Subgroups — C₂×C₆₇×#2591 #30714101

Order 17152.

3438 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2591	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#394×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#388×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#386×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#386×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#121×C₆₇	—	2	8	no	—
16	C₂×#121×C₆₇	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₆₇	—	2	8	no	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
27	#400×C₆₇	C₂²	1	16	yes	complemented
28	#394×C₆₇	C₂²	1	16	yes	complemented
29	#388×C₆₇	C₂²	1	32	yes	complemented
30	#386×C₆₇	C₂²	1	16	yes	complemented
31	#386×C₆₇	C₂²	1	16	yes	complemented
32	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	4	yes	—
35	C₂×#95×C₆₇	C₂²	1	4	yes	—
36	C₂⁵×C₆₇	C₂³	1	2	yes	—
37	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	8	no	—
39	C₂²×D₄×C₆₇	—	2	16	no	—
40	C₂²×D₄×C₆₇	—	2	4	no	—
41	C₂²×D₄×C₆₇	—	4	4	no	—
42	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
43	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	—	2	4	no	—
45	C₂³×C₄×C₆₇	—	2	8	no	—
46	C₂³×C₄×C₆₇	—	2	4	no	—
47	C₂³×C₄×C₆₇	—	4	4	no	—
48	C₂³×C₄×C₆₇	—	4	4	no	—
49	C₂³×C₄×C₆₇	D₄	1	8	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
59	#121×C₆₇	—	2	32	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	32	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
75	C₂×C₄²×C₆₇	—	4	4	no	—
76	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
77	#95×C₆₇	C₂³	1	8	yes	—
78	#95×C₆₇	C₂³	1	16	yes	—
79	#95×C₆₇	C₂³	1	16	yes	—
80	C₂⁴×C₆₇	—	2	4	no	—
81	C₂⁴×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	—	2	4	no	—
83	C₂⁴×C₆₇	—	2	8	no	—
84	C₂⁴×C₆₇	—	2	8	no	—
85	C₂⁴×C₆₇	—	2	8	no	—
86	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
87	C₂⁴×C₆₇	#41	1	8	yes	—
88	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
89	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
90	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂×D₄×C₆₇	—	2	16	no	—
92	C₂×D₄×C₆₇	—	2	32	no	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	4	8	no	—
95	C₂×D₄×C₆₇	—	4	16	no	—
96	C₂×D₄×C₆₇	—	4	32	no	—
97	C₂²×C₄×C₆₇	—	2	32	no	—
98	C₂²×C₄×C₆₇	—	2	8	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	16	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	4	16	no	—
113	C₂²×C₄×C₆₇	—	4	8	no	—
114	C₂²×C₄×C₆₇	—	4	8	no	—
115	C₂²×C₄×C₆₇	—	4	16	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
119	C₂²×C₄×C₆₇	#41	1	8	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
126	C₄⋊C₄×C₆₇	—	2	16	no	—
127	C₄⋊C₄×C₆₇	—	2	16	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₂²⋊C₄×C₆₇	—	2	16	no	—
130	C₂²⋊C₄×C₆₇	—	2	16	no	—
131	C₂²⋊C₄×C₆₇	—	2	32	no	—
132	C₂²⋊C₄×C₆₇	—	4	16	no	—
133	C₄²×C₆₇	—	4	8	no	—
134	C₂³×C₆₇	—	2	8	no	—
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	2	32	no	—
137	C₂³×C₆₇	—	2	4	no	—
138	C₂³×C₆₇	—	2	4	no	—
139	C₂³×C₆₇	—	2	4	no	—
140	C₂³×C₆₇	—	2	8	no	—
141	C₂³×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	2	8	no	—
143	C₂³×C₆₇	—	4	8	no	—
144	C₂³×C₆₇	—	4	8	no	—
145	C₂³×C₆₇	—	4	8	no	—
146	C₂³×C₆₇	—	4	32	no	—
147	C₂³×C₆₇	#124	1	16	yes	—
148	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
149	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
150	C₂³×C₆₇	C₂×#41	1	8	yes	—
151	C₂³×C₆₇	C₂×#41	1	4	yes	—
152	C₂³×C₆₇	C₂×#41	1	4	yes	—
153	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
154	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
155	D₄×C₆₇	—	4	16	no	—
156	D₄×C₆₇	—	4	32	no	—
157	C₂×C₄×C₆₇	—	2	32	no	—
158	C₂×C₄×C₆₇	—	2	16	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	4	16	no	—
173	C₂×C₄×C₆₇	—	4	16	no	—
174	C₂×C₄×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	—	4	8	no	—
176	C₂×C₄×C₆₇	—	4	16	no	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
180	C₂²×C₆₇	—	2	32	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	16	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	—	4	4	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	4	16	no	—
188	C₂²×C₆₇	—	8	16	no	—
189	C₂²×C₆₇	#525	1	4	yes	—
