Subgroups — C₆₇×#32907 #30712158

Order 17152.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#452×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1873	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	C₄×#41×C₆₇	—	2	4	no	—
18	C₄×#41×C₆₇	—	2	4	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	—	2	1	no	—
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₆₇	—	2	1	no	—
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#462×C₆₇	—	2	4	no	—
30	#461×C₆₇	—	2	4	no	—
31	#452×C₆₇	—	2	8	no	—
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₇	C₂²	1	2	yes	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×#103×C₆₇	C₂²	1	2	yes	—
37	C₂×#102×C₆₇	C₂²	1	2	yes	—
38	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	—	2	1	no	—
41	#389×C₆₇	—	2	1	no	—
42	#384×C₆₇	C₂²	1	1	yes	characteristic
43	#380×C₆₇	—	2	1	no	—
44	#380×C₆₇	—	2	1	no	—
45	#340×C₆₇	C₂²	1	2	yes	—
46	#340×C₆₇	C₂²	1	2	yes	—
47	#336×C₆₇	C₂²	1	2	yes	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	D₄	1	4	yes	—
54	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	—	2	1	no	—
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₆₇	—	2	4	no	—
65	C₂×Q₁₆×C₆₇	—	2	4	no	—
66	C₂×Q₁₆×C₆₇	—	2	4	no	—
67	C₂×SD₁₆×C₆₇	—	2	4	no	—
68	C₂×SD₁₆×C₆₇	—	2	4	no	—
69	C₂×SD₁₆×C₆₇	—	2	4	no	—
70	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₄⋊Q₈×C₆₇	—	2	2	no	—
74	C₂²⋊Q₈×C₆₇	—	2	4	no	—
75	C₂²⋊Q₈×C₆₇	—	2	4	no	—
76	C₄×Q₈×C₆₇	—	2	2	no	—
77	C₄×Q₈×C₆₇	—	2	2	no	—
78	C₄×Q₈×C₆₇	—	2	2	no	—
79	C₄×Q₈×C₆₇	D₄	1	4	yes	—
80	C₄×D₄×C₆₇	—	2	8	no	—
81	C₄×D₄×C₆₇	—	2	8	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	D₄	1	4	yes	—
86	#117×C₆₇	—	2	8	no	—
87	#117×C₆₇	—	2	2	no	—
88	#117×C₆₇	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
103	C₂×C₄²×C₆₇	—	2	8	no	—
104	C₂×C₄²×C₆₇	—	2	1	no	—
105	C₂×C₄²×C₆₇	—	2	1	no	—
106	C₂×C₄²×C₆₇	—	2	1	no	—
107	C₂×C₄²×C₆₇	—	2	2	no	—
108	C₂×C₄²×C₆₇	—	2	2	no	—
109	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₆₇	—	2	2	no	—
114	C₄⋊C₈×C₆₇	—	2	2	no	—
115	#103×C₆₇	—	2	4	no	—
116	#103×C₆₇	—	2	4	no	—
117	#103×C₆₇	—	2	4	no	—
118	#102×C₆₇	—	2	4	no	—
119	C₂²⋊C₈×C₆₇	—	2	4	no	—
120	#95×C₆₇	—	2	2	no	—
121	#95×C₆₇	—	2	2	no	—
122	C₂⁴×C₆₇	—	2	1	no	—
123	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
124	#41×C₆₇	—	2	8	no	—
125	#41×C₆₇	—	2	8	no	—
126	#41×C₆₇	—	2	8	no	—
127	#41×C₆₇	—	2	8	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	1	no	—
130	C₂×Q₈×C₆₇	—	2	1	no	—
131	C₂×Q₈×C₆₇	—	2	2	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×Q₈×C₆₇	#41	1	2	yes	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	1	no	—
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	—	2	1	no	—
146	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₆₇	#41	1	2	yes	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	2	no	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	4	1	no	—
169	C₂²×C₄×C₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	—	4	2	no	—
171	C₂²×C₄×C₆₇	—	4	2	no	—
172	C₂²×C₄×C₆₇	—	4	2	no	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	#41	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₆₇	—	4	8	no	—
182	SD₁₆×C₆₇	—	4	8	no	—
183	C₂×C₈×C₆₇	—	2	2	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	—	2	2	no	—
186	C₂×C₈×C₆₇	—	2	2	no	—
187	C₂×C₈×C₆₇	—	2	2	no	—
188	C₂×C₈×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	1	no	—
192	C₄⋊C₄×C₆₇	—	2	1	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	4	2	no	—
196	C₄⋊C₄×C₆₇	—	4	2	no	—
197	C₄⋊C₄×C₆₇	—	4	2	no	—
198	C₄⋊C₄×C₆₇	—	4	4	no	—
199	C₄⋊C₄×C₆₇	—	4	2	no	—
200	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	4	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	—	2	2	no	—
209	C₄²×C₆₇	—	2	4	no	—
