Subgroups — C₂×C₆₇×#2562 #30714147

Order 17152.

3006 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2562	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
6	C₂×#388×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#380×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#380×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
12	C₂×#125×C₆₇	—	2	8	no	—
13	C₂×#125×C₆₇	C₂²	1	8	yes	complemented
14	C₂×C₄×D₄×C₆₇	—	2	8	no	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#400×C₆₇	C₂²	1	16	yes	complemented
28	#394×C₆₇	C₂²	1	16	yes	—
29	#388×C₆₇	C₂²	1	32	yes	complemented
30	#380×C₆₇	C₂²	1	16	yes	complemented
31	#380×C₆₇	C₂²	1	16	yes	complemented
32	C₄×C₄⋊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	C₂²×D₄×C₆₇	—	2	8	no	—
38	C₂²×D₄×C₆₇	—	2	4	no	—
39	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
40	C₂³×C₄×C₆₇	—	2	8	no	—
41	C₂³×C₄×C₆₇	—	2	8	no	—
42	C₂³×C₄×C₆₇	—	2	8	no	—
43	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	#125×C₆₇	—	2	32	no	—
52	C₄×D₄×C₆₇	—	2	32	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
68	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
71	C₂×C₄²×C₆₇	—	4	4	no	—
72	C₂×C₄²×C₆₇	D₄	1	8	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
75	#95×C₆₇	C₂³	1	8	yes	—
76	#95×C₆₇	C₂³	1	16	yes	—
77	#95×C₆₇	C₂³	1	16	yes	—
78	C₂⁴×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	—	2	4	no	—
81	C₂⁴×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	—	2	4	no	—
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	#41	1	8	yes	—
85	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
86	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₂×D₄×C₆₇	—	2	32	no	—
88	C₂×D₄×C₆₇	—	2	16	no	—
89	C₂×D₄×C₆₇	—	2	8	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	2	8	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	8	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	16	no	—
105	C₂²×C₄×C₆₇	—	2	16	no	—
106	C₂²×C₄×C₆₇	—	2	16	no	—
107	C₂²×C₄×C₆₇	—	4	32	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₆₇	#41	1	8	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₆₇	—	2	8	no	—
120	C₄⋊C₄×C₆₇	—	2	16	no	—
121	C₄⋊C₄×C₆₇	—	2	16	no	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₄⋊C₄×C₆₇	—	2	8	no	—
124	C₄⋊C₄×C₆₇	—	2	16	no	—
125	C₂²⋊C₄×C₆₇	—	2	32	no	—
126	C₂²⋊C₄×C₆₇	—	2	16	no	—
127	C₂²⋊C₄×C₆₇	—	4	16	no	—
128	C₄²×C₆₇	—	4	8	no	—
129	C₄²×C₆₇	C₂×D₄	1	16	yes	—
130	C₂³×C₆₇	—	2	16	no	—
131	C₂³×C₆₇	—	2	16	no	—
132	C₂³×C₆₇	—	2	16	no	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	2	4	no	—
135	C₂³×C₆₇	—	2	4	no	—
136	C₂³×C₆₇	—	2	4	no	—
137	C₂³×C₆₇	—	2	4	no	—
138	C₂³×C₆₇	—	2	4	no	—
139	C₂³×C₆₇	—	4	8	no	—
140	C₂³×C₆₇	—	4	8	no	—
141	C₂³×C₆₇	—	4	8	no	—
142	C₂³×C₆₇	—	4	8	no	—
143	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
144	C₂³×C₆₇	C₂×#41	1	8	yes	—
145	C₂³×C₆₇	C₂×#41	1	4	yes	—
146	C₂³×C₆₇	C₂×#41	1	4	yes	—
147	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
148	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
149	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
150	D₄×C₆₇	—	2	32	no	—
151	C₂×C₄×C₆₇	—	2	8	no	—
152	C₂×C₄×C₆₇	—	2	8	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	4	8	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	4	8	no	—
170	C₂×C₄×C₆₇	—	4	16	no	—
171	C₂×C₄×C₆₇	—	4	32	no	—
172	C₂×C₄×C₆₇	#124	1	16	yes	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
176	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
177	C₂²×C₆₇	—	2	8	no	—
178	C₂²×C₆₇	—	2	8	no	—
179	C₂²×C₆₇	—	2	8	no	—
180	C₂²×C₆₇	—	4	32	no	—
181	C₂²×C₆₇	—	4	4	no	—
182	C₂²×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	#525	1	4	yes	—
186	C₂²×C₆₇	C₂×#124	1	4	yes	—
187	C₂²×C₆₇	#537	1	4	yes	—
188	C₂²×C₆₇	#552	1	8	yes	—
189	C₂²×C₆₇	#553	1	8	yes	—
190	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
191	C₂²×C₆₇	C₂²×#41	1	2	yes	—
192	C₂²×C₆₇	C₂²×#41	1	2	yes	—
193	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
195	C₄×C₆₇	—	2	4	no	—
196	C₄×C₆₇	—	2	4	no	—
197	C₄×C₆₇	—	4	4	no	—
198	C₄×C₆₇	—	4	4	no	—
199	C₄×C₆₇	—	4	4	no	—
200	C₄×C₆₇	—	4	8	no	—
201	C₄×C₆₇	C₂×#124	1	8	yes	—
202	C₂×#2562	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	4	8	no	—
204	C₂×C₆₇	#2562	1	8	yes	direct product factor
205	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
