Subgroups — C₆₇×#34385 #30713636

Order 17152.

2090 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2378	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2377	C₂	1	2	yes	maximal
4	C₆₇×#2373	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2355	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2352	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2352	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2337	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2336	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2335	C₂	1	4	yes	maximal, complemented
12	C₆₇×#2317	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2314	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	#529×C₆₇	C₄	1	16	yes	complemented
17	C₂×#126×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#391×C₆₇	C₂²	1	4	yes	—
33	#390×C₆₇	C₂²	1	4	yes	—
34	#389×C₆₇	C₂²	1	4	yes	—
35	#389×C₆₇	C₂²	1	4	yes	—
36	#388×C₆₇	C₂²	1	4	yes	—
37	#388×C₆₇	C₂²	1	4	yes	—
38	#388×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	4	yes	—
41	#388×C₆₇	C₂²	1	4	yes	—
42	#387×C₆₇	C₂²	1	4	yes	—
43	#387×C₆₇	C₂²	1	4	yes	—
44	#387×C₆₇	C₂²	1	4	yes	—
45	#387×C₆₇	C₂²	1	4	yes	—
46	#387×C₆₇	C₂²	1	4	yes	—
47	#387×C₆₇	C₂²	1	4	yes	—
48	#386×C₆₇	C₂²	1	4	yes	—
49	#386×C₆₇	C₂²	1	4	yes	—
50	#385×C₆₇	C₂²	1	4	yes	—
51	#385×C₆₇	C₂²	1	4	yes	—
52	#384×C₆₇	C₂²	1	2	yes	—
53	#384×C₆₇	C₂²	1	2	yes	—
54	#383×C₆₇	C₂²	1	2	yes	—
55	#382×C₆₇	C₂²	1	1	yes	characteristic
56	#382×C₆₇	C₂²	1	1	yes	characteristic
57	#382×C₆₇	C₂²	1	2	yes	—
58	#382×C₆₇	C₂²	1	2	yes	—
59	#381×C₆₇	C₂²	1	2	yes	—
60	#380×C₆₇	C₂²	1	2	yes	—
61	#380×C₆₇	C₂²	1	2	yes	—
62	#380×C₆₇	C₂²	1	2	yes	—
63	#380×C₆₇	C₂²	1	2	yes	—
64	#380×C₆₇	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
72	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
73	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
74	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
75	#376×C₆₇	C₂²	1	2	yes	—
76	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	#126×C₆₇	C₂×C₄	1	16	yes	complemented
85	#125×C₆₇	C₂×C₄	1	8	yes	complemented
86	#124×C₆₇	C₂×C₄	1	8	yes	—
87	#123×C₆₇	C₂×C₄	1	16	yes	complemented
88	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
89	#121×C₆₇	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
93	#117×C₆₇	—	4	2	no	—
94	#117×C₆₇	—	4	4	no	—
95	#117×C₆₇	—	4	2	no	—
96	#117×C₆₇	C₂×C₄	1	16	yes	complemented
97	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
130	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	#95×C₆₇	C₂³	1	4	yes	—
137	#95×C₆₇	C₂³	1	2	yes	—
138	#95×C₆₇	C₂³	1	2	yes	—
139	#95×C₆₇	C₂³	1	4	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	4	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
147	#95×C₆₇	C₂³	1	4	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
151	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
152	C₂×Q₈×C₆₇	—	4	2	no	—
153	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₆₇	—	4	2	no	—
155	C₂×D₄×C₆₇	—	4	2	no	—
156	C₂×D₄×C₆₇	—	4	2	no	—
157	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
158	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
159	C₂²×C₄×C₆₇	—	2	4	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	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	4	no	—
167	C₂²×C₄×C₆₇	—	4	4	no	—
168	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₆₇	—	2	4	no	—
186	C₄⋊C₄×C₆₇	—	2	4	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	4	4	no	—
190	C₄⋊C₄×C₆₇	—	4	4	no	—
191	C₄⋊C₄×C₆₇	—	4	4	no	—
192	C₄⋊C₄×C₆₇	—	4	4	no	—
193	C₄⋊C₄×C₆₇	—	4	4	no	—
194	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
195	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
196	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
197	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
198	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
199	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
200	C₂²⋊C₄×C₆₇	—	4	4	no	—
201	C₂²⋊C₄×C₆₇	—	4	4	no	—
202	C₂²⋊C₄×C₆₇	—	4	4	no	—
203	C₂²⋊C₄×C₆₇	—	4	2	no	—
204	C₂²⋊C₄×C₆₇	—	4	4	no	—
205	C₂²⋊C₄×C₆₇	—	4	2	no	—
206	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
207	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
208	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
209	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
