²×C₆₇	1	2	yes	—
337	C₂×#103	C₂²×C₆₇	1	2	yes	—
338	C₂×#103	C₂²×C₆₇	1	2	yes	—
339	C₂×#102	C₂²×C₆₇	1	2	yes	—
340	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
342	#391	C₂²×C₆₇	1	1	yes	characteristic
343	#390	C₂²×C₆₇	1	1	yes	characteristic
344	#388	—	2	1	no	—
345	#388	—	2	1	no	—
346	#386	—	2	1	no	—
347	#386	—	2	1	no	—
348	#382	C₂²×C₆₇	1	1	yes	characteristic
349	#340	C₂²×C₆₇	1	2	yes	—
350	#340	C₂²×C₆₇	1	2	yes	—
351	#336	C₂²×C₆₇	1	2	yes	—
352	C₂²×Q₈	—	2	4	no	—
353	C₂²×Q₈	—	2	4	no	—
354	C₂²×Q₈	—	2	1	no	—
355	C₂²×Q₈	—	2	2	no	—
356	C₂²×Q₈	D₄×C₆₇	1	4	yes	—
357	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
358	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
361	C₂³×C₄	—	2	1	no	—
362	C₂³×C₄	—	2	1	no	—
363	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
364	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
365	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
366	C₂×Q₁₆	—	2	4	no	—
367	C₂×Q₁₆	—	2	4	no	—
368	C₂×Q₁₆	—	2	4	no	—
369	C₂×SD₁₆	—	2	4	no	—
370	C₂×SD₁₆	—	2	4	no	—
371	C₂×SD₁₆	—	2	4	no	—
372	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
373	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
374	#124	—	2	4	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	#121	—	2	2	no	—
378	#121	—	2	2	no	—
379	C₄×Q₈	—	2	8	no	—
380	C₄×Q₈	—	2	8	no	—
381	C₄×Q₈	—	2	4	no	—
382	C₄×Q₈	—	2	4	no	—
383	C₄×Q₈	—	2	4	no	—
384	C₄×Q₈	D₄×C₆₇	1	8	yes	complemented
385	C₂×C₄⋊C₄	—	2	8	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	4	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₄²	—	2	8	no	—
401	C₂×C₄²	—	2	2	no	—
402	C₂×C₄²	—	2	2	no	—
403	C₂×C₄²	—	2	4	no	—
404	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
408	C₄⋊C₈	—	2	4	no	—
409	#103	—	2	4	no	—
410	#103	—	2	4	no	—
411	#103	—	2	4	no	—
412	#102	—	2	4	no	—
413	C₂²⋊C₈	—	2	2	no	—
414	C₂²⋊C₈	—	2	2	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	C₂⁴	—	2	1	no	—
420	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	2	16	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	2	16	no	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
434	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
435	C₂×Q₈	#41×C₆₇	1	4	yes	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	2	no	—
442	C₂×D₄	—	4	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₂²×C₄	#41×C₆₇	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
470	Q₁₆	—	4	8	no	—
471	SD₁₆	—	4	8	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	2	no	—
478	C₄⋊C₄	—	2	8	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	4	2	no	—
487	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
488	C₂²⋊C₄	—	4	2	no	—
489	C₂²⋊C₄	—	4	2	no	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₂²⋊C₄	—	4	2	no	—
494	C₄²	—	2	8	no	—
495	C₄²	—	2	8	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	C₂×D₄×C₆₇	1	4	yes	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	4	1	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	#121×C₆₇	1	2	yes	—
512	C₂³	#135×C₆₇	1	2	yes	—
513	C₂³	#137×C₆₇	1	2	yes	—
514	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
515	Q₈	—	2	16	no	—
516	Q₈	—	2	4	no	—
517	Q₈	—	2	4	no	—
518	Q₈	—	2	4	no	—
519	Q₈	—	4	2	no	—
520	Q₈	—	4	2	no	—
521	Q₈	#121×C₆₇	1	8	yes	—
522	D₄	—	4	2	no	—
523	D₄	—	4	2	no	—
524	D₄	—	8	4	no	—
525	C₂×C₄	—	2	16	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	#120×C₆₇	1	2	yes	—
558	C₂×C₄	#120×C₆₇	1	2	yes	—
559	C₂×C₄	#121×C₆₇	1	2	yes	—
560	C₂×C₄	#137×C₆₇	1	2	yes	—
561	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
563	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
564	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
565	C₈	—	4	2	no	—
566	C₈	—	4	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	8	1	no	—
574	C₂²	—	8	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	—	8	1	no	—
579	C₂²	—	8	1	no	—
580	C₂²	#464×C₆₇	1	4	yes	complemented
581	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
582	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
583	C₂²	#525×C₆₇	1	1	yes	characteristic
584	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
585	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
586	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
587	C₂²	#575×C₆₇	1	1	yes	characteristic
588	C₄	—	2	8	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	4	1	no	—
594	C₄	—	4	4	no	—
595	C₄	—	8	1	no	—
596	C₄	—	8	1	no	—
597	C₄	—	8	1	no	—
598	C₄	C₂×#121×C₆₇	1	2	yes	—
599	C₄	#525×C₆₇	1	2	yes	—
600	C₂	—	2	4	no	—
601	C₂	—	16	1	no	—
602	C₂	C₆₇×#2847	1	1	yes	characteristic
603	C₂	C₆₇×#3002	1	1	yes	characteristic
604	C₂	C₆₇×#3004	1	1	yes	characteristic
605	C₂	C₂×#464×C₆₇	1	1	yes	characteristic
606	C₂	C₆₇×#3037	1	1	yes	characteristic
607	C₂	C₆₇×#3108	1	1	yes	characteristic
608	C₂	C₆₇×#3117	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
2. C₂×#464×C₆₇	C₂	semidirect
3. C₂×#451×C₆₇	C₂	semidirect
6. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1878	C₂	semidirect
9. C₆₇×#1872	C₂	semidirect
10. C₆₇×#1805	C₂	semidirect
11. C₆₇×#1797	C₂	semidirect
79. C₄×Q₈×C₆₇	D₄	semidirect
275. C₂²×C₆₇	#464	semidirect
295. #32909	C₆₇	direct (N × H)
305. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
306. C₂×#464	C₂×C₆₇	semidirect
307. C₂×#451	C₂×C₆₇	semidirect
310. #1880	C₂×C₆₇	semidirect
311. #1878	C₂×C₆₇	semidirect
313. #1872	C₂×C₆₇	semidirect
314. #1805	C₂×C₆₇	semidirect
315. #1797	C₂×C₆₇	semidirect
316. C₆₇	#32909	direct (N × H)
384. C₄×Q₈	D₄×C₆₇	semidirect
580. C₂²	#464×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#464×C₆₇	2
C₂×#451×C₆₇	2
C₂×#448×C₆₇	2
C₆₇×#2850	2
C₆₇×#1880	2
C₆₇×#1878	2
C₆₇×#1874	2
C₆₇×#1872	2
C₆₇×#1805	2
C₆₇×#1797	2
#32909	67

Subgroups — C67×#32909 #30712160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32909 #30712160