Subgroups — C₂²×C₇×#470 #13237656

Order 1792.

2878 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₂³×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₂×C₇×#470	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×#106	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	C₂²×C₇×M₁₆	—	2	8	no	—
12	C₂²×C₇×M₁₆	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×#470	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#106	C₂²	1	12	yes	complemented
22	C₂×C₇×#103	C₂²	1	24	yes	complemented
23	C₂²×#470	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	C₂×C₇×M₁₆	—	2	48	no	—
40	C₂×C₇×M₁₆	C₂³	1	16	yes	—
41	C₂²×C₇×C₈	—	2	4	no	—
42	C₂²×C₇×C₈	D₄	1	8	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×#106	C₂³	1	16	yes	complemented
52	C₇×#103	C₂³	1	32	yes	complemented
53	C₂³×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂³×M₁₆	C₂×C₇	1	1	yes	characteristic
55	C₂²×C₂²⋊Q₈	C₂×C₇	1	2	yes	—
56	C₂×#470	C₂×C₇	1	24	yes	direct product factor
57	C₂²×#106	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	8	no	—
61	C₂⁴×C₇	C₂×D₄	1	6	yes	—
62	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₇×Q₈	—	2	12	no	—
64	C₂×C₇×Q₈	—	2	24	no	—
65	C₂×C₇×Q₈	—	2	24	no	—
66	C₂×C₇×Q₈	—	4	24	no	—
67	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₇	—	2	4	no	—
69	C₂²×C₄×C₇	—	2	24	no	—
70	C₂²×C₄×C₇	—	2	12	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	—	4	8	no	—
75	C₂²×C₄×C₇	—	4	8	no	—
76	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₇	#41	1	4	yes	—
81	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
83	C₇×Q₁₆	—	2	32	no	—
84	C₇×M₁₆	—	2	32	no	—
85	C₂×C₇×C₈	—	2	12	no	—
86	C₂×C₇×C₈	C₂×D₄	1	24	yes	—
87	C₂×C₇×C₈	C₂⁴	1	4	yes	—
88	C₂×C₇×C₈	C₂⁴	1	4	yes	—
89	C₇×C₄⋊C₄	—	2	32	no	—
90	C₇×C₄⋊C₄	—	2	12	no	—
91	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
92	C₇×C₂²⋊C₄	—	2	32	no	—
93	C₂³×Q₈	C₂²×C₇	1	2	yes	—
94	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
95	C₂²×Q₁₆	—	2	8	no	—
96	C₂²×Q₁₆	C₂²×C₇	1	12	yes	complemented
97	C₂²×M₁₆	—	2	8	no	—
98	C₂²×M₁₆	C₂²×C₇	1	12	yes	—
99	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
100	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₂²⋊Q₈	—	2	12	no	—
102	C₂×C₂²⋊Q₈	C₂²×C₇	1	24	yes	—
103	C₂²×C₄⋊C₄	—	2	2	no	—
104	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
105	C₂²×C₂²⋊C₄	—	2	2	no	—
106	#470	C₂²×C₇	1	64	yes	direct product factor
107	C₂×#106	C₂²×C₇	1	12	yes	complemented
108	C₂×#103	C₂²×C₇	1	24	yes	complemented
109	C₂³×C₇	—	2	12	no	—
110	C₂³×C₇	—	2	24	no	—
111	C₂³×C₇	—	4	16	no	—
112	C₂³×C₇	#121	1	4	yes	—
113	C₂³×C₇	#137	1	8	yes	—
114	C₂³×C₇	C₂²×D₄	1	4	yes	—
115	C₂³×C₇	C₂²×D₄	1	3	yes	—
116	C₇×Q₈	—	2	8	no	—
117	C₇×Q₈	—	2	8	no	—
118	C₇×Q₈	—	4	24	no	—
