Subgroups — C₂²×C₇×#467 #13237652

Order 1792.

3006 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₂×C₇×#467	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#103	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₇×Q₁₆	—	2	8	no	—
10	C₂²×C₇×Q₁₆	C₂²	1	12	yes	complemented
11	C₂³×C₇×C₈	—	2	8	no	—
12	C₂³×C₇×C₈	C₂²	1	12	yes	—
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₂²⋊Q₈	—	2	12	no	—
16	C₂×C₇×C₂²⋊Q₈	C₂²	1	24	yes	—
17	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
18	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
20	C₇×#467	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#107	C₂²	1	12	yes	complemented
22	C₂×C₇×#103	C₂²	1	24	yes	complemented
23	C₂²×#467	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₇	D₄	1	1	yes	characteristic
25	C₂²×C₇×Q₈	—	2	6	no	—
26	C₂²×C₇×Q₈	—	2	8	no	—
27	C₂²×C₇×Q₈	—	2	8	no	—
28	C₂²×C₇×Q₈	C₂³	1	12	yes	—
29	C₂³×C₄×C₇	—	2	4	no	—
30	C₂³×C₄×C₇	—	2	4	no	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂³×C₄×C₇	—	2	2	no	—
33	C₂³×C₄×C₇	D₄	1	6	yes	—
34	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₇	C₂³	1	6	yes	—
36	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂×C₇×Q₁₆	—	2	48	no	—
38	C₂×C₇×Q₁₆	C₂³	1	16	yes	complemented
39	C₂²×C₇×C₈	—	2	4	no	—
40	C₂²×C₇×C₈	—	2	48	no	—
41	C₂²×C₇×C₈	D₄	1	8	yes	—
42	C₂²×C₇×C₈	C₂³	1	16	yes	—
43	C₂²×C₇×C₈	C₂³	1	6	yes	—
44	C₂²×C₇×C₈	C₂³	1	6	yes	—
45	C₇×C₂²⋊Q₈	—	2	48	no	—
46	C₇×C₂²⋊Q₈	C₂³	1	32	yes	—
47	C₂×C₇×C₄⋊C₄	—	2	24	no	—
48	C₂×C₇×C₄⋊C₄	—	2	6	no	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
50	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
51	C₇×#107	C₂³	1	16	yes	complemented
52	C₇×#103	C₂³	1	32	yes	complemented
53	C₂³×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic
55	C₂²×C₂²⋊Q₈	C₂×C₇	1	2	yes	—
56	C₂×#467	C₂×C₇	1	24	yes	direct product factor
57	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂²×#103	C₂×C₇	1	2	yes	complemented
59	C₂⁴×C₇	—	2	4	no	—
60	C₂⁴×C₇	—	2	4	no	—
61	C₂⁴×C₇	Q₁₆	1	8	yes	complemented
62	C₂⁴×C₇	C₂×D₄	1	6	yes	—
63	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×Q₈	—	2	12	no	—
65	C₂×C₇×Q₈	—	2	24	no	—
66	C₂×C₇×Q₈	—	2	24	no	—
67	C₂×C₇×Q₈	—	4	24	no	—
68	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	—	2	24	no	—
71	C₂²×C₄×C₇	—	2	12	no	—
72	C₂²×C₄×C₇	—	2	12	no	—
73	C₂²×C₄×C₇	—	2	12	no	—
74	C₂²×C₄×C₇	—	2	12	no	—
75	C₂²×C₄×C₇	—	4	8	no	—
76	C₂²×C₄×C₇	—	4	8	no	—
77	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₇	#41	1	4	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
84	C₇×Q₁₆	—	2	32	no	—
85	C₂×C₇×C₈	—	2	12	no	—
86	C₂×C₇×C₈	—	2	32	no	—
87	C₂×C₇×C₈	C₂×D₄	1	24	yes	—
88	C₂×C₇×C₈	C₂⁴	1	4	yes	—
89	C₂×C₇×C₈	C₂⁴	1	4	yes	—
90	C₇×C₄⋊C₄	—	2	32	no	—
91	C₇×C₄⋊C₄	—	2	12	no	—
92	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
93	C₇×C₂²⋊C₄	—	2	32	no	—
94	C₂³×Q₈	C₂²×C₇	1	2	yes	—
95	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
96	C₂²×Q₁₆	—	2	8	no	—
97	C₂²×Q₁₆	C₂²×C₇	1	12	yes	complemented
98	C₂³×C₈	—	2	8	no	—
99	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
100	C₂³×C₈	C₂²×C₇	1	12	yes	—
101	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₂²⋊Q₈	—	2	12	no	—
103	C₂×C₂²⋊Q₈	C₂²×C₇	1	24	yes	—
104	C₂²×C₄⋊C₄	—	2	2	no	—