210	C₄²×C₆₇	—	2	2	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₄²×C₆₇	—	2	1	no	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	C₂×D₄	1	2	yes	—
215	C₂³×C₆₇	—	2	1	no	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	1	no	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	2	1	no	—
221	C₂³×C₆₇	—	2	1	no	—
222	C₂³×C₆₇	—	2	1	no	—
223	C₂³×C₆₇	—	2	1	no	—
224	C₂³×C₆₇	—	4	1	no	—
225	C₂³×C₆₇	—	4	1	no	—
226	C₂³×C₆₇	—	4	1	no	—
227	C₂³×C₆₇	—	4	1	no	—
228	C₂³×C₆₇	#121	1	2	yes	—
229	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₆₇	—	2	2	no	—
231	Q₈×C₆₇	—	2	2	no	—
232	Q₈×C₆₇	—	2	2	no	—
233	Q₈×C₆₇	—	4	2	no	—
234	Q₈×C₆₇	—	4	2	no	—
235	Q₈×C₆₇	—	4	2	no	—
236	Q₈×C₆₇	—	4	2	no	—
237	Q₈×C₆₇	#121	1	4	yes	—
238	D₄×C₆₇	—	2	16	no	—
239	D₄×C₆₇	—	2	2	no	—
240	D₄×C₆₇	—	2	2	no	—
241	D₄×C₆₇	—	2	2	no	—
242	D₄×C₆₇	#121	1	4	yes	—
243	C₂×C₄×C₆₇	—	2	16	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	4	4	no	—
263	C₂×C₄×C₆₇	—	4	8	no	—
264	C₂×C₄×C₆₇	—	4	1	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	1	no	—
267	C₂×C₄×C₆₇	—	4	1	no	—
268	C₂×C₄×C₆₇	—	4	4	no	—
269	C₂×C₄×C₆₇	—	4	8	no	—
270	C₂×C₄×C₆₇	—	4	2	no	—
271	C₂×C₄×C₆₇	—	4	2	no	—
272	C₂×C₄×C₆₇	—	4	2	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	2	no	—
275	C₂×C₄×C₆₇	—	4	2	no	—
276	C₂×C₄×C₆₇	#120	1	2	yes	—
277	C₂×C₄×C₆₇	#120	1	2	yes	—
278	C₂×C₄×C₆₇	#121	1	2	yes	—
279	C₂×C₄×C₆₇	#135	1	2	yes	—
280	C₂×C₄×C₆₇	#137	1	2	yes	—
281	C₂×C₄×C₆₇	#137	1	2	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
288	C₈×C₆₇	—	4	2	no	—
289	C₈×C₆₇	—	4	2	no	—
290	C₂²×C₆₇	—	2	2	no	—
291	C₂²×C₆₇	—	2	4	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	4	1	no	—
295	C₂²×C₆₇	—	4	2	no	—
296	C₂²×C₆₇	—	4	1	no	—
297	C₂²×C₆₇	—	4	2	no	—
298	C₂²×C₆₇	—	4	1	no	—
299	C₂²×C₆₇	—	4	2	no	—
300	C₂²×C₆₇	—	4	1	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	2	no	—
304	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
305	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
306	C₂²×C₆₇	#525	1	1	yes	characteristic
307	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
308	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
310	C₂²×C₆₇	#575	1	1	yes	characteristic
311	C₄×C₆₇	—	2	2	no	—
312	C₄×C₆₇	—	2	4	no	—
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	4	1	no	—
316	C₄×C₆₇	—	4	2	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	8	2	no	—
319	C₄×C₆₇	—	8	2	no	—
320	C₄×C₆₇	#452	1	4	yes	complemented
321	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
322	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
323	C₄×C₆₇	#525	1	1	yes	characteristic
324	C₄×C₆₇	#525	1	1	yes	characteristic
325	#32907	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
326	C₂×C₆₇	—	4	2	no	—
327	C₂×C₆₇	—	4	4	no	—
328	C₂×C₆₇	#2847	1	1	yes	characteristic
329	C₂×C₆₇	C₂×#452	1	1	yes	characteristic
330	C₂×C₆₇	#3007	1	1	yes	characteristic
331	C₂×C₆₇	#3035	1	1	yes	characteristic
332	C₂×C₆₇	#3039	1	1	yes	characteristic
333	C₂×C₆₇	#3107	1	1	yes	characteristic
334	C₂×C₆₇	#3116	1	1	yes	characteristic
335	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
336	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
337	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
338	C₂×#452	C₂×C₆₇	1	2	yes	—
339	#2857	C₂×C₆₇	1	1	yes	characteristic, complemented
340	#1878	C₂×C₆₇	1	2	yes	complemented
341	#1877	C₂×C₆₇	1	2	yes	complemented
342	#1873	C₂×C₆₇	1	2	yes	complemented
343	#1803	C₂×C₆₇	1	1	yes	characteristic, complemented
344	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
346	C₆₇	#32907	1	1	yes	characteristic, direct product factor, Sylow
347	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
348	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
349	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
350	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
352	C₄×#41	—	2	4	no	—
