Subgroups — C₆₇×#24042 #30703293

Order 17152.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#484×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2487	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2043	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2036	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1892	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1869	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1787	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	D₄×Q₈×C₆₇	—	2	4	no	—
16	D₄×Q₈×C₆₇	—	2	4	no	—
17	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	—	2	1	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	#484×C₆₇	—	2	4	no	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	—	2	4	no	—
30	C₂×#103×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×#103×C₆₇	C₂²	1	2	yes	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#400×C₆₇	—	2	1	no	—
39	#399×C₆₇	—	2	1	no	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	#386×C₆₇	—	2	1	no	—
42	#385×C₆₇	—	2	1	no	—
43	#380×C₆₇	C₂²	1	2	yes	—
44	#380×C₆₇	C₂²	1	1	yes	characteristic
45	#340×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₆₇	—	2	2	no	—
48	C₂²×Q₈×C₆₇	—	2	2	no	—
49	C₂²×Q₈×C₆₇	—	2	1	no	—
50	C₂²×Q₈×C₆₇	—	2	1	no	—
51	C₂²×Q₈×C₆₇	—	2	2	no	—
52	C₂²×Q₈×C₆₇	—	2	4	no	—
53	C₂²×Q₈×C₆₇	—	2	4	no	—
54	C₂²×Q₈×C₆₇	—	4	2	no	—
55	C₂²×Q₈×C₆₇	—	4	2	no	—
56	C₂²×Q₈×C₆₇	D₄	1	4	yes	—
57	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	—	2	1	no	—
62	C₂³×C₄×C₆₇	—	2	1	no	—
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	—	2	2	no	—
67	C₂²×C₈×C₆₇	—	4	2	no	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₆₇	—	2	8	no	—
71	C₄⋊Q₈×C₆₇	—	2	2	no	—
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	8	no	—
74	C₂²⋊Q₈×C₆₇	—	2	8	no	—
75	C₂²⋊Q₈×C₆₇	—	2	2	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	2	no	—
78	C₂²⋊Q₈×C₆₇	D₄	1	4	yes	—
79	C₄×Q₈×C₆₇	—	2	2	no	—
80	C₄×Q₈×C₆₇	—	2	2	no	—
81	C₄×D₄×C₆₇	—	2	8	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	—	2	1	no	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	#107×C₆₇	—	2	4	no	—
106	#103×C₆₇	—	2	4	no	—
107	#103×C₆₇	—	2	4	no	—
108	#103×C₆₇	—	2	4	no	—
109	#103×C₆₇	—	2	4	no	—
110	#103×C₆₇	—	4	2	no	—
111	#103×C₆₇	—	4	2	no	—
112	C₂²⋊C₈×C₆₇	—	2	2	no	—
113	C₂²⋊C₈×C₆₇	—	4	2	no	—
114	#95×C₆₇	—	2	2	no	—
115	#95×C₆₇	—	2	1	no	—
116	#95×C₆₇	—	2	2	no	—
117	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Q₈×C₆₇	—	2	4	no	—
120	C₂×Q₈×C₆₇	—	2	4	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	—	2	2	no	—
125	C₂×Q₈×C₆₇	—	2	8	no	—
126	C₂×Q₈×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	—	2	2	no	—
128	C₂×Q₈×C₆₇	—	2	2	no	—
129	C₂×Q₈×C₆₇	—	2	2	no	—
130	C₂×Q₈×C₆₇	—	4	4	no	—
131	C₂×Q₈×C₆₇	—	4	4	no	—
132	C₂×Q₈×C₆₇	—	4	4	no	—
133	C₂×Q₈×C₆₇	—	4	8	no	—
134	C₂×Q₈×C₆₇	—	4	8	no	—
135	C₂×Q₈×C₆₇	—	4	4	no	—
136	C₂×Q₈×C₆₇	—	4	4	no	—
137	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×Q₈×C₆₇	#41	1	4	yes	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	Q₁₆	1	4	yes	—
144	C₂²×C₄×C₆₇	—	2	1	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	4	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	1	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	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	4	4	no	—
159	C₂²×C₄×C₆₇	—	4	4	no	—
160	C₂²×C₄×C₆₇	—	4	2	no	—
161	C₂²×C₄×C₆₇	—	8	2	no	—
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
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	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	4	4	no	—
176	C₄⋊C₄×C₆₇	—	2	4	no	—
177	C₄⋊C₄×C₆₇	—	2	4	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	4	2	no	—
188	C₄⋊C₄×C₆₇	—	8	2	no	—
189	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	—	2	4	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₂²⋊C₄×C₆₇	—	4	2	no	—
195	C₂²⋊C₄×C₆₇	—	8	2	no	—
196	C₄²×C₆₇	—	2	4	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	—	2	1	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	2	1	no	—
204	C₂³×C₆₇	—	2	2	no	—
205	C₂³×C₆₇	#120	1	2	yes	—
206	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
