Subgroups — C₂²×C₇×#475 #13237692

Order 1792.

2358 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#475	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#106	C₂	1	2	yes	maximal
5	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×#103	C₂	1	2	yes	maximal
7	C₂³×C₇×Q₈	—	2	1	no	—
8	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊Q₈	—	2	6	no	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	12	yes	—
12	C₂×C₄×C₇×Q₈	—	2	6	no	—
13	C₂×C₄×C₇×Q₈	C₂²	1	12	yes	—
14	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
15	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇	—	2	1	no	—
19	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₇×#475	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#106	C₂²	1	24	yes	—
22	C₂×C₇×C₄⋊C₈	C₂²	1	12	yes	—
23	C₂×C₇×#103	C₂²	1	24	yes	—
24	C₂²×#475	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₇×Q₈	—	2	3	no	—
26	C₂²×C₇×Q₈	—	2	12	no	—
27	C₂²×C₇×Q₈	—	2	4	no	—
28	C₂²×C₇×Q₈	—	4	8	no	—
29	C₂²×C₇×Q₈	Q₈	1	8	yes	complemented
30	C₂²×C₇×Q₈	C₂³	1	6	yes	—
31	C₂³×C₄×C₇	—	2	1	no	—
32	C₂³×C₄×C₇	—	2	2	no	—
33	C₂³×C₄×C₇	—	2	1	no	—
34	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	—	2	8	no	—
38	C₂²×C₇×C₈	C₂³	1	12	yes	—
39	C₇×C₄⋊Q₈	—	2	24	no	—
40	C₇×C₄⋊Q₈	C₂³	1	16	yes	—
41	C₄×C₇×Q₈	—	2	24	no	—
42	C₄×C₇×Q₈	C₂³	1	16	yes	—
43	C₂×C₇×C₄⋊C₄	—	2	12	no	—
44	C₂×C₇×C₄⋊C₄	—	2	12	no	—
45	C₂×C₇×C₄⋊C₄	—	2	3	no	—
46	C₂×C₇×C₄⋊C₄	—	2	6	no	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
49	C₂×C₄²×C₇	—	2	12	no	—
50	C₂×C₄²×C₇	D₄	1	6	yes	—
51	C₂×C₄²×C₇	C₂³	1	6	yes	—
52	C₇×#106	C₂³	1	32	yes	—
53	C₇×C₄⋊C₈	C₂³	1	16	yes	—
54	C₇×#103	C₂³	1	32	yes	—
55	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
56	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
57	C₂×#475	C₂×C₇	1	24	yes	direct product factor
58	C₂²×#106	C₂×C₇	1	2	yes	—
59	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
60	C₂²×#103	C₂×C₇	1	2	yes	—
61	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₇×Q₈	—	2	24	no	—
63	C₂×C₇×Q₈	—	2	6	no	—
64	C₂×C₇×Q₈	—	2	16	no	—
65	C₂×C₇×Q₈	—	2	12	no	—
66	C₂×C₇×Q₈	—	4	48	no	—
67	C₂×C₇×Q₈	C₂×Q₈	1	24	yes	complemented
68	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₇	—	2	8	no	—
70	C₂²×C₄×C₇	—	2	4	no	—
71	C₂²×C₄×C₇	—	2	6	no	—
72	C₂²×C₄×C₇	—	2	6	no	—
73	C₂²×C₄×C₇	—	2	12	no	—
74	C₂²×C₄×C₇	—	4	8	no	—
75	C₂²×C₄×C₇	—	4	4	no	—
76	C₂²×C₄×C₇	SD₁₆	1	8	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₇	#41	1	4	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
83	C₂×C₇×C₈	—	2	24	no	—
84	C₂×C₇×C₈	C₂⁴	1	8	yes	—
85	C₇×C₄⋊C₄	—	2	16	no	—
86	C₇×C₄⋊C₄	—	2	16	no	—
87	C₇×C₄⋊C₄	—	2	6	no	—
88	C₇×C₄⋊C₄	—	2	12	no	—
89	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
90	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
91	C₄²×C₇	—	2	16	no	—
92	C₄²×C₇	C₂×D₄	1	12	yes	—
93	C₄²×C₇	C₂⁴	1	4	yes	—
94	C₂³×Q₈	—	2	1	no	—
95	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
96	C₂³×C₈	C₂²×C₇	1	2	yes	—
97	C₂×C₄⋊Q₈	—	2	6	no	—