353	C₄×#41	—	2	4	no	—
354	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
356	C₂×C₄×D₄	—	2	1	no	—
357	C₂×C₄×D₄	—	2	1	no	—
358	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×#117	—	2	1	no	—
361	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
362	C₂²×C₄²	—	2	1	no	—
363	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
364	#462	—	2	4	no	—
365	#461	—	2	4	no	—
366	#452	—	2	8	no	—
367	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
368	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
369	C₂×#103	C₂²×C₆₇	1	2	yes	—
370	C₂×#103	C₂²×C₆₇	1	2	yes	—
371	C₂×#103	C₂²×C₆₇	1	2	yes	—
372	C₂×#102	C₂²×C₆₇	1	2	yes	—
373	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
374	#391	C₂²×C₆₇	1	2	yes	—
375	#389	—	2	1	no	—
376	#389	—	2	1	no	—
377	#384	C₂²×C₆₇	1	1	yes	characteristic
378	#380	—	2	1	no	—
379	#380	—	2	1	no	—
380	#340	C₂²×C₆₇	1	2	yes	—
381	#340	C₂²×C₆₇	1	2	yes	—
382	#336	C₂²×C₆₇	1	2	yes	—
383	C₂×#41	—	2	2	no	—
384	C₂×#41	—	2	2	no	—
385	C₂×#41	—	2	2	no	—
386	C₂×#41	—	2	2	no	—
387	C₂×#41	—	2	2	no	—
388	C₂×#41	D₄×C₆₇	1	4	yes	—
389	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
390	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
391	C₂²×D₄	—	2	1	no	—
392	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂³×C₄	—	2	1	no	—
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	—	2	1	no	—
396	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
397	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
399	C₂×Q₁₆	—	2	4	no	—
400	C₂×Q₁₆	—	2	4	no	—
401	C₂×Q₁₆	—	2	4	no	—
402	C₂×SD₁₆	—	2	4	no	—
403	C₂×SD₁₆	—	2	4	no	—
404	C₂×SD₁₆	—	2	4	no	—
405	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
406	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
407	C₄⋊Q₈	—	2	2	no	—
408	C₄⋊Q₈	—	2	2	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	—	2	2	no	—
413	C₄×Q₈	—	2	2	no	—
414	C₄×Q₈	D₄×C₆₇	1	4	yes	—
415	C₄×D₄	—	2	8	no	—
416	C₄×D₄	—	2	8	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	D₄×C₆₇	1	4	yes	—
421	#117	—	2	8	no	—
422	#117	—	2	2	no	—
423	#117	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	2	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	2	no	—
438	C₂×C₄²	—	2	8	no	—
439	C₂×C₄²	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	—	2	2	no	—
443	C₂×C₄²	—	2	2	no	—
444	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
445	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
448	C₄⋊C₈	—	2	2	no	—
449	C₄⋊C₈	—	2	2	no	—
450	#103	—	2	4	no	—
451	#103	—	2	4	no	—
452	#103	—	2	4	no	—
453	#102	—	2	4	no	—
454	C₂²⋊C₈	—	2	4	no	—
455	#95	—	2	2	no	—
456	#95	—	2	2	no	—
457	C₂⁴	—	2	1	no	—
458	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
459	#41	—	2	8	no	—
460	#41	—	2	8	no	—
461	#41	—	2	8	no	—
462	#41	—	2	8	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	2	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	2	no	—
470	C₂×Q₈	—	2	2	no	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
473	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
474	C₂×Q₈	#41×C₆₇	1	2	yes	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
482	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
483	C₂×D₄	#41×C₆₇	1	2	yes	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	4	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	4	1	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	—	4	2	no	—
508	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
511	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
512	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
513	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
514	C₂²×C₄	#41×C₆₇	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
516	Q₁₆	—	4	8	no	—
517	SD₁₆	—	4	8	no	—
518	C₂×C₈	—	2	2	no	—
519	C₂×C₈	—	2	2	no	—