210	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
211	C₄²×C₆₇	—	4	4	no	—
212	C₄²×C₆₇	—	4	4	no	—
213	C₄²×C₆₇	—	4	4	no	—
214	C₄²×C₆₇	—	4	2	no	—
215	C₄²×C₆₇	—	4	2	no	—
216	C₄²×C₆₇	—	4	4	no	—
217	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
218	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
219	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
220	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
221	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
222	C₂³×C₆₇	—	2	1	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	1	no	—
225	C₂³×C₆₇	—	2	2	no	—
226	C₂³×C₆₇	—	2	2	no	—
227	C₂³×C₆₇	—	2	2	no	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	2	2	no	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
233	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
234	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
235	Q₈×C₆₇	—	4	4	no	—
236	D₄×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	4	8	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	16	no	—
251	C₂×C₄×C₆₇	—	2	8	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	4	4	no	—
265	C₂×C₄×C₆₇	—	4	4	no	—
266	C₂×C₄×C₆₇	—	8	8	no	—
267	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
270	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
272	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
273	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
274	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
275	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
276	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
277	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
278	C₂²×C₆₇	—	2	2	no	—
279	C₂²×C₆₇	—	2	4	no	—
280	C₂²×C₆₇	—	2	2	no	—
281	C₂²×C₆₇	—	2	2	no	—
282	C₂²×C₆₇	—	2	4	no	—
283	C₂²×C₆₇	—	2	4	no	—
284	C₂²×C₆₇	—	4	2	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	2	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	4	2	no	—
290	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
291	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
293	C₂²×C₆₇	#585	1	2	yes	—
294	C₂²×C₆₇	#585	1	2	yes	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	4	no	—
297	C₄×C₆₇	—	2	2	no	—
298	C₄×C₆₇	—	2	2	no	—
299	C₄×C₆₇	—	2	2	no	—
300	C₄×C₆₇	—	2	2	no	—
301	C₄×C₆₇	—	2	4	no	—
302	C₄×C₆₇	—	2	4	no	—
303	C₄×C₆₇	—	4	4	no	—
304	C₄×C₆₇	—	4	4	no	—
305	C₄×C₆₇	—	4	4	no	—
306	C₄×C₆₇	—	4	4	no	—
307	C₄×C₆₇	#520	1	4	yes	—
308	#34385	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
309	C₂×C₆₇	—	4	2	no	—
310	C₂×C₆₇	—	4	4	no	—
311	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
312	C₂×C₆₇	#3428	1	1	yes	characteristic
313	C₂×C₆₇	#3428	1	1	yes	characteristic
314	C₂×C₆₇	#3562	1	2	yes	—
315	C₂×C₆₇	#3565	1	2	yes	—
316	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
317	#2378	C₂×C₆₇	1	2	yes	complemented
318	#2377	C₂×C₆₇	1	2	yes	—
319	#2373	C₂×C₆₇	1	4	yes	complemented
320	#2355	C₂×C₆₇	1	2	yes	complemented
321	#2354	C₂×C₆₇	1	2	yes	complemented
322	#2352	C₂×C₆₇	1	2	yes	complemented
323	#2352	C₂×C₆₇	1	2	yes	complemented
324	#2337	C₂×C₆₇	1	2	yes	complemented
325	#2336	C₂×C₆₇	1	2	yes	complemented
326	#2335	C₂×C₆₇	1	4	yes	complemented
327	#2317	C₂×C₆₇	1	2	yes	complemented
328	#2314	C₂×C₆₇	1	2	yes	complemented
329	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₆₇	#34385	1	1	yes	characteristic, direct product factor, Sylow
332	#529	C₄×C₆₇	1	16	yes	complemented
333	C₂×#126	C₂²×C₆₇	1	2	yes	—
334	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×#123	C₂²×C₆₇	1	2	yes	—
337	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
342	C₂×#117	C₂²×C₆₇	1	2	yes	—
343	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×#117	C₂²×C₆₇	1	2	yes	—
346	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
347	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
348	#391	C₂²×C₆₇	1	4	yes	—
349	#390	C₂²×C₆₇	1	4	yes	—
350	#389	C₂²×C₆₇	1	4	yes	—
351	#389	C₂²×C₆₇	1	4	yes	—
352	#388	C₂²×C₆₇	1	4	yes	—
353	#388	C₂²×C₆₇	1	4	yes	—
354	#388	C₂²×C₆₇	1	4	yes	—
355	#388	C₂²×C₆₇	1	4	yes	—
356	#388	C₂²×C₆₇	1	4	yes	—
357	#388	C₂²×C₆₇	1	4	yes	—
358	#387	C₂²×C₆₇	1	4	yes	—
359	#387	C₂²×C₆₇	1	4	yes	—
360	#387	C₂²×C₆₇	1	4	yes	—
361	#387	C₂²×C₆₇	1	4	yes	—
362	#387	C₂²×C₆₇	1	4	yes	—
363	#387	C₂²×C₆₇	1	4	yes	—
364	#386	C₂²×C₆₇	1	4	yes	—