105	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
106	C₂²×C₂²⋊C₄	—	2	2	no	—
107	#467	C₂²×C₇	1	64	yes	direct product factor
108	C₂×#107	C₂²×C₇	1	12	yes	complemented
109	C₂×#103	C₂²×C₇	1	24	yes	complemented
110	C₂³×C₇	—	2	12	no	—
111	C₂³×C₇	—	2	12	no	—
112	C₂³×C₇	—	2	32	no	—
113	C₂³×C₇	#121	1	4	yes	—
114	C₂³×C₇	C₂×Q₁₆	1	4	yes	—
115	C₂³×C₇	C₂×Q₁₆	1	24	yes	complemented
116	C₂³×C₇	#135	1	4	yes	—
117	C₂³×C₇	C₂²×D₄	1	4	yes	—
118	C₂³×C₇	C₂²×D₄	1	3	yes	—
119	C₇×Q₈	—	2	8	no	—
120	C₇×Q₈	—	2	8	no	—
121	C₇×Q₈	—	4	24	no	—
122	C₂×C₄×C₇	—	2	12	no	—
123	C₂×C₄×C₇	—	2	24	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	8	no	—
127	C₂×C₄×C₇	—	2	8	no	—
128	C₂×C₄×C₇	—	4	24	no	—
129	C₂×C₄×C₇	—	4	24	no	—
130	C₂×C₄×C₇	#121	1	12	yes	—
131	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
132	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
133	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₇	C₂×#41	1	6	yes	—
135	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
136	C₇×C₈	—	2	4	no	—
137	C₇×C₈	C₂²×D₄	1	8	yes	—
138	C₂⁵	C₇×D₄	1	1	yes	characteristic
139	C₂²×Q₈	—	2	6	no	—
140	C₂²×Q₈	—	2	8	no	—
141	C₂²×Q₈	—	2	8	no	—
142	C₂²×Q₈	C₂³×C₇	1	12	yes	—
143	C₂³×C₄	—	2	4	no	—
144	C₂³×C₄	—	2	4	no	—
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
148	C₂³×C₄	C₇×D₄	1	6	yes	—
149	C₂³×C₄	C₂³×C₇	1	6	yes	—
150	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
151	C₂×Q₁₆	—	2	48	no	—
152	C₂×Q₁₆	C₂³×C₇	1	16	yes	complemented
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	2	48	no	—
155	C₂²×C₈	C₇×D₄	1	8	yes	—
156	C₂²×C₈	C₂³×C₇	1	6	yes	—
157	C₂²×C₈	C₂³×C₇	1	16	yes	—
158	C₂²×C₈	C₂³×C₇	1	6	yes	—
159	C₂²⋊Q₈	—	2	48	no	—
160	C₂²⋊Q₈	C₂³×C₇	1	32	yes	—
161	C₂×C₄⋊C₄	—	2	24	no	—
162	C₂×C₄⋊C₄	—	2	6	no	—
163	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
164	C₂×C₂²⋊C₄	—	2	24	no	—
165	#107	C₂³×C₇	1	16	yes	complemented
166	#103	C₂³×C₇	1	32	yes	complemented
167	C₂²×C₇	—	2	4	no	—
168	C₂²×C₇	—	2	48	no	—
169	C₂²×C₇	—	2	4	no	—
170	C₂²×C₇	#467	1	16	yes	direct product factor
171	C₂²×C₇	C₂×#121	1	6	yes	—
172	C₂²×C₇	C₂²×Q₁₆	1	6	yes	—
173	C₂²×C₇	C₂²×Q₁₆	1	8	yes	complemented
174	C₂²×C₇	C₂×#135	1	6	yes	—
175	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
176	C₄×C₇	—	2	4	no	—
177	C₄×C₇	—	4	8	no	—
178	C₄×C₇	—	4	8	no	—
179	C₄×C₇	C₂×#121	1	6	yes	—
180	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
181	C₄×C₇	C₂²×#41	1	1	yes	characteristic
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	C₇×Q₁₆	1	8	yes	complemented
185	C₂⁴	C₂×C₇×D₄	1	6	yes	—
186	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
187	C₂×Q₈	—	2	12	no	—
188	C₂×Q₈	—	2	24	no	—
189	C₂×Q₈	—	2	24	no	—
190	C₂×Q₈	—	4	24	no	—
191	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	12	no	—
195	C₂²×C₄	—	2	12	no	—
196	C₂²×C₄	—	2	12	no	—
197	C₂²×C₄	—	2	12	no	—
198	C₂²×C₄	—	4	8	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
201	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
202	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
203	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
204	C₂²×C₄	C₇×#41	1	4	yes	—
205	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
206	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