520	C₂×C₈	—	2	2	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	2	no	—
523	C₂×C₈	—	2	2	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	2	no	—
533	C₄⋊C₄	—	4	4	no	—
534	C₄⋊C₄	—	4	2	no	—
535	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
536	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	4	4	no	—
541	C₄²	—	2	4	no	—
542	C₄²	—	2	4	no	—
543	C₄²	—	2	2	no	—
544	C₄²	—	2	4	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	C₂×D₄×C₆₇	1	2	yes	—
549	C₄²	C₂×D₄×C₆₇	1	2	yes	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	—	4	1	no	—
562	C₂³	—	4	1	no	—
563	C₂³	#121×C₆₇	1	2	yes	—
564	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	4	2	no	—
570	Q₈	—	4	2	no	—
571	Q₈	—	4	2	no	—
572	Q₈	#121×C₆₇	1	4	yes	—
573	D₄	—	2	16	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	2	no	—
577	D₄	#121×C₆₇	1	4	yes	—
578	C₂×C₄	—	2	16	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	4	4	no	—
598	C₂×C₄	—	4	8	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	—	4	8	no	—
605	C₂×C₄	—	4	2	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	#120×C₆₇	1	2	yes	—
612	C₂×C₄	#120×C₆₇	1	2	yes	—
613	C₂×C₄	#121×C₆₇	1	2	yes	—
614	C₂×C₄	#135×C₆₇	1	2	yes	—
615	C₂×C₄	#137×C₆₇	1	2	yes	—
616	C₂×C₄	#137×C₆₇	1	2	yes	—
617	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
618	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
620	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
621	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
622	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
623	C₈	—	4	2	no	—
624	C₈	—	4	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	4	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	2	no	—
639	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
640	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
641	C₂²	#525×C₆₇	1	1	yes	characteristic
642	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
643	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
644	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
645	C₂²	#575×C₆₇	1	1	yes	characteristic
646	C₄	—	2	2	no	—
647	C₄	—	2	4	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	8	2	no	—
654	C₄	—	8	2	no	—
655	C₄	#452×C₆₇	1	4	yes	complemented
656	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
657	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
658	C₄	#525×C₆₇	1	1	yes	characteristic
659	C₄	#525×C₆₇	1	1	yes	characteristic
660	C₂	—	4	2	no	—
661	C₂	—	4	4	no	—
662	C₂	C₆₇×#2847	1	1	yes	characteristic
663	C₂	C₂×#452×C₆₇	1	1	yes	characteristic
664	C₂	C₆₇×#3007	1	1	yes	characteristic
665	C₂	C₆₇×#3035	1	1	yes	characteristic
666	C₂	C₆₇×#3039	1	1	yes	characteristic
667	C₂	C₆₇×#3107	1	1	yes	characteristic
668	C₂	C₆₇×#3116	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×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
2. C₂×#462×C₆₇	C₂	semidirect
3. C₂×#461×C₆₇	C₂	semidirect
5. C₆₇×#2857	C₂	semidirect
6. C₆₇×#1878	C₂	semidirect
7. C₆₇×#1877	C₂	semidirect
8. C₆₇×#1873	C₂	semidirect
9. C₆₇×#1803	C₂	semidirect
10. C₆₇×#1802	C₂	semidirect
11. C₆₇×#1797	C₂	semidirect
320. C₄×C₆₇	#452	semidirect
325. #32907	C₆₇	direct (N × H)
336. C₂×#462	C₂×C₆₇	semidirect
337. C₂×#461	C₂×C₆₇	semidirect
339. #2857	C₂×C₆₇	semidirect
340. #1878	C₂×C₆₇	semidirect
341. #1877	C₂×C₆₇	semidirect
342. #1873	C₂×C₆₇	semidirect
343. #1803	C₂×C₆₇	semidirect
344. #1802	C₂×C₆₇	semidirect
345. #1797	C₂×C₆₇	semidirect
346. C₆₇	#32907	direct (N × H)
655. C₄	#452×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂×#462×C₆₇	2
C₂×#461×C₆₇	2
C₂×#452×C₆₇	2
C₆₇×#2857	2
C₆₇×#1878	2
C₆₇×#1877	2
C₆₇×#1873	2
C₆₇×#1803	2
C₆₇×#1802	2
C₆₇×#1797	2
#32907	67

Subgroups — C67×#32907 #30712158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32907 #30712158