Subgroups — C₂×C₆₇×#2322 #30713313

Order 17152.

3718 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2322	C₂	1	16	yes	maximal, direct product factor
5	C₂×#389×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#388×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#386×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#385×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#385×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#381×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#123×C₆₇	C₄	1	16	yes	complemented
19	C₂×#123×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄×D₄×C₆₇	—	2	8	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄×C₆₇	—	2	8	no	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	C₂²	1	16	yes	complemented
38	#388×C₆₇	C₂²	1	16	yes	complemented
39	#387×C₆₇	C₂²	1	16	yes	complemented
40	#386×C₆₇	C₂²	1	8	yes	—
41	#385×C₆₇	C₂²	1	8	yes	complemented
42	#385×C₆₇	C₂²	1	16	yes	complemented
43	#381×C₆₇	C₂²	1	8	yes	complemented
44	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	2	yes	—
47	C₂×#95×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂×#95×C₆₇	C₂²	1	2	yes	—
50	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
51	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	—	2	4	no	—
54	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₆₇	—	2	4	no	—
57	C₂³×C₄×C₆₇	—	2	4	no	—
58	C₂³×C₄×C₆₇	—	2	4	no	—
59	C₂³×C₄×C₆₇	—	2	4	no	—
60	C₂³×C₄×C₆₇	—	4	4	no	—
61	C₂³×C₄×C₆₇	—	4	4	no	—
62	C₂³×C₄×C₆₇	—	4	4	no	—
63	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	complemented
64	C₂³×C₄×C₆₇	D₄	1	8	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
69	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
78	#123×C₆₇	C₂×C₄	1	64	yes	complemented
79	C₄×D₄×C₆₇	—	2	32	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	32	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
96	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₄²×C₆₇	—	2	4	no	—
104	C₂×C₄²×C₆₇	—	4	4	no	—
105	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
107	#95×C₆₇	C₂³	1	8	yes	—
108	#95×C₆₇	C₂³	1	8	yes	—
109	#95×C₆₇	C₂³	1	8	yes	—
110	#95×C₆₇	C₂³	1	8	yes	—
111	#95×C₆₇	C₂³	1	8	yes	—
112	C₂⁴×C₆₇	—	2	4	no	—
113	C₂⁴×C₆₇	—	2	4	no	—
114	C₂⁴×C₆₇	—	2	2	no	—
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	—	2	4	no	—
117	C₂⁴×C₆₇	—	2	4	no	—
118	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
120	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
121	C₂⁴×C₆₇	#41	1	4	yes	—
122	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
123	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
124	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂×D₄×C₆₇	—	2	32	no	—
126	C₂×D₄×C₆₇	—	2	8	no	—
127	C₂×D₄×C₆₇	—	2	8	no	—
128	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
129	C₂²×C₄×C₆₇	—	2	32	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	8	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	8	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	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	8	no	—
152	C₂²×C₄×C₆₇	—	4	16	no	—
153	C₂²×C₄×C₆₇	—	4	8	no	—
154	C₂²×C₄×C₆₇	—	4	8	no	—
155	C₂²×C₄×C₆₇	—	4	8	no	—
156	C₂²×C₄×C₆₇	—	4	16	no	—
157	C₂²×C₄×C₆₇	—	4	16	no	—
158	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
160	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
162	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₄⋊C₄×C₆₇	—	2	16	no	—
175	C₄⋊C₄×C₆₇	—	2	16	no	—
176	C₄⋊C₄×C₆₇	—	2	8	no	—
177	C₄⋊C₄×C₆₇	—	4	8	no	—
178	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	complemented
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	2	8	no	—
182	C₂²⋊C₄×C₆₇	—	4	8	no	—
183	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
184	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	complemented
185	C₄²×C₆₇	—	2	8	no	—
186	C₄²×C₆₇	—	4	8	no	—
187	C₂³×C₆₇	—	2	16	no	—
188	C₂³×C₆₇	—	2	8	no	—
189	C₂³×C₆₇	—	2	8	no	—
190	C₂³×C₆₇	—	2	16	no	—
191	C₂³×C₆₇	—	2	4	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	2	8	no	—
194	C₂³×C₆₇	—	2	16	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	—	2	4	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	—	4	4	no	—
201	C₂³×C₆₇	—	4	8	no	—
202	C₂³×C₆₇	C₄×D₄	1	16	yes	—
203	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
204	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
205	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
206	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
207	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
