Subgroups — C₂×C₆₇×#2566 #30714142

Order 17152.

3094 conjugacy classes of subgroups.

The subgroup lattice has 432 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×#2566	C₂	1	16	yes	maximal, direct product factor
4	C₂×#396×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#385×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
13	C₂×#125×C₆₇	—	2	8	no	—
14	C₂×#125×C₆₇	C₂²	1	8	yes	complemented
15	C₂×C₄×D₄×C₆₇	—	2	8	no	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	#396×C₆₇	C₂²	1	32	yes	complemented
31	#390×C₆₇	C₂²	1	8	yes	—
32	#390×C₆₇	C₂²	1	8	yes	—
33	#385×C₆₇	C₂²	1	32	yes	complemented
34	#384×C₆₇	C₂²	1	8	yes	complemented
35	#382×C₆₇	C₂²	1	8	yes	complemented
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#95×C₆₇	C₂²	1	2	yes	—
38	C₂×#95×C₆₇	C₂²	1	4	yes	—
39	C₂⁵×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	—	2	4	no	—
41	C₂²×D₄×C₆₇	—	2	4	no	—
42	C₂²×D₄×C₆₇	Q₈	1	8	yes	—
43	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	—	2	8	no	—
45	C₂³×C₄×C₆₇	—	4	8	no	—
46	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	#125×C₆₇	—	2	32	no	—
56	C₄×D₄×C₆₇	—	2	32	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
77	C₂×C₄²×C₆₇	—	2	4	no	—
78	C₂×C₄²×C₆₇	—	2	4	no	—
79	C₂×C₄²×C₆₇	D₄	1	8	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
83	#95×C₆₇	C₂³	1	8	yes	—
84	#95×C₆₇	C₂³	1	16	yes	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	8	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	8	no	—
89	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
90	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
91	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₂×D₄×C₆₇	—	2	32	no	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	2	8	no	—
95	C₂×D₄×C₆₇	C₂×Q₈	1	16	yes	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	2	4	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	4	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	4	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	16	no	—
110	C₂²×C₄×C₆₇	—	4	16	no	—
111	C₂²×C₄×C₆₇	—	4	32	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
116	C₂²×C₄×C₆₇	#41	1	8	yes	—
117	C₂²×C₄×C₆₇	#41	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	8	no	—
127	C₄⋊C₄×C₆₇	—	2	16	no	—
128	C₄⋊C₄×C₆₇	—	2	16	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	16	no	—
132	C₄⋊C₄×C₆₇	—	4	16	no	—
133	C₂²⋊C₄×C₆₇	—	2	16	no	—
134	C₂²⋊C₄×C₆₇	—	2	32	no	—
135	C₂²⋊C₄×C₆₇	—	4	16	no	—
136	C₄²×C₆₇	—	2	8	no	—
137	C₄²×C₆₇	—	2	8	no	—
138	C₄²×C₆₇	C₂×D₄	1	16	yes	—
139	C₂³×C₆₇	—	2	16	no	—
140	C₂³×C₆₇	—	2	32	no	—
141	C₂³×C₆₇	—	2	4	no	—
142	C₂³×C₆₇	—	2	8	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	8	no	—
145	C₂³×C₆₇	—	4	8	no	—
146	C₂³×C₆₇	—	4	8	no	—
147	C₂³×C₆₇	—	4	16	no	—
148	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
149	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
150	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
151	C₂³×C₆₇	C₂×#41	1	2	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	—	2	32	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	16	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	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	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	4	32	no	—
173	C₂×C₄×C₆₇	—	4	8	no	—
174	C₂×C₄×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	—	4	8	no	—
176	C₂×C₄×C₆₇	—	4	8	no	—
177	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
181	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
182	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
183	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
184	C₂²×C₆₇	—	2	8	no	—
185	C₂²×C₆₇	—	2	16	no	—
186	C₂²×C₆₇	—	4	32	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	4	8	no	—
190	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
191	C₂²×C₆₇	#532	1	4	yes	—
192	C₂²×C₆₇	#535	1	4	yes	—
193	C₂²×C₆₇	#553	1	8	yes	—
194	C₂²×C₆₇	#554	1	8	yes	—
195	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
196	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
197	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
198	C₂²×C₆₇	C₂²×#41	1	2	yes	—
