Subgroups — C₆₇×#32965 #30712216

Order 17152.

1282 conjugacy classes of subgroups.

The subgroup lattice has 588 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
2	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2628	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1853	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1804	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1788	C₂	1	2	yes	maximal
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#520×C₆₇	—	2	4	no	—
15	#520×C₆₇	—	2	4	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	—	2	1	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	—	2	1	no	—
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	#476×C₆₇	—	2	4	no	—
27	#475×C₆₇	—	2	4	no	—
28	C₂×#106×C₆₇	C₂²	1	2	yes	—
29	C₂×#106×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
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×#102×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
37	#400×C₆₇	—	2	1	no	—
38	#399×C₆₇	—	2	1	no	—
39	#395×C₆₇	—	2	1	no	—
40	#393×C₆₇	—	2	1	no	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#382×C₆₇	C₂²	1	1	yes	characteristic
43	#380×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#340×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	C₂×#41×C₆₇	—	2	2	no	—
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	Q₈	1	4	yes	complemented
53	C₂²×Q₈×C₆₇	—	2	1	no	—
54	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₆₇	—	2	2	no	—
65	C₄⋊Q₈×C₆₇	—	2	2	no	—
66	C₄×Q₈×C₆₇	—	2	2	no	—
67	C₄×Q₈×C₆₇	—	2	2	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	D₄	1	4	yes	—
70	C₄×D₄×C₆₇	—	2	8	no	—
71	C₄×D₄×C₆₇	—	2	8	no	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	—	2	2	no	—
74	C₄×D₄×C₆₇	—	2	2	no	—
75	C₄×D₄×C₆₇	D₄	1	4	yes	—
76	#117×C₆₇	—	2	8	no	—
77	#117×C₆₇	—	2	2	no	—
78	#117×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	—	2	1	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#106×C₆₇	—	2	4	no	—
103	#106×C₆₇	—	2	4	no	—
104	C₄⋊C₈×C₆₇	—	2	2	no	—
105	C₄⋊C₈×C₆₇	—	2	2	no	—
106	#103×C₆₇	—	2	4	no	—
107	#103×C₆₇	—	2	4	no	—
108	#102×C₆₇	—	2	4	no	—
109	#102×C₆₇	—	2	4	no	—
110	C₂²⋊C₈×C₆₇	—	2	4	no	—
111	#95×C₆₇	—	2	2	no	—
112	#95×C₆₇	—	2	1	no	—
113	#95×C₆₇	—	2	1	no	—
114	#95×C₆₇	—	2	2	no	—
115	C₂⁴×C₆₇	—	2	1	no	—
116	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
117	#41×C₆₇	—	2	8	no	—
118	#41×C₆₇	—	2	8	no	—
119	#41×C₆₇	—	2	8	no	—
120	#41×C₆₇	—	2	8	no	—
121	C₂×Q₈×C₆₇	—	2	1	no	—
122	C₂×Q₈×C₆₇	—	2	1	no	—
123	C₂×Q₈×C₆₇	—	2	1	no	—
124	C₂×Q₈×C₆₇	—	4	2	no	—
125	C₂×Q₈×C₆₇	—	8	2	no	—
126	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
128	C₂×Q₈×C₆₇	#41	1	2	yes	—
129	C₂×D₄×C₆₇	—	2	4	no	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	2	1	no	—
133	C₂×D₄×C₆₇	—	2	1	no	—
134	C₂×D₄×C₆₇	—	2	1	no	—
135	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
137	C₂×D₄×C₆₇	#41	1	2	yes	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	4	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	8	2	no	—
154	C₂²×C₄×C₆₇	—	8	2	no	—
155	C₂²×C₄×C₆₇	—	8	2	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
162	C₂²×C₄×C₆₇	#41	1	2	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	—	2	2	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	4	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	4	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	2	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	4	2	no	—
182	C₄⋊C₄×C₆₇	—	8	2	no	—
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	—	8	2	no	—
190	C₄²×C₆₇	—	2	4	no	—
191	C₄²×C₆₇	—	2	4	no	—
192	C₄²×C₆₇	—	2	1	no	—
193	C₄²×C₆₇	—	2	1	no	—
194	C₄²×C₆₇	C₂×D₄	1	2	yes	—
195	C₄²×C₆₇	C₂×D₄	1	2	yes	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	1	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	—	2	2	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	4	2	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
206	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
207	Q₈×C₆₇	—	2	2	no	—
208	Q₈×C₆₇	—	2	2	no	—
209	Q₈×C₆₇	—	2	2	no	—
210	Q₈×C₆₇	—	8	4	no	—
211	Q₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
212	D₄×C₆₇	—	2	16	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	2	2	no	—
215	D₄×C₆₇	—	2	2	no	—
216	D₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
217	C₂×C₄×C₆₇	—	2	16	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	—	4	2	no	—
242	C₂×C₄×C₆₇	—	4	2	no	—
243	C₂×C₄×C₆₇	—	8	4	no	—
244	C₂×C₄×C₆₇	—	8	4	no	—
245	C₂×C₄×C₆₇	—	8	4	no	—
246	C₂×C₄×C₆₇	#120	1	2	yes	—
247	C₂×C₄×C₆₇	#120	1	2	yes	—
248	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
249	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
255	C₈×C₆₇	—	4	2	no	—
256	C₈×C₆₇	—	4	2	no	—
257	C₂²×C₆₇	—	2	4	no	—
258	C₂²×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	4	4	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	4	no	—
263	C₂²×C₆₇	—	4	1	no	—
264	C₂²×C₆₇	—	4	2	no	—
265	C₂²×C₆₇	—	4	1	no	—
266	C₂²×C₆₇	—	4	2	no	—
267	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
268	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂²×C₆₇	#525	1	1	yes	characteristic
270	C₂²×C₆₇	#577	1	2	yes	—
271	C₂²×C₆₇	#577	1	2	yes	—
272	C₄×C₆₇	—	2	2	no	—
273	C₄×C₆₇	—	2	4	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	4	1	no	—
277	C₄×C₆₇	—	4	2	no	—
278	C₄×C₆₇	—	4	2	no	—
279	C₄×C₆₇	—	4	1	no	—
280	C₄×C₆₇	—	8	2	no	—
281	C₄×C₆₇	—	8	2	no	—
282	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
283	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
284	C₄×C₆₇	#525	1	1	yes	characteristic
285	C₄×C₆₇	#525	1	1	yes	characteristic
286	#32965	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₆₇	—	4	2	no	—