208	C₂³×C₆₇	C₂×#41	1	4	yes	—
209	C₂³×C₆₇	C₂×#41	1	2	yes	—
210	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
211	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
212	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
213	D₄×C₆₇	—	2	32	no	—
214	C₂×C₄×C₆₇	—	2	16	no	—
215	C₂×C₄×C₆₇	—	2	32	no	—
216	C₂×C₄×C₆₇	—	2	8	no	—
217	C₂×C₄×C₆₇	—	2	16	no	—
218	C₂×C₄×C₆₇	—	2	8	no	—
219	C₂×C₄×C₆₇	—	2	8	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	4	16	no	—
239	C₂×C₄×C₆₇	—	4	16	no	—
240	C₂×C₄×C₆₇	—	4	16	no	—
241	C₂×C₄×C₆₇	—	4	4	no	—
242	C₂×C₄×C₆₇	—	4	8	no	—
243	C₂×C₄×C₆₇	—	4	4	no	—
244	C₂×C₄×C₆₇	—	4	8	no	—
245	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
246	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
247	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
248	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
250	C₂²×C₆₇	—	2	32	no	—
251	C₂²×C₆₇	—	2	8	no	—
252	C₂²×C₆₇	—	2	4	no	—
253	C₂²×C₆₇	—	2	4	no	—
254	C₂²×C₆₇	—	2	8	no	—
255	C₂²×C₆₇	—	2	4	no	—
256	C₂²×C₆₇	—	2	8	no	—
257	C₂²×C₆₇	—	4	16	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	4	4	no	—
261	C₂²×C₆₇	C₂×C₄×D₄	1	8	yes	—
262	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
263	C₂²×C₆₇	#520	1	8	yes	—
264	C₂²×C₆₇	#534	1	4	yes	—
265	C₂²×C₆₇	#536	1	4	yes	—
266	C₂²×C₆₇	#539	1	8	yes	—
267	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
268	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
269	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
270	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
271	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
272	C₄×C₆₇	—	2	8	no	—
273	C₄×C₆₇	—	2	4	no	—
274	C₄×C₆₇	—	2	8	no	—
275	C₄×C₆₇	—	2	4	no	—
276	C₄×C₆₇	—	2	4	no	—
277	C₄×C₆₇	—	4	2	no	—
278	C₄×C₆₇	—	4	4	no	—
279	C₄×C₆₇	—	4	2	no	—
280	C₄×C₆₇	—	4	4	no	—
281	C₂×#2322	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₂×C₆₇	—	2	8	no	—
283	C₂×C₆₇	—	4	4	no	—
284	C₂×C₆₇	#2322	1	8	yes	direct product factor
285	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
286	C₂×C₆₇	C₂×#520	1	2	yes	—
287	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
288	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
289	C₂×C₆₇	C₂×#539	1	2	yes	—
290	C₂²×#123	C₂×C₆₇	1	1	yes	characteristic
291	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
292	C₂³×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
293	#2322	C₂×C₆₇	1	16	yes	direct product factor
294	C₂×#389	C₂×C₆₇	1	2	yes	complemented
295	C₂×#388	C₂×C₆₇	1	2	yes	complemented
296	C₂×#387	C₂×C₆₇	1	2	yes	complemented
297	C₂×#386	C₂×C₆₇	1	1	yes	characteristic
298	C₂×#385	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₂×#385	C₂×C₆₇	1	2	yes	complemented
300	C₂×#381	C₂×C₆₇	1	1	yes	characteristic, complemented
301	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
302	C₆₇	C₂×#2322	1	1	yes	characteristic, direct product factor, Sylow
303	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
304	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
305	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
306	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
307	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×#123	C₄×C₆₇	1	16	yes	complemented
309	C₂×#123	C₂²×C₆₇	1	8	yes	—
310	C₂×C₄×D₄	—	2	8	no	—
311	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
312	C₂²×C₄⋊C₄	—	2	8	no	—
313	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
314	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
315	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
316	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
317	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
318	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
319	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
320	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
321	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
323	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
324	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
325	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
326	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
327	#389	C₂²×C₆₇	1	16	yes	complemented
328	#388	C₂²×C₆₇	1	16	yes	complemented
329	#387	C₂²×C₆₇	1	16	yes	complemented
330	#386	C₂²×C₆₇	1	8	yes	—
331	#385	C₂²×C₆₇	1	16	yes	complemented
332	#385	C₂²×C₆₇	1	8	yes	complemented
333	#381	C₂²×C₆₇	1	8	yes	complemented
334	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
335	C₂×#95	C₂²×C₆₇	1	2	yes	—
336	C₂×#95	C₂²×C₆₇	1	2	yes	—
