Subgroups — C₇×#22734 #13199950

Order 1792.

876 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1939	C₂	1	6	yes	maximal
4	C₇×#1831	C₂	1	3	yes	maximal
5	C₄×C₇×C₄⋊C₈	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×C₈×Q₈	—	2	8	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	6	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	6	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	6	yes	—
13	C₄×C₇×C₄⋊C₄	C₄	1	3	yes	—
14	C₄×C₇×C₄⋊C₄	C₄	1	3	yes	—
15	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
16	C₄³×C₇	C₂²	1	3	yes	—
17	C₇×#336	C₂²	1	1	yes	characteristic
18	C₇×#336	C₂²	1	3	yes	—
19	C₇×#336	C₂²	1	6	yes	—
20	#22734	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	6	yes	—
23	C₂²×C₇×C₈	C₂³	1	2	yes	—
24	C₄×C₇×Q₈	—	2	4	no	—
25	C₄×C₇×Q₈	C₈	1	8	yes	complemented
26	C₄×C₇×Q₈	D₄	1	4	yes	—
27	C₄×C₇×Q₈	Q₈	1	4	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂³	1	3	yes	—
37	C₂×C₄²×C₇	C₂³	1	3	yes	—
38	C₇×C₄⋊C₈	—	2	12	no	—
39	C₇×C₄⋊C₈	—	2	12	no	—
40	C₄×C₇×C₈	—	2	12	no	—
41	C₇×#95	C₂×C₄	1	6	yes	—
42	C₂×C₈×Q₈	C₂×C₇	1	2	yes	—
43	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
44	#1939	C₂×C₇	1	6	yes	—
45	#1831	C₂×C₇	1	3	yes	—
46	C₄×C₄⋊C₈	C₂×C₇	1	3	yes	—
47	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
49	C₂×C₇×Q₈	M₁₆	1	4	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	2	no	—
59	C₂×C₇×C₈	—	2	6	no	—
60	C₂×C₇×C₈	—	2	6	no	—
61	C₂×C₇×C₈	—	2	6	no	—
62	C₂×C₇×C₈	#41	1	4	yes	—
63	C₇×C₄⋊C₄	—	2	6	no	—
64	C₇×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
65	C₇×C₄⋊C₄	C₂×C₈	1	12	yes	—
66	C₄²×C₇	—	2	6	no	—
67	C₄²×C₇	—	2	6	no	—
68	C₄²×C₇	—	2	6	no	—
69	C₄²×C₇	—	2	2	no	—
70	C₄²×C₇	—	4	3	no	—
71	C₄²×C₇	C₂×C₈	1	12	yes	—
72	C₄²×C₇	C₂×D₄	1	6	yes	—
73	C₄²×C₇	C₂×Q₈	1	6	yes	—
74	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
75	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
76	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
77	C₈×Q₈	—	2	8	no	—
78	C₂×C₄⋊C₈	C₂²×C₇	1	6	yes	—
79	C₂×C₄⋊C₈	C₂²×C₇	1	6	yes	—
80	C₂×C₄×C₈	C₂²×C₇	1	6	yes	—
81	C₄×C₄⋊C₄	C₄×C₇	1	3	yes	—
82	C₄×C₄⋊C₄	C₄×C₇	1	3	yes	—
83	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
84	C₄³	C₂²×C₇	1	3	yes	—
85	#336	C₂²×C₇	1	3	yes	—
86	#336	C₂²×C₇	1	6	yes	—
87	#336	C₂²×C₇	1	1	yes	characteristic
88	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
89	C₇×Q₈	C₄⋊C₈	1	16	yes	complemented
90	C₂×C₄×C₇	—	2	3	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	3	no	—
95	C₂×C₄×C₇	—	2	3	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	3	no	—
98	C₂×C₄×C₇	—	2	3	no	—
99	C₂×C₄×C₇	—	2	3	no	—
100	C₂×C₄×C₇	—	2	3	no	—
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
105	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
106	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
107	C₂×C₄×C₇	#131	1	2	yes	—
108	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×#41	1	2	yes	—
111	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
112	C₇×C₈	—	2	4	no	—
113	C₇×C₈	—	4	6	no	—
114	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₇	1	6	yes	—
116	C₂²×C₈	C₂³×C₇	1	2	yes	—
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	C₇×C₈	1	8	yes	complemented
119	C₄×Q₈	C₇×D₄	1	4	yes	—
120	C₄×Q₈	C₇×Q₈	1	4	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
123	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
124	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
125	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
126	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
127	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