288	C₂×C₆₇	—	4	4	no	—
289	C₂×C₆₇	#2848	1	1	yes	characteristic
290	C₂×C₆₇	#3009	1	1	yes	characteristic
291	C₂×C₆₇	#3010	1	1	yes	characteristic
292	C₂×C₆₇	#3084	1	2	yes	—
293	C₂×C₆₇	#3125	1	2	yes	—
294	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
295	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
296	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
297	#2628	C₂×C₆₇	1	1	yes	characteristic, complemented
298	#1865	C₂×C₆₇	1	2	yes	complemented
299	#1864	C₂×C₆₇	1	2	yes	complemented
300	#1853	C₂×C₆₇	1	2	yes	complemented
301	#1805	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1804	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1788	C₂×C₆₇	1	2	yes	—
305	C₆₇	#32965	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
308	#520	—	2	4	no	—
309	#520	—	2	4	no	—
310	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₄×D₄	—	2	1	no	—
313	C₂×C₄×D₄	—	2	1	no	—
314	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#117	—	2	1	no	—
317	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	—	2	1	no	—
319	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
320	#476	—	2	4	no	—
321	#475	—	2	4	no	—
322	C₂×#106	C₂²×C₆₇	1	2	yes	—
323	C₂×#106	C₂²×C₆₇	1	2	yes	—
324	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×#103	C₂²×C₆₇	1	2	yes	—
327	C₂×#103	C₂²×C₆₇	1	2	yes	—
328	C₂×#102	C₂²×C₆₇	1	2	yes	—
329	C₂×#102	C₂²×C₆₇	1	2	yes	—
330	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
331	#400	—	2	1	no	—
332	#399	—	2	1	no	—
333	#395	—	2	1	no	—
334	#393	—	2	1	no	—
335	#389	C₂²×C₆₇	1	2	yes	—
336	#382	C₂²×C₆₇	1	1	yes	characteristic
337	#380	C₂²×C₆₇	1	2	yes	—
338	#340	C₂²×C₆₇	1	2	yes	—
339	#340	C₂²×C₆₇	1	2	yes	—
340	#336	C₂²×C₆₇	1	2	yes	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	2	2	no	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	Q₈×C₆₇	1	4	yes	complemented
347	C₂²×Q₈	—	2	1	no	—
348	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
349	C₂²×D₄	—	2	1	no	—
350	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	—	2	1	no	—
353	C₂³×C₄	—	2	1	no	—
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
357	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
358	C₄⋊Q₈	—	2	2	no	—
359	C₄⋊Q₈	—	2	2	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	D₄×C₆₇	1	4	yes	—
364	C₄×D₄	—	2	8	no	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	D₄×C₆₇	1	4	yes	—
370	#117	—	2	8	no	—
371	#117	—	2	2	no	—
372	#117	—	2	2	no	—
373	C₂×C₄⋊C₄	—	2	8	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	#106	—	2	4	no	—
397	#106	—	2	4	no	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	#103	—	2	4	no	—
401	#103	—	2	4	no	—
402	#102	—	2	4	no	—
403	#102	—	2	4	no	—
404	C₂²⋊C₈	—	2	4	no	—
405	#95	—	2	2	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	2	no	—
409	C₂⁴	—	2	1	no	—
410	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
411	#41	—	2	8	no	—
412	#41	—	2	8	no	—
413	#41	—	2	8	no	—
414	#41	—	2	8	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	4	2	no	—
419	C₂×Q₈	—	8	2	no	—
420	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
421	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
422	C₂×Q₈	#41×C₆₇	1	2	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
430	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
431	C₂×D₄	#41×C₆₇	1	2	yes	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	8	2	no	—
448	C₂²×C₄	—	8	2	no	—
449	C₂²×C₄	—	8	2	no	—
450	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
456	C₂²×C₄	#41×C₆₇	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	4	2	no	—
476	C₄⋊C₄	—	8	2	no	—
477	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	8	2	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	4	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	C₂×D₄×C₆₇	1	2	yes	—
489	C₄²	C₂×D₄×C₆₇	1	2	yes	—
490	C₂³	—	2	1	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
500	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
501	Q₈	—	2	2	no	—
502	Q₈	—	2	2	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	8	4	no	—
505	Q₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
506	D₄	—	2	16	no	—
507	D₄	—	2	2	no	—
508	D₄	—	2	2	no	—
509	D₄	—	2	2	no	—
510	D₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
511	C₂×C₄	—	2	16	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	8	4	no	—
538	C₂×C₄	—	8	4	no	—
539	C₂×C₄	—	8	4	no	—
540	C₂×C₄	#120×C₆₇	1	2	yes	—
541	C₂×C₄	#120×C₆₇	1	2	yes	—
542	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
543	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
545	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
546	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
547	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
548	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
549	C₈	—	4	2	no	—
550	C₈	—	4	2	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
562	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
563	C₂²	#525×C₆₇	1	1	yes	characteristic
564	C₂²	#577×C₆₇	1	2	yes	—
565	C₂²	#577×C₆₇	1	2	yes	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	4	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	2	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	1	no	—
574	C₄	—	8	2	no	—
575	C₄	—	8	2	no	—
576	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
577	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
578	C₄	#525×C₆₇	1	1	yes	characteristic
579	C₄	#525×C₆₇	1	1	yes	characteristic
580	C₂	—	4	2	no	—
581	C₂	—	4	4	no	—
582	C₂	C₆₇×#2848	1	1	yes	characteristic
583	C₂	C₆₇×#3009	1	1	yes	characteristic
584	C₂	C₆₇×#3010	1	1	yes	characteristic
585	C₂	C₆₇×#3084	1	2	yes	—
586	C₂	C₆₇×#3125	1	2	yes	—