365	#386	C₂²×C₆₇	1	4	yes	—
366	#385	C₂²×C₆₇	1	4	yes	—
367	#385	C₂²×C₆₇	1	4	yes	—
368	#384	C₂²×C₆₇	1	2	yes	—
369	#384	C₂²×C₆₇	1	2	yes	—
370	#383	C₂²×C₆₇	1	2	yes	—
371	#382	C₂²×C₆₇	1	1	yes	characteristic
372	#382	C₂²×C₆₇	1	1	yes	characteristic
373	#382	C₂²×C₆₇	1	2	yes	—
374	#382	C₂²×C₆₇	1	2	yes	—
375	#381	C₂²×C₆₇	1	2	yes	—
376	#380	C₂²×C₆₇	1	2	yes	—
377	#380	C₂²×C₆₇	1	2	yes	—
378	#380	C₂²×C₆₇	1	2	yes	—
379	#380	C₂²×C₆₇	1	2	yes	—
380	#380	C₂²×C₆₇	1	2	yes	—
381	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
382	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
383	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
384	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
385	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
386	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
387	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
388	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
389	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
390	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
391	#376	C₂²×C₆₇	1	2	yes	—
392	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
394	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
396	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
398	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
399	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
400	#126	C₂×C₄×C₆₇	1	16	yes	complemented
401	#125	C₂×C₄×C₆₇	1	8	yes	complemented
402	#124	C₂×C₄×C₆₇	1	8	yes	—
403	#123	C₂×C₄×C₆₇	1	16	yes	complemented
404	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
405	#121	C₂×C₄×C₆₇	1	8	yes	—
406	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
407	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
408	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
409	#117	—	4	2	no	—
410	#117	—	4	4	no	—
411	#117	—	4	2	no	—
412	#117	C₂×C₄×C₆₇	1	16	yes	complemented
413	C₂×C₄⋊C₄	—	4	4	no	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
438	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
439	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
440	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
443	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
445	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
446	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
447	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
449	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
452	#95	C₂³×C₆₇	1	4	yes	—
453	#95	C₂³×C₆₇	1	2	yes	—
454	#95	C₂³×C₆₇	1	2	yes	—
455	#95	C₂³×C₆₇	1	4	yes	—
456	#95	C₂³×C₆₇	1	4	yes	—
457	#95	C₂³×C₆₇	1	2	yes	—
458	#95	C₂³×C₆₇	1	2	yes	—
459	#95	C₂³×C₆₇	1	4	yes	—
460	#95	C₂³×C₆₇	1	4	yes	—
461	#95	C₂³×C₆₇	1	4	yes	—
462	#95	C₂³×C₆₇	1	4	yes	—
463	#95	C₂³×C₆₇	1	4	yes	—
464	#95	C₂³×C₆₇	1	4	yes	—
465	#95	C₂³×C₆₇	1	4	yes	—
466	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
467	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
468	C₂×Q₈	—	4	2	no	—
469	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
470	C₂×D₄	—	4	2	no	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	—	4	2	no	—
473	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
474	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	4	2	no	—
480	C₂²×C₄	—	4	2	no	—
481	C₂²×C₄	—	4	4	no	—
482	C₂²×C₄	—	4	4	no	—
483	C₂²×C₄	—	4	4	no	—
484	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
485	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
486	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
492	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
496	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
500	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	4	4	no	—
506	C₄⋊C₄	—	4	4	no	—
507	C₄⋊C₄	—	4	4	no	—
508	C₄⋊C₄	—	4	4	no	—
509	C₄⋊C₄	—	4	4	no	—
510	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
511	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
512	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
513	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
514	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
515	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
516	C₂²⋊C₄	—	4	2	no	—
517	C₂²⋊C₄	—	4	4	no	—
518	C₂²⋊C₄	—	4	2	no	—
519	C₂²⋊C₄	—	4	4	no	—
520	C₂²⋊C₄	—	4	4	no	—
521	C₂²⋊C₄	—	4	4	no	—