190	C₂²×C₆₇	C₂×#124	1	8	yes	—
191	C₂²×C₆₇	C₂×#124	1	4	yes	—
192	C₂²×C₆₇	#534	1	4	yes	—
193	C₂²×C₆₇	#553	1	16	yes	—
194	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
195	C₂²×C₆₇	C₂²×#41	1	2	yes	—
196	C₂²×C₆₇	C₂²×#41	1	2	yes	—
197	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
198	C₄×C₆₇	—	2	8	no	—
199	C₄×C₆₇	—	2	4	no	—
200	C₄×C₆₇	—	2	4	no	—
201	C₄×C₆₇	—	4	4	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	—	4	8	no	—
204	C₂×#2591	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂×C₆₇	—	2	8	no	—
206	C₂×C₆₇	—	8	4	no	—
207	C₂×C₆₇	#2591	1	8	yes	direct product factor
208	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
209	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
210	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#553	1	4	yes	—
212	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
213	C₂³×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#2591	C₂×C₆₇	1	16	yes	direct product factor
215	C₂×#400	C₂×C₆₇	1	2	yes	complemented
216	C₂×#394	C₂×C₆₇	1	2	yes	complemented
217	C₂×#388	C₂×C₆₇	1	4	yes	complemented
218	C₂×#386	C₂×C₆₇	1	2	yes	complemented
219	C₂×#386	C₂×C₆₇	1	2	yes	complemented
220	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₆₇	C₂×#2591	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
223	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
225	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
226	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#121	—	2	8	no	—
228	C₂×#121	C₂²×C₆₇	1	8	yes	—
229	C₂²×C₄⋊C₄	—	2	8	no	—
230	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
231	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
233	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
234	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
235	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
236	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
237	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
238	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
239	#400	C₂²×C₆₇	1	16	yes	complemented
240	#394	C₂²×C₆₇	1	16	yes	complemented
241	#388	C₂²×C₆₇	1	32	yes	complemented
242	#386	C₂²×C₆₇	1	16	yes	complemented
243	#386	C₂²×C₆₇	1	16	yes	complemented
244	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
245	C₂×#95	C₂²×C₆₇	1	2	yes	—
246	C₂×#95	C₂²×C₆₇	1	4	yes	—
247	C₂×#95	C₂²×C₆₇	1	4	yes	—
248	C₂⁵	C₂³×C₆₇	1	2	yes	—
249	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
250	C₂²×D₄	—	2	8	no	—
251	C₂²×D₄	—	2	16	no	—
252	C₂²×D₄	—	2	4	no	—
253	C₂²×D₄	—	4	4	no	—
254	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
255	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
256	C₂³×C₄	—	2	4	no	—
257	C₂³×C₄	—	2	8	no	—
258	C₂³×C₄	—	2	4	no	—
259	C₂³×C₄	—	4	4	no	—
260	C₂³×C₄	—	4	4	no	—
261	C₂³×C₄	D₄×C₆₇	1	8	yes	—
262	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
263	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
269	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
271	#121	—	2	32	no	—
272	C₂×C₄⋊C₄	—	2	8	no	—
273	C₂×C₄⋊C₄	—	2	8	no	—
274	C₂×C₄⋊C₄	—	2	32	no	—
275	C₂×C₄⋊C₄	—	2	4	no	—
276	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
279	C₂×C₂²⋊C₄	—	2	8	no	—
280	C₂×C₂²⋊C₄	—	2	8	no	—
281	C₂×C₂²⋊C₄	—	2	16	no	—
282	C₂×C₂²⋊C₄	—	4	8	no	—
283	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
287	C₂×C₄²	—	4	4	no	—
288	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
289	#95	C₂³×C₆₇	1	8	yes	—
290	#95	C₂³×C₆₇	1	16	yes	—
291	#95	C₂³×C₆₇	1	16	yes	—
292	C₂⁴	—	2	4	no	—
293	C₂⁴	—	2	4	no	—
294	C₂⁴	—	2	4	no	—
295	C₂⁴	—	2	8	no	—
296	C₂⁴	—	2	8	no	—
297	C₂⁴	—	2	8	no	—
298	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
299	C₂⁴	#41×C₆₇	1	8	yes	—
300	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
301	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
302	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
303	C₂×D₄	—	2	16	no	—
304	C₂×D₄	—	2	32	no	—
305	C₂×D₄	—	2	8	no	—
306	C₂×D₄	—	4	8	no	—
307	C₂×D₄	—	4	16	no	—
308	C₂×D₄	—	4	32	no	—
309	C₂²×C₄	—	2	32	no	—