207	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
208	C₂³×C₆₇	#137	1	2	yes	—
209	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₆₇	—	2	2	no	—
211	Q₈×C₆₇	—	2	4	no	—
212	Q₈×C₆₇	—	2	2	no	—
213	Q₈×C₆₇	—	2	2	no	—
214	Q₈×C₆₇	—	2	4	no	—
215	Q₈×C₆₇	—	4	8	no	—
216	Q₈×C₆₇	—	4	8	no	—
217	Q₈×C₆₇	—	4	1	no	—
218	Q₈×C₆₇	—	4	1	no	—
219	Q₈×C₆₇	—	4	2	no	—
220	Q₈×C₆₇	#123	1	4	yes	—
221	D₄×C₆₇	—	2	8	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	4	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	1	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	4	8	no	—
238	C₂×C₄×C₆₇	—	4	8	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	1	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	2	no	—
245	C₂×C₄×C₆₇	—	4	2	no	—
246	C₂×C₄×C₆₇	—	4	2	no	—
247	C₂×C₄×C₆₇	—	4	2	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	8	4	no	—
252	C₂×C₄×C₆₇	#120	1	2	yes	—
253	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
257	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
258	C₈×C₆₇	—	4	2	no	—
259	C₈×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	4	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	4	1	no	—
267	C₂²×C₆₇	—	4	1	no	—
268	C₂²×C₆₇	—	4	2	no	—
269	C₂²×C₆₇	#451	1	4	yes	—
270	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
271	C₂²×C₆₇	#546	1	2	yes	—
272	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
274	C₂²×C₆₇	#577	1	2	yes	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	2	4	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	—	4	2	no	—
280	C₄×C₆₇	—	4	2	no	—
281	C₄×C₆₇	—	4	1	no	—
282	C₄×C₆₇	—	4	2	no	—
283	C₄×C₆₇	—	8	2	no	—
284	C₄×C₆₇	—	8	2	no	—
285	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
286	C₄×C₆₇	#534	1	1	yes	characteristic
287	C₄×C₆₇	#546	1	2	yes	—
288	#24042	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂×C₆₇	—	2	4	no	—
290	C₂×C₆₇	—	4	2	no	—
291	C₂×C₆₇	#2405	1	1	yes	characteristic
292	C₂×C₆₇	C₂×#451	1	1	yes	characteristic
293	C₂×C₆₇	#3010	1	1	yes	characteristic
294	C₂×C₆₇	#3299	1	2	yes	—
295	C₂×C₆₇	#3320	1	2	yes	—
296	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
297	C₂×#484	C₂×C₆₇	1	1	yes	characteristic, complemented
298	C₂²×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
299	#2487	C₂×C₆₇	1	1	yes	characteristic
300	#2043	C₂×C₆₇	1	2	yes	complemented
301	#2036	C₂×C₆₇	1	2	yes	complemented
302	#1892	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1869	C₂×C₆₇	1	1	yes	characteristic
304	#1863	C₂×C₆₇	1	2	yes	complemented
305	#1787	C₂×C₆₇	1	2	yes	complemented
306	#1787	C₂×C₆₇	1	1	yes	characteristic, complemented
307	C₆₇	#24042	1	1	yes	characteristic, direct product factor, Sylow
308	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
310	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
311	D₄×Q₈	—	2	4	no	—
312	D₄×Q₈	—	2	4	no	—
313	C₂×C₄⋊Q₈	—	2	1	no	—
314	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₂²⋊Q₈	—	2	1	no	—
316	C₂×C₂²⋊Q₈	—	2	1	no	—
317	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×C₄×D₄	—	2	1	no	—
321	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
323	#484	—	2	4	no	—
324	C₂×#107	C₂²×C₆₇	1	2	yes	—
325	C₂×#103	—	2	4	no	—
326	C₂×#103	C₂²×C₆₇	1	2	yes	—
327	C₂×#103	C₂²×C₆₇	1	2	yes	—
328	C₂×#103	C₂²×C₆₇	1	2	yes	—
329	C₂×#103	C₂²×C₆₇	1	2	yes	—
330	C₂×#103	C₂²×C₆₇	1	2	yes	—
331	C₂×#103	C₂²×C₆₇	1	2	yes	—
332	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
334	#400	—	2	1	no	—
335	#399	—	2	1	no	—
336	#389	C₂²×C₆₇	1	2	yes	—
337	#386	—	2	1	no	—
338	#385	—	2	1	no	—
339	#380	C₂²×C₆₇	1	1	yes	characteristic
340	#380	C₂²×C₆₇	1	2	yes	—
341	#340	C₂²×C₆₇	1	2	yes	—
342	#336	C₂²×C₆₇	1	2	yes	—
343	C₂²×Q₈	—	2	2	no	—
344	C₂²×Q₈	—	2	2	no	—
345	C₂²×Q₈	—	2	1	no	—
346	C₂²×Q₈	—	2	1	no	—
347	C₂²×Q₈	—	2	2	no	—
348	C₂²×Q₈	—	2	4	no	—
349	C₂²×Q₈	—	2	4	no	—
350	C₂²×Q₈	—	4	2	no	—
351	C₂²×Q₈	—	4	2	no	—
352	C₂²×Q₈	D₄×C₆₇	1	4	yes	—
353	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
354	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
356	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
357	C₂³×C₄	—	2	1	no	—
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
362	C₂²×C₈	—	2	2	no	—
363	C₂²×C₈	—	4	2	no	—