206	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
207	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
208	C₂×C₆₇	C₂×#552	1	2	yes	—
209	C₂×C₆₇	C₂×#553	1	2	yes	—
210	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
211	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
212	#2562	C₂×C₆₇	1	16	yes	direct product factor
213	C₂×#400	C₂×C₆₇	1	2	yes	complemented
214	C₂×#394	C₂×C₆₇	1	2	yes	—
215	C₂×#388	C₂×C₆₇	1	4	yes	complemented
216	C₂×#380	C₂×C₆₇	1	2	yes	complemented
217	C₂×#380	C₂×C₆₇	1	2	yes	complemented
218	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₆₇	C₂×#2562	1	1	yes	characteristic, direct product factor, Sylow
220	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
221	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
222	C₂×#125	—	2	8	no	—
223	C₂×#125	C₂²×C₆₇	1	8	yes	complemented
224	C₂×C₄×D₄	—	2	8	no	—
225	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
226	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
227	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
228	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
229	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
231	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
232	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
233	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
234	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
235	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
236	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
237	#400	C₂²×C₆₇	1	16	yes	complemented
238	#394	C₂²×C₆₇	1	16	yes	—
239	#388	C₂²×C₆₇	1	32	yes	complemented
240	#380	C₂²×C₆₇	1	16	yes	complemented
241	#380	C₂²×C₆₇	1	16	yes	complemented
242	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
243	C₂×#95	C₂²×C₆₇	1	2	yes	—
244	C₂×#95	C₂²×C₆₇	1	4	yes	—
245	C₂×#95	C₂²×C₆₇	1	4	yes	—
246	C₂⁵	C₂³×C₆₇	1	2	yes	—
247	C₂²×D₄	—	2	8	no	—
248	C₂²×D₄	—	2	4	no	—
249	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
250	C₂³×C₄	—	2	8	no	—
251	C₂³×C₄	—	2	8	no	—
252	C₂³×C₄	—	2	8	no	—
253	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
254	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
255	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
256	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
257	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
258	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
259	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
260	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
261	#125	—	2	32	no	—
262	C₄×D₄	—	2	32	no	—
263	C₂×C₄⋊C₄	—	2	4	no	—
264	C₂×C₄⋊C₄	—	2	8	no	—
265	C₂×C₄⋊C₄	—	2	8	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₄⋊C₄	—	2	4	no	—
268	C₂×C₄⋊C₄	—	2	8	no	—
269	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
270	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
271	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
272	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
273	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
274	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
275	C₂×C₂²⋊C₄	—	2	16	no	—
276	C₂×C₂²⋊C₄	—	2	8	no	—
277	C₂×C₂²⋊C₄	—	4	8	no	—
278	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
279	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
280	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
281	C₂×C₄²	—	4	4	no	—
282	C₂×C₄²	D₄×C₆₇	1	8	yes	—
283	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
284	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
285	#95	C₂³×C₆₇	1	8	yes	—
286	#95	C₂³×C₆₇	1	16	yes	—
287	#95	C₂³×C₆₇	1	16	yes	—
288	C₂⁴	—	2	4	no	—
289	C₂⁴	—	2	4	no	—
290	C₂⁴	—	2	4	no	—
291	C₂⁴	—	2	4	no	—
292	C₂⁴	—	2	4	no	—
293	C₂⁴	—	2	4	no	—
294	C₂⁴	#41×C₆₇	1	8	yes	—
295	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
296	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
297	C₂×D₄	—	2	32	no	—
298	C₂×D₄	—	2	16	no	—
299	C₂×D₄	—	2	8	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	8	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	8	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	16	no	—
315	C₂²×C₄	—	2	16	no	—
316	C₂²×C₄	—	2	16	no	—
317	C₂²×C₄	—	4	32	no	—
318	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