310	C₂²×C₄	—	2	8	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	16	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	8	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	8	no	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	4	16	no	—
325	C₂²×C₄	—	4	8	no	—
326	C₂²×C₄	—	4	8	no	—
327	C₂²×C₄	—	4	16	no	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
331	C₂²×C₄	#41×C₆₇	1	8	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
338	C₄⋊C₄	—	2	16	no	—
339	C₄⋊C₄	—	2	16	no	—
340	C₄⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	—	2	16	no	—
342	C₂²⋊C₄	—	2	16	no	—
343	C₂²⋊C₄	—	2	32	no	—
344	C₂²⋊C₄	—	4	16	no	—
345	C₄²	—	4	8	no	—
346	C₂³	—	2	8	no	—
347	C₂³	—	2	8	no	—
348	C₂³	—	2	32	no	—
349	C₂³	—	2	4	no	—
350	C₂³	—	2	4	no	—
351	C₂³	—	2	4	no	—
352	C₂³	—	2	8	no	—
353	C₂³	—	2	8	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	4	8	no	—
356	C₂³	—	4	8	no	—
357	C₂³	—	4	8	no	—
358	C₂³	—	4	32	no	—
359	C₂³	#124×C₆₇	1	16	yes	—
360	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
361	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
362	C₂³	C₂×#41×C₆₇	1	8	yes	—
363	C₂³	C₂×#41×C₆₇	1	4	yes	—
364	C₂³	C₂×#41×C₆₇	1	4	yes	—
365	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
366	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
367	D₄	—	4	16	no	—
368	D₄	—	4	32	no	—
369	C₂×C₄	—	2	32	no	—
370	C₂×C₄	—	2	16	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	8	no	—
380	C₂×C₄	—	2	8	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	4	16	no	—
385	C₂×C₄	—	4	16	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	8	no	—
388	C₂×C₄	—	4	16	no	—
389	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
390	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
391	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
392	C₂²	—	2	32	no	—
393	C₂²	—	2	4	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	16	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	4	16	no	—
400	C₂²	—	8	16	no	—
401	C₂²	#525×C₆₇	1	4	yes	—
402	C₂²	C₂×#124×C₆₇	1	8	yes	—
403	C₂²	C₂×#124×C₆₇	1	4	yes	—
404	C₂²	#534×C₆₇	1	4	yes	—
405	C₂²	#553×C₆₇	1	16	yes	—
406	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
407	C₂²	C₂²×#41×C₆₇	1	2	yes	—
408	C₂²	C₂²×#41×C₆₇	1	2	yes	—
409	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
410	C₄	—	2	8	no	—
411	C₄	—	2	4	no	—
412	C₄	—	2	4	no	—
413	C₄	—	4	4	no	—
414	C₄	—	4	4	no	—
415	C₄	—	4	8	no	—
416	C₂	—	2	8	no	—
417	C₂	—	8	4	no	—
418	C₂	C₆₇×#2591	1	8	yes	direct product factor
419	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
420	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
421	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
422	C₂	C₂×#553×C₆₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
3. C₆₇×#2591	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
5. C₂×#394×C₆₇	C₂	semidirect
6. C₂×#388×C₆₇	C₂	semidirect
7. C₂×#386×C₆₇	C₂	semidirect
8. C₂×#386×C₆₇	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
21. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
27. #400×C₆₇	C₂²	semidirect
28. #394×C₆₇	C₂²	semidirect
29. #388×C₆₇	C₂²	semidirect
30. #386×C₆₇	C₂²	semidirect
31. #386×C₆₇	C₂²	semidirect
32. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
204. C₂×#2591	C₆₇	direct (N × H)
207. C₂×C₆₇	#2591	direct (N × H)
213. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
214. #2591	C₂×C₆₇	direct (N × H)
215. C₂×#400	C₂×C₆₇	semidirect
216. C₂×#394	C₂×C₆₇	semidirect
217. C₂×#388	C₂×C₆₇	semidirect
218. C₂×#386	C₂×C₆₇	semidirect
219. C₂×#386	C₂×C₆₇	semidirect
220. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
221. C₆₇	C₂×#2591	direct (N × H)
233. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
239. #400	C₂²×C₆₇	semidirect
240. #394	C₂²×C₆₇	semidirect
241. #388	C₂²×C₆₇	semidirect
242. #386	C₂²×C₆₇	semidirect
243. #386	C₂²×C₆₇	semidirect
244. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
418. C₂	C₆₇×#2591	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×#121×C₆₇	2
C₂³×C₄⋊C₄×C₆₇	2
C₆₇×#2591	2
C₂×#400×C₆₇	2
C₂×#394×C₆₇	2
C₂×#388×C₆₇	2
C₂×#386×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2591	67

Subgroups — C2×C67×#2591 #30714101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2591 #30714101