98	C₂×C₄⋊Q₈	C₂²×C₇	1	12	yes	—
99	C₂×C₄×Q₈	—	2	6	no	—
100	C₂×C₄×Q₈	C₂²×C₇	1	12	yes	—
101	C₂²×C₄⋊C₄	—	2	1	no	—
102	C₂²×C₄⋊C₄	—	2	1	no	—
103	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
104	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
105	C₂²×C₄²	—	2	1	no	—
106	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
107	#475	C₂²×C₇	1	64	yes	direct product factor
108	C₂×#106	C₂²×C₇	1	24	yes	—
109	C₂×C₄⋊C₈	C₂²×C₇	1	12	yes	—
110	C₂×#103	C₂²×C₇	1	24	yes	—
111	C₂³×C₇	C₂²⋊Q₈	1	4	yes	—
112	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
113	C₂³×C₇	#137	1	4	yes	—
114	C₂³×C₇	C₂²×D₄	1	3	yes	—
115	C₇×Q₈	—	2	24	no	—
116	C₇×Q₈	—	2	4	no	—
117	C₇×Q₈	—	4	32	no	—
118	C₇×Q₈	C₂²×Q₈	1	8	yes	complemented
119	C₂×C₄×C₇	—	2	24	no	—
120	C₂×C₄×C₇	—	2	12	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	—	4	24	no	—
125	C₂×C₄×C₇	—	4	12	no	—
126	C₂×C₄×C₇	C₂²⋊Q₈	1	12	yes	—
127	C₂×C₄×C₇	C₂×SD₁₆	1	24	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	4	yes	—
131	C₂×C₄×C₇	C₂²×Q₈	1	6	yes	—
132	C₂×C₄×C₇	C₂×#41	1	6	yes	—
133	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
134	C₇×C₈	—	2	8	no	—
135	C₂²×Q₈	—	2	3	no	—
136	C₂²×Q₈	—	2	12	no	—
137	C₂²×Q₈	—	2	4	no	—
138	C₂²×Q₈	—	4	8	no	—
139	C₂²×Q₈	C₇×Q₈	1	8	yes	complemented
140	C₂²×Q₈	C₂³×C₇	1	6	yes	—
141	C₂³×C₄	—	2	1	no	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	—	2	1	no	—
144	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
145	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
146	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
147	C₂²×C₈	—	2	8	no	—
148	C₂²×C₈	C₂³×C₇	1	12	yes	—
149	C₄⋊Q₈	—	2	24	no	—
150	C₄⋊Q₈	C₂³×C₇	1	16	yes	—
151	C₄×Q₈	—	2	24	no	—
152	C₄×Q₈	C₂³×C₇	1	16	yes	—
153	C₂×C₄⋊C₄	—	2	12	no	—
154	C₂×C₄⋊C₄	—	2	12	no	—
155	C₂×C₄⋊C₄	—	2	3	no	—
156	C₂×C₄⋊C₄	—	2	6	no	—
157	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
158	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
159	C₂×C₄²	—	2	12	no	—
160	C₂×C₄²	C₇×D₄	1	6	yes	—
161	C₂×C₄²	C₂³×C₇	1	6	yes	—
162	#106	C₂³×C₇	1	32	yes	—
163	C₄⋊C₈	C₂³×C₇	1	16	yes	—
164	#103	C₂³×C₇	1	32	yes	—
165	C₂²×C₇	#475	1	16	yes	direct product factor
166	C₂²×C₇	C₂×C₂²⋊Q₈	1	6	yes	—
167	C₂²×C₇	C₂²×SD₁₆	1	6	yes	—
168	C₂²×C₇	C₂×#137	1	6	yes	—
169	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
170	C₄×C₇	—	2	8	no	—
171	C₄×C₇	—	2	4	no	—
172	C₄×C₇	—	4	8	no	—
173	C₄×C₇	—	4	4	no	—
174	C₄×C₇	C₂×C₂²⋊Q₈	1	6	yes	—
175	C₄×C₇	C₂²×SD₁₆	1	8	yes	—
176	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
177	C₄×C₇	C₂²×#41	1	1	yes	characteristic
178	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
179	C₂×Q₈	—	2	24	no	—
180	C₂×Q₈	—	2	6	no	—
181	C₂×Q₈	—	2	16	no	—
182	C₂×Q₈	—	2	12	no	—
183	C₂×Q₈	—	4	48	no	—
184	C₂×Q₈	C₂×C₇×Q₈	1	24	yes	complemented
185	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	6	no	—
189	C₂²×C₄	—	2	6	no	—
190	C₂²×C₄	—	2	12	no	—
191	C₂²×C₄	—	4	8	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₇×SD₁₆	1	8	yes	—