207	Q₁₆	—	2	32	no	—
208	C₂×C₈	—	2	12	no	—
209	C₂×C₈	—	2	32	no	—
210	C₂×C₈	C₂×C₇×D₄	1	24	yes	—
211	C₂×C₈	C₂⁴×C₇	1	4	yes	—
212	C₂×C₈	C₂⁴×C₇	1	4	yes	—
213	C₄⋊C₄	—	2	32	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
216	C₂²⋊C₄	—	2	32	no	—
217	C₂×C₇	—	2	8	no	—
218	C₂×C₇	C₂×#467	1	12	yes	direct product factor
219	C₂×C₇	C₂²×#121	1	1	yes	characteristic
220	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
221	C₂×C₇	C₂²×#135	1	1	yes	characteristic
222	C₂³	—	2	32	no	—
223	C₂³	—	2	12	no	—
224	C₂³	—	2	12	no	—
225	C₂³	C₇×#121	1	4	yes	—
226	C₂³	C₂×C₇×Q₁₆	1	4	yes	—
227	C₂³	C₂×C₇×Q₁₆	1	24	yes	complemented
228	C₂³	C₇×#135	1	4	yes	—
229	C₂³	C₂²×C₇×D₄	1	4	yes	—
230	C₂³	C₂²×C₇×D₄	1	3	yes	—
231	Q₈	—	2	8	no	—
232	Q₈	—	2	8	no	—
233	Q₈	—	4	24	no	—
234	C₂×C₄	—	2	12	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	24	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	24	no	—
241	C₂×C₄	—	4	24	no	—
242	C₂×C₄	C₇×#121	1	12	yes	—
243	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
244	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
245	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
246	C₂×C₄	C₂×C₇×#41	1	6	yes	—
247	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	C₂²×C₇×D₄	1	8	yes	—
250	C₇	C₂²×#467	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	48	no	—
254	C₂²	C₇×#467	1	16	yes	direct product factor
255	C₂²	C₂×C₇×#121	1	6	yes	—
256	C₂²	C₂²×C₇×Q₁₆	1	6	yes	—
257	C₂²	C₂²×C₇×Q₁₆	1	8	yes	complemented
258	C₂²	C₂×C₇×#135	1	6	yes	—
259	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	4	8	no	—
263	C₄	C₂×C₇×#121	1	6	yes	—
264	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
265	C₄	C₂²×C₇×#41	1	1	yes	characteristic
266	C₂	—	2	8	no	—
267	C₂	C₂×C₇×#467	1	12	yes	direct product factor
268	C₂	C₂²×C₇×#121	1	1	yes	characteristic
269	C₂	C₂³×C₇×Q₁₆	1	1	yes	characteristic
270	C₂	C₂²×C₇×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×Q₁₆	C₂	semidirect
4. C₂×C₇×#467	C₂	direct (N × H)
5. C₂²×C₇×#107	C₂	semidirect
6. C₂²×C₇×#103	C₂	semidirect
10. C₂²×C₇×Q₁₆	C₂²	semidirect
20. C₇×#467	C₂²	direct (N × H)
21. C₂×C₇×#107	C₂²	semidirect
22. C₂×C₇×#103	C₂²	semidirect
23. C₂²×#467	C₇	direct (N × H)
38. C₂×C₇×Q₁₆	C₂³	semidirect
51. C₇×#107	C₂³	semidirect
52. C₇×#103	C₂³	semidirect
53. C₂³×Q₁₆	C₂×C₇	semidirect
56. C₂×#467	C₂×C₇	direct (N × H)
57. C₂²×#107	C₂×C₇	semidirect
58. C₂²×#103	C₂×C₇	semidirect
61. C₂⁴×C₇	Q₁₆	semidirect
97. C₂²×Q₁₆	C₂²×C₇	semidirect
107. #467	C₂²×C₇	direct (N × H)
108. C₂×#107	C₂²×C₇	semidirect
109. C₂×#103	C₂²×C₇	semidirect
115. C₂³×C₇	C₂×Q₁₆	semidirect
152. C₂×Q₁₆	C₂³×C₇	semidirect
165. #107	C₂³×C₇	semidirect
166. #103	C₂³×C₇	semidirect
170. C₂²×C₇	#467	direct (N × H)
173. C₂²×C₇	C₂²×Q₁₆	semidirect
184. C₂⁴	C₇×Q₁₆	semidirect
218. C₂×C₇	C₂×#467	direct (N × H)
227. C₂³	C₂×C₇×Q₁₆	semidirect
250. C₇	C₂²×#467	direct (N × H)
254. C₂²	C₇×#467	direct (N × H)
257. C₂²	C₂²×C₇×Q₁₆	semidirect
267. C₂	C₂×C₇×#467	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×Q₁₆	2
C₂⁴×C₇×C₈	2
C₂²×C₇×C₂²⋊Q₈	2
C₂×C₇×#467	2
C₂²×C₇×#107	2
C₂²×C₇×#103	2
C₂²×#467	7

Subgroups — C22×C7×#467 #13237652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#467 #13237652