337	C₂×#95	C₂²×C₆₇	1	2	yes	—
338	C₂×#95	C₂²×C₆₇	1	2	yes	—
339	C₂×#95	C₂²×C₆₇	1	2	yes	—
340	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
341	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
342	C₂²×D₄	—	2	4	no	—
343	C₂²×D₄	—	2	4	no	—
344	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
345	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
346	C₂³×C₄	—	2	4	no	—
347	C₂³×C₄	—	2	4	no	—
348	C₂³×C₄	—	2	4	no	—
349	C₂³×C₄	—	2	4	no	—
350	C₂³×C₄	—	4	4	no	—
351	C₂³×C₄	—	4	4	no	—
352	C₂³×C₄	—	4	4	no	—
353	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	complemented
354	C₂³×C₄	D₄×C₆₇	1	8	yes	—
355	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
357	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
359	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
360	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
361	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
362	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
363	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
364	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
365	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
366	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
367	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
368	#123	C₂×C₄×C₆₇	1	64	yes	complemented
369	C₄×D₄	—	2	32	no	—
370	C₂×C₄⋊C₄	—	2	8	no	—
371	C₂×C₄⋊C₄	—	2	8	no	—
372	C₂×C₄⋊C₄	—	2	32	no	—
373	C₂×C₄⋊C₄	—	2	4	no	—
374	C₂×C₄⋊C₄	—	4	4	no	—
375	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
381	C₂×C₂²⋊C₄	—	2	4	no	—
382	C₂×C₂²⋊C₄	—	2	4	no	—
383	C₂×C₂²⋊C₄	—	2	4	no	—
384	C₂×C₂²⋊C₄	—	4	4	no	—
385	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
386	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
387	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
393	C₂×C₄²	—	2	4	no	—
394	C₂×C₄²	—	4	4	no	—
395	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
396	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
397	#95	C₂³×C₆₇	1	8	yes	—
398	#95	C₂³×C₆₇	1	8	yes	—
399	#95	C₂³×C₆₇	1	8	yes	—
400	#95	C₂³×C₆₇	1	8	yes	—
401	#95	C₂³×C₆₇	1	8	yes	—
402	C₂⁴	—	2	4	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	2	2	no	—
405	C₂⁴	—	2	2	no	—
406	C₂⁴	—	2	4	no	—
407	C₂⁴	—	2	4	no	—
408	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
409	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
410	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
411	C₂⁴	#41×C₆₇	1	4	yes	—
412	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
413	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
414	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₂×D₄	—	2	32	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
419	C₂²×C₄	—	2	32	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	—	4	16	no	—
443	C₂²×C₄	—	4	8	no	—
444	C₂²×C₄	—	4	8	no	—
445	C₂²×C₄	—	4	8	no	—
446	C₂²×C₄	—	4	16	no	—
447	C₂²×C₄	—	4	16	no	—
448	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
449	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
450	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
451	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
452	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
453	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
454	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
455	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
464	C₄⋊C₄	—	2	16	no	—
465	C₄⋊C₄	—	2	16	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	4	8	no	—
468	C₄⋊C₄	C₂²×C₄×C₆₇	1	32	yes	complemented
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	4	8	no	—
473	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
474	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	complemented
475	C₄²	—	2	8	no	—
476	C₄²	—	4	8	no	—
477	C₂³	—	2	16	no	—
478	C₂³	—	2	8	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	2	16	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	2	16	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	8	no	—
492	C₂³	C₄×D₄×C₆₇	1	16	yes	—
493	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
494	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
495	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
496	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
497	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
498	C₂³	C₂×#41×C₆₇	1	4	yes	—
499	C₂³	C₂×#41×C₆₇	1	2	yes	—
500	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
501	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
502	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