320	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
321	C₂²×C₄	#41×C₆₇	1	8	yes	—
322	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
323	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
324	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
325	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
326	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
327	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
328	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	16	no	—
331	C₄⋊C₄	—	2	16	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	16	no	—
335	C₂²⋊C₄	—	2	32	no	—
336	C₂²⋊C₄	—	2	16	no	—
337	C₂²⋊C₄	—	4	16	no	—
338	C₄²	—	4	8	no	—
339	C₄²	C₂×D₄×C₆₇	1	16	yes	—
340	C₂³	—	2	16	no	—
341	C₂³	—	2	16	no	—
342	C₂³	—	2	16	no	—
343	C₂³	—	2	4	no	—
344	C₂³	—	2	4	no	—
345	C₂³	—	2	4	no	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	4	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	4	8	no	—
350	C₂³	—	4	8	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	4	8	no	—
353	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
354	C₂³	C₂×#41×C₆₇	1	8	yes	—
355	C₂³	C₂×#41×C₆₇	1	4	yes	—
356	C₂³	C₂×#41×C₆₇	1	4	yes	—
357	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
358	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
359	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
360	D₄	—	2	32	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	8	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	4	8	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	—	4	16	no	—
381	C₂×C₄	—	4	32	no	—
382	C₂×C₄	#124×C₆₇	1	16	yes	—
383	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
384	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
385	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
386	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	8	no	—
389	C₂²	—	2	8	no	—
390	C₂²	—	4	32	no	—
391	C₂²	—	4	4	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	4	4	no	—
394	C₂²	—	4	4	no	—
395	C₂²	#525×C₆₇	1	4	yes	—
396	C₂²	C₂×#124×C₆₇	1	4	yes	—
397	C₂²	#537×C₆₇	1	4	yes	—
398	C₂²	#552×C₆₇	1	8	yes	—
399	C₂²	#553×C₆₇	1	8	yes	—
400	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
401	C₂²	C₂²×#41×C₆₇	1	2	yes	—
402	C₂²	C₂²×#41×C₆₇	1	2	yes	—
403	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
404	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
405	C₄	—	2	4	no	—
406	C₄	—	2	4	no	—
407	C₄	—	4	4	no	—
408	C₄	—	4	4	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	8	no	—
411	C₄	C₂×#124×C₆₇	1	8	yes	—
412	C₂	—	4	8	no	—
413	C₂	C₆₇×#2562	1	8	yes	direct product factor
414	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
415	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
416	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
417	C₂	C₂×#552×C₆₇	1	2	yes	—
418	C₂	C₂×#553×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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
1. C₂²×#125×C₆₇	C₂	semidirect
3. C₆₇×#2562	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
6. C₂×#388×C₆₇	C₂	semidirect
7. C₂×#380×C₆₇	C₂	semidirect
8. C₂×#380×C₆₇	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
13. C₂×#125×C₆₇	C₂²	semidirect
27. #400×C₆₇	C₂²	semidirect
29. #388×C₆₇	C₂²	semidirect
30. #380×C₆₇	C₂²	semidirect
31. #380×C₆₇	C₂²	semidirect
32. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
202. C₂×#2562	C₆₇	direct (N × H)
204. C₂×C₆₇	#2562	direct (N × H)
210. C₂²×#125	C₂×C₆₇	semidirect
212. #2562	C₂×C₆₇	direct (N × H)
213. C₂×#400	C₂×C₆₇	semidirect
215. C₂×#388	C₂×C₆₇	semidirect
216. C₂×#380	C₂×C₆₇	semidirect
217. C₂×#380	C₂×C₆₇	semidirect
218. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
219. C₆₇	C₂×#2562	direct (N × H)
223. C₂×#125	C₂²×C₆₇	semidirect
237. #400	C₂²×C₆₇	semidirect
239. #388	C₂²×C₆₇	semidirect
240. #380	C₂²×C₆₇	semidirect
241. #380	C₂²×C₆₇	semidirect
242. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
413. C₂	C₆₇×#2562	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×#125×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2562	2
C₂×#400×C₆₇	2
C₂×#394×C₆₇	2
C₂×#388×C₆₇	2
C₂×#380×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2562	67

Subgroups — C2×C67×#2562 #30714147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2562 #30714147