	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 ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#475×C₆₇	C₂	semidirect
3. C₂×#476×C₆₇	C₂	semidirect
4. C₆₇×#2628	C₂	semidirect
5. C₆₇×#1865	C₂	semidirect
6. C₆₇×#1864	C₂	semidirect
7. C₆₇×#1853	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
9. C₆₇×#1804	C₂	semidirect
10. C₆₇×#1798	C₂	semidirect
52. C₂×#41×C₆₇	Q₈	semidirect
286. #32965	C₆₇	direct (N × H)
295. C₂×#475	C₂×C₆₇	semidirect
296. C₂×#476	C₂×C₆₇	semidirect
297. #2628	C₂×C₆₇	semidirect
298. #1865	C₂×C₆₇	semidirect
299. #1864	C₂×C₆₇	semidirect
300. #1853	C₂×C₆₇	semidirect
301. #1805	C₂×C₆₇	semidirect
302. #1804	C₂×C₆₇	semidirect
303. #1798	C₂×C₆₇	semidirect
305. C₆₇	#32965	direct (N × H)
346. C₂×#41	Q₈×C₆₇	semidirect

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×#475×C₆₇	2
C₂×#476×C₆₇	2
C₆₇×#2628	2
C₆₇×#1865	2
C₆₇×#1864	2
C₆₇×#1853	2
C₆₇×#1805	2
C₆₇×#1804	2
C₆₇×#1798	2
C₆₇×#1788	2
#32965	67

Subgroups — C67×#32965 #30712216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32965 #30712216