364	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
365	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
366	C₄⋊Q₈	—	2	8	no	—
367	C₄⋊Q₈	—	2	2	no	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	8	no	—
370	C₂²⋊Q₈	—	2	8	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	C₂²⋊Q₈	D₄×C₆₇	1	4	yes	—
375	C₄×Q₈	—	2	2	no	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×D₄	—	2	8	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
401	#107	—	2	4	no	—
402	#103	—	2	4	no	—
403	#103	—	2	4	no	—
404	#103	—	2	4	no	—
405	#103	—	2	4	no	—
406	#103	—	4	2	no	—
407	#103	—	4	2	no	—
408	C₂²⋊C₈	—	2	2	no	—
409	C₂²⋊C₈	—	4	2	no	—
410	#95	—	2	2	no	—
411	#95	—	2	1	no	—
412	#95	—	2	2	no	—
413	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
414	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	8	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	4	4	no	—
427	C₂×Q₈	—	4	4	no	—
428	C₂×Q₈	—	4	4	no	—
429	C₂×Q₈	—	4	8	no	—
430	C₂×Q₈	—	4	8	no	—
431	C₂×Q₈	—	4	4	no	—
432	C₂×Q₈	—	4	4	no	—
433	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
434	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
435	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
436	C₂×Q₈	#41×C₆₇	1	4	yes	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	Q₁₆×C₆₇	1	4	yes	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	8	2	no	—
458	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	4	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	—	8	2	no	—
485	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₂²⋊C₄	—	8	2	no	—
492	C₄²	—	2	4	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	2	no	—
501	C₂³	#120×C₆₇	1	2	yes	—
502	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
503	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
504	C₂³	#137×C₆₇	1	2	yes	—
505	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
506	Q₈	—	2	2	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	2	2	no	—
510	Q₈	—	2	4	no	—
511	Q₈	—	4	8	no	—
512	Q₈	—	4	8	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	2	no	—
516	Q₈	#123×C₆₇	1	4	yes	—
517	D₄	—	2	8	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	—	4	8	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	8	4	no	—
548	C₂×C₄	#120×C₆₇	1	2	yes	—
549	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
553	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
554	C₈	—	4	2	no	—
555	C₈	—	4	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	1	no	—
565	C₂²	#451×C₆₇	1	4	yes	—
566	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
567	C₂²	#546×C₆₇	1	2	yes	—
568	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
569	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
570	C₂²	#577×C₆₇	1	2	yes	—
571	C₄	—	2	2	no	—
572	C₄	—	2	4	no	—
573	C₄	—	2	2	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	1	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	8	2	no	—
580	C₄	—	8	2	no	—
581	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
582	C₄	#534×C₆₇	1	1	yes	characteristic
583	C₄	#546×C₆₇	1	2	yes	—
584	C₂	—	2	4	no	—
585	C₂	—	4	2	no	—
586	C₂	C₆₇×#2405	1	1	yes	characteristic
587	C₂	C₂×#451×C₆₇	1	1	yes	characteristic
588	C₂	C₆₇×#3010	1	1	yes	characteristic
589	C₂	C₆₇×#3299	1	2	yes	—
590	C₂	C₆₇×#3320	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#484×C₆₇	C₂	semidirect
3. C₂²×#103×C₆₇	C₂	semidirect
5. C₆₇×#2043	C₂	semidirect
6. C₆₇×#2036	C₂	semidirect
7. C₆₇×#1892	C₂	semidirect
9. C₆₇×#1863	C₂	semidirect
10. C₆₇×#1787	C₂	semidirect
11. C₆₇×#1787	C₂	semidirect
288. #24042	C₆₇	direct (N × H)
297. C₂×#484	C₂×C₆₇	semidirect
298. C₂²×#103	C₂×C₆₇	semidirect
300. #2043	C₂×C₆₇	semidirect
301. #2036	C₂×C₆₇	semidirect
302. #1892	C₂×C₆₇	semidirect
304. #1863	C₂×C₆₇	semidirect
305. #1787	C₂×C₆₇	semidirect
306. #1787	C₂×C₆₇	semidirect
307. C₆₇	#24042	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×D₄×Q₈×C₆₇	2
C₂×#484×C₆₇	2
C₂²×#103×C₆₇	2
C₆₇×#2487	2
C₆₇×#2043	2
C₆₇×#2036	2
C₆₇×#1892	2
C₆₇×#1869	2
C₆₇×#1863	2
C₆₇×#1787	2
#24042	67

Subgroups — C67×#24042 #30703293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#24042 #30703293