199	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
200	C₄×C₆₇	—	2	4	no	—
201	C₄×C₆₇	—	2	4	no	—
202	C₄×C₆₇	—	2	8	no	—
203	C₄×C₆₇	—	4	4	no	—
204	C₄×C₆₇	—	4	4	no	—
205	C₄×C₆₇	—	4	4	no	—
206	C₄×C₆₇	—	4	4	no	—
207	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
208	C₂×#2566	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	4	8	no	—
210	C₂×C₆₇	#2566	1	8	yes	direct product factor
211	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
213	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
214	C₂×C₆₇	C₂×#553	1	2	yes	—
215	C₂×C₆₇	C₂×#554	1	2	yes	—
216	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
218	#2566	C₂×C₆₇	1	16	yes	direct product factor
219	C₂×#396	C₂×C₆₇	1	4	yes	complemented
220	C₂×#390	C₂×C₆₇	1	1	yes	characteristic
221	C₂×#390	C₂×C₆₇	1	1	yes	characteristic
222	C₂×#385	C₂×C₆₇	1	4	yes	complemented
223	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₂×#382	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₆₇	C₂×#2566	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
228	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
229	C₂×#125	—	2	8	no	—
230	C₂×#125	C₂²×C₆₇	1	8	yes	complemented
231	C₂×C₄×D₄	—	2	8	no	—
232	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
233	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
234	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
235	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
236	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
238	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
239	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
240	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
241	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
242	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
243	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
244	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
245	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
246	#396	C₂²×C₆₇	1	32	yes	complemented
247	#390	C₂²×C₆₇	1	8	yes	—
248	#390	C₂²×C₆₇	1	8	yes	—
249	#385	C₂²×C₆₇	1	32	yes	complemented
250	#384	C₂²×C₆₇	1	8	yes	complemented
251	#382	C₂²×C₆₇	1	8	yes	complemented
252	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
253	C₂×#95	C₂²×C₆₇	1	4	yes	—
254	C₂×#95	C₂²×C₆₇	1	2	yes	—
255	C₂⁵	C₂³×C₆₇	1	2	yes	—
256	C₂²×D₄	—	2	4	no	—
257	C₂²×D₄	—	2	4	no	—
258	C₂²×D₄	Q₈×C₆₇	1	8	yes	—
259	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
260	C₂³×C₄	—	2	8	no	—
261	C₂³×C₄	—	4	8	no	—
262	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
263	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	2	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	8	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	#125	—	2	32	no	—
272	C₄×D₄	—	2	32	no	—
273	C₂×C₄⋊C₄	—	2	4	no	—
274	C₂×C₄⋊C₄	—	2	8	no	—
275	C₂×C₄⋊C₄	—	2	8	no	—
276	C₂×C₄⋊C₄	—	2	4	no	—
277	C₂×C₄⋊C₄	—	2	4	no	—
278	C₂×C₄⋊C₄	—	2	8	no	—
279	C₂×C₄⋊C₄	—	4	8	no	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
287	C₂×C₂²⋊C₄	—	2	8	no	—
288	C₂×C₂²⋊C₄	—	2	16	no	—
289	C₂×C₂²⋊C₄	—	4	8	no	—
290	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
293	C₂×C₄²	—	2	4	no	—
294	C₂×C₄²	—	2	4	no	—
295	C₂×C₄²	D₄×C₆₇	1	8	yes	—
296	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
297	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
298	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
299	#95	C₂³×C₆₇	1	8	yes	—
300	#95	C₂³×C₆₇	1	16	yes	—
301	C₂⁴	—	2	4	no	—
302	C₂⁴	—	2	8	no	—
303	C₂⁴	—	2	4	no	—
304	C₂⁴	—	2	8	no	—
305	C₂⁴	C₂×Q₈×C₆₇	1	8	yes	—
306	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
307	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
308	C₂×D₄	—	2	32	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	C₂×Q₈×C₆₇	1	16	yes	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	2	16	no	—
326	C₂²×C₄	—	4	16	no	—
327	C₂²×C₄	—	4	32	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	4	yes	—
331	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
332	C₂²×C₄	#41×C₆₇	1	8	yes	—
333	C₂²×C₄	#41×C₆₇	1	4	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
337	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