194	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
195	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
196	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
197	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
198	C₂²×C₄	C₇×#41	1	4	yes	—
199	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
200	C₂×C₈	—	2	24	no	—
201	C₂×C₈	C₂⁴×C₇	1	8	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	6	no	—
205	C₄⋊C₄	—	2	12	no	—
206	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
207	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
208	C₄²	—	2	16	no	—
209	C₄²	C₂×C₇×D₄	1	12	yes	—
210	C₄²	C₂⁴×C₇	1	4	yes	—
211	C₂×C₇	C₂×#475	1	12	yes	direct product factor
212	C₂×C₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
214	C₂×C₇	C₂²×#137	1	1	yes	characteristic
215	C₂³	C₇×C₂²⋊Q₈	1	4	yes	—
216	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
217	C₂³	C₇×#137	1	4	yes	—
218	C₂³	C₂²×C₇×D₄	1	3	yes	—
219	Q₈	—	2	4	no	—
220	Q₈	—	2	24	no	—
221	Q₈	—	4	32	no	—
222	Q₈	C₂²×C₇×Q₈	1	8	yes	complemented
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	4	24	no	—
229	C₂×C₄	—	4	12	no	—
230	C₂×C₄	C₇×C₂²⋊Q₈	1	12	yes	—
231	C₂×C₄	C₂×C₇×SD₁₆	1	24	yes	—
232	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
233	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
234	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
235	C₂×C₄	C₂²×C₇×Q₈	1	6	yes	—
236	C₂×C₄	C₂×C₇×#41	1	6	yes	—
237	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
238	C₈	—	2	8	no	—
239	C₇	C₂²×#475	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	C₇×#475	1	16	yes	direct product factor
241	C₂²	C₂×C₇×C₂²⋊Q₈	1	6	yes	—
242	C₂²	C₂²×C₇×SD₁₆	1	6	yes	—
243	C₂²	C₂×C₇×#137	1	6	yes	—
244	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	8	no	—
247	C₄	—	4	8	no	—
248	C₄	—	4	4	no	—
249	C₄	C₂×C₇×C₂²⋊Q₈	1	6	yes	—
250	C₄	C₂²×C₇×SD₁₆	1	8	yes	—
251	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
252	C₄	C₂²×C₇×#41	1	1	yes	characteristic
253	C₂	C₂×C₇×#475	1	12	yes	direct product factor
254	C₂	C₂²×C₇×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic
256	C₂	C₂²×C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#475	C₂	direct (N × H)
20. C₇×#475	C₂²	direct (N × H)
24. C₂²×#475	C₇	direct (N × H)
29. C₂²×C₇×Q₈	Q₈	semidirect
57. C₂×#475	C₂×C₇	direct (N × H)
67. C₂×C₇×Q₈	C₂×Q₈	semidirect
107. #475	C₂²×C₇	direct (N × H)
118. C₇×Q₈	C₂²×Q₈	semidirect
139. C₂²×Q₈	C₇×Q₈	semidirect
165. C₂²×C₇	#475	direct (N × H)
184. C₂×Q₈	C₂×C₇×Q₈	semidirect
211. C₂×C₇	C₂×#475	direct (N × H)
222. Q₈	C₂²×C₇×Q₈	semidirect
239. C₇	C₂²×#475	direct (N × H)
240. C₂²	C₇×#475	direct (N × H)
253. C₂	C₂×C₇×#475	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×C₇×C₄⋊Q₈	2
C₂²×C₄×C₇×Q₈	2
C₂×C₇×#475	2
C₂²×C₇×#106	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₇×#103	2
C₂²×#475	7

Subgroups — C22×C7×#475 #13237692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#475 #13237692