119	C₂×C₄×C₇	—	2	12	no	—
120	C₂×C₄×C₇	—	2	24	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	8	no	—
124	C₂×C₄×C₇	—	2	8	no	—
125	C₂×C₄×C₇	—	4	24	no	—
126	C₂×C₄×C₇	—	4	24	no	—
127	C₂×C₄×C₇	#121	1	12	yes	—
128	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
129	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
130	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
131	C₂×C₄×C₇	C₂×#41	1	6	yes	—
132	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
133	C₇×C₈	—	2	4	no	—
134	C₇×C₈	C₂²×D₄	1	8	yes	—
135	C₂⁵	C₇×D₄	1	1	yes	characteristic
136	C₂²×Q₈	—	2	6	no	—
137	C₂²×Q₈	—	2	8	no	—
138	C₂²×Q₈	—	2	8	no	—
139	C₂²×Q₈	C₂³×C₇	1	12	yes	—
140	C₂³×C₄	—	2	4	no	—
141	C₂³×C₄	—	2	4	no	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	—	2	2	no	—
144	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
145	C₂³×C₄	C₇×D₄	1	6	yes	—
146	C₂³×C₄	C₂³×C₇	1	6	yes	—
147	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
148	C₂×Q₁₆	—	2	48	no	—
149	C₂×Q₁₆	C₂³×C₇	1	16	yes	complemented
150	C₂×M₁₆	—	2	48	no	—
151	C₂×M₁₆	C₂³×C₇	1	16	yes	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₇×D₄	1	8	yes	—
154	C₂²×C₈	C₂³×C₇	1	6	yes	—
155	C₂²×C₈	C₂³×C₇	1	6	yes	—
156	C₂²⋊Q₈	—	2	48	no	—
157	C₂²⋊Q₈	C₂³×C₇	1	32	yes	—
158	C₂×C₄⋊C₄	—	2	24	no	—
159	C₂×C₄⋊C₄	—	2	6	no	—
160	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
161	C₂×C₂²⋊C₄	—	2	24	no	—
162	#106	C₂³×C₇	1	16	yes	complemented
163	#103	C₂³×C₇	1	32	yes	complemented
164	C₂²×C₇	—	2	4	no	—
165	C₂²×C₇	—	2	8	no	—
166	C₂²×C₇	—	4	24	no	—
167	C₂²×C₇	#470	1	16	yes	direct product factor
168	C₂²×C₇	C₂×#121	1	6	yes	—
169	C₂²×C₇	C₂×#137	1	12	yes	—
170	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
171	C₄×C₇	—	2	4	no	—
172	C₄×C₇	—	4	8	no	—
173	C₄×C₇	—	4	8	no	—
174	C₄×C₇	C₂×#121	1	6	yes	—
175	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
176	C₄×C₇	C₂²×#41	1	1	yes	characteristic
177	C₂⁴	—	2	4	no	—
178	C₂⁴	—	2	8	no	—
179	C₂⁴	C₂×C₇×D₄	1	6	yes	—
180	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂×Q₈	—	2	12	no	—
182	C₂×Q₈	—	2	24	no	—
183	C₂×Q₈	—	2	24	no	—
184	C₂×Q₈	—	4	24	no	—
185	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	—	2	12	no	—
189	C₂²×C₄	—	2	12	no	—
190	C₂²×C₄	—	2	12	no	—
191	C₂²×C₄	—	2	12	no	—
192	C₂²×C₄	—	4	8	no	—
193	C₂²×C₄	—	4	8	no	—
194	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
195	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
196	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
197	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
198	C₂²×C₄	C₇×#41	1	4	yes	—
199	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
201	Q₁₆	—	2	32	no	—
202	M₁₆	—	2	32	no	—
203	C₂×C₈	—	2	12	no	—
204	C₂×C₈	C₂×C₇×D₄	1	24	yes	—
205	C₂×C₈	C₂⁴×C₇	1	4	yes	—