503	D₄	—	2	32	no	—
504	C₂×C₄	—	2	16	no	—
505	C₂×C₄	—	2	32	no	—
506	C₂×C₄	—	2	8	no	—
507	C₂×C₄	—	2	16	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	4	16	no	—
529	C₂×C₄	—	4	16	no	—
530	C₂×C₄	—	4	16	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	4	no	—
534	C₂×C₄	—	4	8	no	—
535	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
536	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
537	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
538	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
539	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
540	C₂²	—	2	32	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	8	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	2	8	no	—
547	C₂²	—	4	16	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	4	no	—
551	C₂²	C₂×C₄×D₄×C₆₇	1	8	yes	—
552	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
553	C₂²	#520×C₆₇	1	8	yes	—
554	C₂²	#534×C₆₇	1	4	yes	—
555	C₂²	#536×C₆₇	1	4	yes	—
556	C₂²	#539×C₆₇	1	8	yes	—
557	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
558	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
559	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
560	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
561	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
562	C₄	—	2	8	no	—
563	C₄	—	2	4	no	—
564	C₄	—	2	8	no	—
565	C₄	—	2	4	no	—
566	C₄	—	2	4	no	—
567	C₄	—	4	2	no	—
568	C₄	—	4	4	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	4	no	—
571	C₂	—	2	8	no	—
572	C₂	—	4	4	no	—
573	C₂	C₆₇×#2322	1	8	yes	direct product factor
574	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
575	C₂	C₂×#520×C₆₇	1	2	yes	—
576	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
577	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
578	C₂	C₂×#539×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#2322	C₂	direct (N × H)
5. C₂×#389×C₆₇	C₂	semidirect
6. C₂×#388×C₆₇	C₂	semidirect
7. C₂×#387×C₆₇	C₂	semidirect
9. C₂×#385×C₆₇	C₂	semidirect
10. C₂×#385×C₆₇	C₂	semidirect
11. C₂×#381×C₆₇	C₂	semidirect
12. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
18. C₂×#123×C₆₇	C₄	semidirect
28. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
37. #389×C₆₇	C₂²	semidirect
38. #388×C₆₇	C₂²	semidirect
39. #387×C₆₇	C₂²	semidirect
41. #385×C₆₇	C₂²	semidirect
42. #385×C₆₇	C₂²	semidirect
43. #381×C₆₇	C₂²	semidirect
44. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
63. C₂³×C₄×C₆₇	C₂×C₄	semidirect
78. #123×C₆₇	C₂×C₄	semidirect
85. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
95. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
96. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
161. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
178. C₄⋊C₄×C₆₇	C₂²×C₄	semidirect
183. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
184. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
281. C₂×#2322	C₆₇	direct (N × H)
284. C₂×C₆₇	#2322	direct (N × H)
292. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
293. #2322	C₂×C₆₇	direct (N × H)
294. C₂×#389	C₂×C₆₇	semidirect
295. C₂×#388	C₂×C₆₇	semidirect
296. C₂×#387	C₂×C₆₇	semidirect
298. C₂×#385	C₂×C₆₇	semidirect
299. C₂×#385	C₂×C₆₇	semidirect
300. C₂×#381	C₂×C₆₇	semidirect
301. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
302. C₆₇	C₂×#2322	direct (N × H)
308. C₂×#123	C₄×C₆₇	semidirect
318. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
327. #389	C₂²×C₆₇	semidirect
328. #388	C₂²×C₆₇	semidirect
329. #387	C₂²×C₆₇	semidirect
331. #385	C₂²×C₆₇	semidirect
332. #385	C₂²×C₆₇	semidirect
333. #381	C₂²×C₆₇	semidirect
334. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
353. C₂³×C₄	C₂×C₄×C₆₇	semidirect
368. #123	C₂×C₄×C₆₇	semidirect
375. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
385. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
386. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
451. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
468. C₄⋊C₄	C₂²×C₄×C₆₇	semidirect
473. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
474. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
573. C₂	C₆₇×#2322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#123×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₂³×C₄⋊C₄×C₆₇	2
C₆₇×#2322	2
C₂×#389×C₆₇	2
C₂×#388×C₆₇	2
C₂×#387×C₆₇	2
C₂×#386×C₆₇	2
C₂×#385×C₆₇	2
C₂×#381×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2322	67

Subgroups — C2×C67×#2322 #30713313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2322 #30713313