522	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
523	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
524	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
525	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
526	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
527	C₄²	—	4	2	no	—
528	C₄²	—	4	2	no	—
529	C₄²	—	4	4	no	—
530	C₄²	—	4	4	no	—
531	C₄²	—	4	4	no	—
532	C₄²	—	4	4	no	—
533	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
534	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
535	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
536	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
537	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	2	1	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
549	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
550	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
551	Q₈	—	4	4	no	—
552	D₄	—	4	4	no	—
553	D₄	—	4	8	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	16	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	8	8	no	—
583	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
584	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
585	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
586	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
587	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
588	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
589	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
590	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
591	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
592	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
593	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	4	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	4	no	—
599	C₂²	—	2	4	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
607	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
608	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
609	C₂²	#585×C₆₇	1	2	yes	—
610	C₂²	#585×C₆₇	1	2	yes	—
611	C₄	—	2	2	no	—
612	C₄	—	2	4	no	—
613	C₄	—	2	2	no	—
614	C₄	—	2	2	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	4	no	—
618	C₄	—	2	4	no	—
619	C₄	—	4	4	no	—
620	C₄	—	4	4	no	—
621	C₄	—	4	4	no	—
622	C₄	—	4	4	no	—
623	C₄	#520×C₆₇	1	4	yes	—
624	C₂	—	4	2	no	—
625	C₂	—	4	4	no	—
626	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
627	C₂	C₆₇×#3428	1	1	yes	characteristic
628	C₂	C₆₇×#3428	1	1	yes	characteristic
629	C₂	C₆₇×#3562	1	2	yes	—
630	C₂	C₆₇×#3565	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2378	C₂	semidirect
4. C₆₇×#2373	C₂	semidirect
5. C₆₇×#2355	C₂	semidirect
6. C₆₇×#2354	C₂	semidirect
7. C₆₇×#2352	C₂	semidirect
8. C₆₇×#2352	C₂	semidirect
9. C₆₇×#2337	C₂	semidirect
10. C₆₇×#2336	C₂	semidirect
11. C₆₇×#2335	C₂	semidirect
12. C₆₇×#2317	C₂	semidirect
13. C₆₇×#2314	C₂	semidirect
14. C₆₇×#2285	C₂	semidirect
15. C₆₇×#2283	C₂	semidirect
16. #529×C₆₇	C₄	semidirect
84. #126×C₆₇	C₂×C₄	semidirect
85. #125×C₆₇	C₂×C₄	semidirect
87. #123×C₆₇	C₂×C₄	semidirect
88. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
90. C₄×Q₈×C₆₇	C₂×C₄	semidirect
92. C₄×D₄×C₆₇	C₂×C₄	semidirect
96. #117×C₆₇	C₂×C₄	semidirect
124. C₂×C₄²×C₆₇	C₂×C₄	semidirect
308. #34385	C₆₇	direct (N × H)
317. #2378	C₂×C₆₇	semidirect
319. #2373	C₂×C₆₇	semidirect
320. #2355	C₂×C₆₇	semidirect
321. #2354	C₂×C₆₇	semidirect
322. #2352	C₂×C₆₇	semidirect
323. #2352	C₂×C₆₇	semidirect
324. #2337	C₂×C₆₇	semidirect
325. #2336	C₂×C₆₇	semidirect
326. #2335	C₂×C₆₇	semidirect
327. #2317	C₂×C₆₇	semidirect
328. #2314	C₂×C₆₇	semidirect
329. #2285	C₂×C₆₇	semidirect
330. #2283	C₂×C₆₇	semidirect
331. C₆₇	#34385	direct (N × H)
332. #529	C₄×C₆₇	semidirect
400. #126	C₂×C₄×C₆₇	semidirect
401. #125	C₂×C₄×C₆₇	semidirect
403. #123	C₂×C₄×C₆₇	semidirect
404. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
406. C₄×Q₈	C₂×C₄×C₆₇	semidirect
408. C₄×D₄	C₂×C₄×C₆₇	semidirect
412. #117	C₂×C₄×C₆₇	semidirect
440. C₂×C₄²	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₆₇×#2378	2
C₆₇×#2377	2
C₆₇×#2373	2
C₆₇×#2355	2
C₆₇×#2354	2
C₆₇×#2352	2
C₆₇×#2337	2
C₆₇×#2336	2
C₆₇×#2335	2
C₆₇×#2317	2
C₆₇×#2314	2
C₆₇×#2285	2
C₆₇×#2283	2
#34385	67

Subgroups — C67×#34385 #30713636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34385 #30713636