206	C₂×C₈	C₂⁴×C₇	1	4	yes	—
207	C₄⋊C₄	—	2	32	no	—
208	C₄⋊C₄	—	2	12	no	—
209	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
210	C₂²⋊C₄	—	2	32	no	—
211	C₂×C₇	—	4	4	no	—
212	C₂×C₇	C₂×#470	1	12	yes	direct product factor
213	C₂×C₇	C₂²×#121	1	1	yes	characteristic
214	C₂×C₇	C₂²×#137	1	2	yes	—
215	C₂³	—	2	12	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	4	16	no	—
218	C₂³	C₇×#121	1	4	yes	—
219	C₂³	C₇×#137	1	8	yes	—
220	C₂³	C₂²×C₇×D₄	1	4	yes	—
221	C₂³	C₂²×C₇×D₄	1	3	yes	—
222	Q₈	—	2	8	no	—
223	Q₈	—	2	8	no	—
224	Q₈	—	4	24	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	24	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	4	24	no	—
232	C₂×C₄	—	4	24	no	—
233	C₂×C₄	C₇×#121	1	12	yes	—
234	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
235	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
236	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
237	C₂×C₄	C₂×C₇×#41	1	6	yes	—
238	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	C₂²×C₇×D₄	1	8	yes	—
241	C₇	C₂²×#470	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²	—	2	8	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	24	no	—
245	C₂²	C₇×#470	1	16	yes	direct product factor
246	C₂²	C₂×C₇×#121	1	6	yes	—
247	C₂²	C₂×C₇×#137	1	12	yes	—
248	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	4	8	no	—
251	C₄	—	4	8	no	—
252	C₄	C₂×C₇×#121	1	6	yes	—
253	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
254	C₄	C₂²×C₇×#41	1	1	yes	characteristic
255	C₂	—	4	4	no	—
256	C₂	C₂×C₇×#470	1	12	yes	direct product factor
257	C₂	C₂²×C₇×#121	1	1	yes	characteristic
258	C₂	C₂²×C₇×#137	1	2	yes	—

	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×#470	C₂	direct (N × H)
5. C₂²×C₇×#106	C₂	semidirect
6. C₂²×C₇×#103	C₂	semidirect
10. C₂²×C₇×Q₁₆	C₂²	semidirect
20. C₇×#470	C₂²	direct (N × H)
21. C₂×C₇×#106	C₂²	semidirect
22. C₂×C₇×#103	C₂²	semidirect
23. C₂²×#470	C₇	direct (N × H)
38. C₂×C₇×Q₁₆	C₂³	semidirect
51. C₇×#106	C₂³	semidirect
52. C₇×#103	C₂³	semidirect
53. C₂³×Q₁₆	C₂×C₇	semidirect
56. C₂×#470	C₂×C₇	direct (N × H)
57. C₂²×#106	C₂×C₇	semidirect
58. C₂²×#103	C₂×C₇	semidirect
96. C₂²×Q₁₆	C₂²×C₇	semidirect
106. #470	C₂²×C₇	direct (N × H)
107. C₂×#106	C₂²×C₇	semidirect
108. C₂×#103	C₂²×C₇	semidirect
149. C₂×Q₁₆	C₂³×C₇	semidirect
162. #106	C₂³×C₇	semidirect
163. #103	C₂³×C₇	semidirect
167. C₂²×C₇	#470	direct (N × H)
212. C₂×C₇	C₂×#470	direct (N × H)
241. C₇	C₂²×#470	direct (N × H)
245. C₂²	C₇×#470	direct (N × H)
256. C₂	C₂×C₇×#470	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
C₂³×C₇×M₁₆	2
C₂²×C₇×C₂²⋊Q₈	2
C₂×C₇×#470	2
C₂²×C₇×#106	2
C₂²×C₇×#103	2
C₂²×#470	7

Subgroups — C22×C7×#470 #13237656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#470 #13237656