128	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
129	C₂×C₄²	C₂³×C₇	1	3	yes	—
130	C₂×C₄²	C₂³×C₇	1	3	yes	—
131	C₄⋊C₈	—	2	12	no	—
132	C₄⋊C₈	—	2	12	no	—
133	C₄×C₈	—	2	12	no	—
134	#95	C₂×C₄×C₇	1	6	yes	—
135	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₇	C₄×#41	1	1	yes	characteristic
137	C₂²×C₇	#519	1	1	yes	characteristic
138	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
139	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
140	C₂²×C₇	C₂×#131	1	1	yes	characteristic
141	C₂²×C₇	#568	1	1	yes	characteristic
142	C₄×C₇	—	2	6	no	—
143	C₄×C₇	—	2	6	no	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	2	2	no	—
146	C₄×C₇	—	4	3	no	—
147	C₄×C₇	C₂×C₄⋊C₈	1	12	yes	—
148	C₄×C₇	#548	1	2	yes	—
149	C₄×C₇	#557	1	2	yes	—
150	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
151	C₂×Q₈	C₂×C₇×C₈	1	4	yes	—
152	C₂×Q₈	C₇×M₁₆	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
154	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
155	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
156	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
159	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	6	no	—
163	C₂×C₈	—	2	6	no	—
164	C₂×C₈	—	2	6	no	—
165	C₂×C₈	C₇×#41	1	4	yes	—
166	C₄⋊C₄	—	2	6	no	—
167	C₄⋊C₄	C₇×C₄⋊C₄	1	12	yes	—
168	C₄⋊C₄	C₂×C₇×C₈	1	12	yes	—
169	C₄²	—	2	6	no	—
170	C₄²	—	2	6	no	—
171	C₄²	—	2	6	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	4	3	no	—
174	C₄²	C₂×C₇×C₈	1	12	yes	—
175	C₄²	C₂×C₇×D₄	1	6	yes	—
176	C₄²	C₂×C₇×Q₈	1	6	yes	—
177	C₂×C₇	#2351	1	1	yes	characteristic
178	C₂×C₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
179	C₂×C₇	#2906	1	1	yes	characteristic
180	C₂×C₇	C₈×#41	1	1	yes	characteristic
181	C₂×C₇	#2971	1	1	yes	characteristic
182	C₂×C₇	#2982	1	1	yes	characteristic
183	C₂×C₇	#2984	1	1	yes	characteristic
184	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
185	Q₈	C₇×C₄⋊C₈	1	16	yes	complemented
186	C₂×C₄	—	2	3	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
198	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
199	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
200	C₂×C₄	C₂²×C₇×C₈	1	6	yes	—
201	C₂×C₄	C₂²×C₇×C₈	1	6	yes	—
202	C₂×C₄	C₂×C₇×M₁₆	1	6	yes	—
203	C₂×C₄	C₇×#131	1	2	yes	—
204	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
206	C₂×C₄	C₂×C₇×#41	1	2	yes	—
207	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	4	6	no	—
210	C₇	#22734	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
212	C₂²	C₄×C₇×#41	1	1	yes	characteristic
213	C₂²	C₇×#519	1	1	yes	characteristic
214	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
215	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
216	C₂²	C₂×C₇×#131	1	1	yes	characteristic
217	C₂²	C₇×#568	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	6	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₇×C₄⋊C₈	1	12	yes	—
224	C₄	C₇×#548	1	2	yes	—
225	C₄	C₇×#557	1	2	yes	—
226	C₂	C₇×#2351	1	1	yes	characteristic
227	C₂	C₂²×C₇×C₄⋊C₈	1	1	yes	characteristic
228	C₂	C₇×#2906	1	1	yes	characteristic
229	C₂	C₇×C₈×#41	1	1	yes	characteristic
230	C₂	C₇×#2971	1	1	yes	characteristic
231	C₂	C₇×#2982	1	1	yes	characteristic
232	C₂	C₇×#2984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
20. #22734	C₇	direct (N × H)
25. C₄×C₇×Q₈	C₈	semidirect
89. C₇×Q₈	C₄⋊C₈	semidirect
118. C₄×Q₈	C₇×C₈	semidirect
185. Q₈	C₇×C₄⋊C₈	semidirect
210. C₇	#22734	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
C₄²×C₇×Q₈	2
C₇×#1939	2
C₇×#1831	2
C₄×C₇×C₄⋊C₈	2
#22734	7

Subgroups — C7×#22734 #13199950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22734 #13199950