338	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
339	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
341	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
342	C₄⋊C₄	—	2	8	no	—
343	C₄⋊C₄	—	2	16	no	—
344	C₄⋊C₄	—	2	16	no	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	16	no	—
348	C₄⋊C₄	—	4	16	no	—
349	C₂²⋊C₄	—	2	16	no	—
350	C₂²⋊C₄	—	2	32	no	—
351	C₂²⋊C₄	—	4	16	no	—
352	C₄²	—	2	8	no	—
353	C₄²	—	2	8	no	—
354	C₄²	C₂×D₄×C₆₇	1	16	yes	—
355	C₂³	—	2	16	no	—
356	C₂³	—	2	32	no	—
357	C₂³	—	2	4	no	—
358	C₂³	—	2	8	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	8	no	—
361	C₂³	—	4	8	no	—
362	C₂³	—	4	8	no	—
363	C₂³	—	4	16	no	—
364	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
365	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
366	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
367	C₂³	C₂×#41×C₆₇	1	2	yes	—
368	C₂³	C₂×#41×C₆₇	1	4	yes	—
369	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
370	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
371	D₄	—	2	32	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	2	16	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	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	8	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	4	32	no	—
389	C₂×C₄	—	4	8	no	—
390	C₂×C₄	—	4	8	no	—
391	C₂×C₄	—	4	8	no	—
392	C₂×C₄	—	4	8	no	—
393	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
394	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
395	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
396	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
397	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
398	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
399	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
400	C₂²	—	2	8	no	—
401	C₂²	—	2	16	no	—
402	C₂²	—	4	32	no	—
403	C₂²	—	4	4	no	—
404	C₂²	—	4	4	no	—
405	C₂²	—	4	8	no	—
406	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
407	C₂²	#532×C₆₇	1	4	yes	—
408	C₂²	#535×C₆₇	1	4	yes	—
409	C₂²	#553×C₆₇	1	8	yes	—
410	C₂²	#554×C₆₇	1	8	yes	—
411	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
412	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
413	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
414	C₂²	C₂²×#41×C₆₇	1	2	yes	—
415	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
416	C₄	—	2	4	no	—
417	C₄	—	2	4	no	—
418	C₄	—	2	8	no	—
419	C₄	—	4	4	no	—
420	C₄	—	4	4	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	4	no	—
423	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
424	C₂	—	4	8	no	—
425	C₂	C₆₇×#2566	1	8	yes	direct product factor
426	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
427	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
428	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
429	C₂	C₂×#553×C₆₇	1	2	yes	—
430	C₂	C₂×#554×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×#2566	C₂	direct (N × H)
4. C₂×#396×C₆₇	C₂	semidirect
7. C₂×#385×C₆₇	C₂	semidirect
8. C₂×#384×C₆₇	C₂	semidirect
9. C₂×#382×C₆₇	C₂	semidirect
10. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
14. C₂×#125×C₆₇	C₂²	semidirect
30. #396×C₆₇	C₂²	semidirect
33. #385×C₆₇	C₂²	semidirect
34. #384×C₆₇	C₂²	semidirect
35. #382×C₆₇	C₂²	semidirect
36. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
208. C₂×#2566	C₆₇	direct (N × H)
210. C₂×C₆₇	#2566	direct (N × H)
216. C₂²×#125	C₂×C₆₇	semidirect
218. #2566	C₂×C₆₇	direct (N × H)
219. C₂×#396	C₂×C₆₇	semidirect
222. C₂×#385	C₂×C₆₇	semidirect
223. C₂×#384	C₂×C₆₇	semidirect
224. C₂×#382	C₂×C₆₇	semidirect
225. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
226. C₆₇	C₂×#2566	direct (N × H)
230. C₂×#125	C₂²×C₆₇	semidirect
246. #396	C₂²×C₆₇	semidirect
249. #385	C₂²×C₆₇	semidirect
250. #384	C₂²×C₆₇	semidirect
251. #382	C₂²×C₆₇	semidirect
252. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
425. C₂	C₆₇×#2566	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×#2566	2
C₂×#396×C₆₇	2
C₂×#390×C₆₇	2
C₂×#385×C₆₇	2
C₂×#384×C₆₇	2
C₂×#382×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2566	67

Subgroups — C2×C67×#2566 #30714142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2566 #30714142