Subgroups — C₇×#23801 #13201017

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	4	yes	maximal
2	C₄×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₇×#2095	C₂	1	4	yes	maximal
4	C₇×#2016	C₂	1	4	yes	maximal
5	C₇×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
9	C₄×C₇×Q₁₆	—	2	16	no	—
10	C₂×C₇×#107	C₂²	1	4	yes	—
11	C₂×C₇×#103	C₂²	1	8	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
13	C₇×#391	C₂²	1	4	yes	—
14	C₇×#389	—	2	4	no	—
15	C₇×#384	C₂²	1	2	yes	—
16	C₄×C₇×C₄⋊C₄	—	2	4	no	—
17	C₄³×C₇	C₂²	1	1	yes	characteristic
18	C₇×#336	C₂²	1	4	yes	—
19	#23801	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	C₂³	1	4	yes	—
21	C₂×C₇×Q₁₆	—	2	16	no	—
22	C₂×C₇×Q₁₆	—	2	8	no	—
23	C₂²×C₇×C₈	C₂³	1	4	yes	—
24	C₇×C₄⋊Q₈	—	2	8	no	—
25	C₄×C₇×Q₈	—	2	16	no	—
26	C₂×C₇×C₄⋊C₄	—	2	4	no	—
27	C₂×C₇×C₄⋊C₄	—	2	4	no	—
28	C₂×C₇×C₄⋊C₄	—	2	4	no	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
30	C₂×C₄²×C₇	—	2	4	no	—
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₇×#107	—	2	8	no	—
36	C₇×#103	—	2	16	no	—
37	C₄×C₇×C₈	D₄	1	8	yes	—
38	C₇×#95	—	2	8	no	—
39	C₂×C₄×Q₁₆	C₂×C₇	1	4	yes	—
40	C₄×C₄⋊Q₈	C₂×C₇	1	2	yes	—
41	#2095	C₂×C₇	1	4	yes	—
42	#2016	C₂×C₇	1	4	yes	—
43	#1744	C₂×C₇	1	1	yes	characteristic
44	C₂×C₇×Q₈	—	2	4	no	—
45	C₂×C₇×Q₈	—	2	8	no	—
46	C₂×C₇×Q₈	—	2	8	no	—
47	C₂×C₇×Q₈	#41	1	8	yes	—
48	C₂²×C₄×C₇	—	2	4	no	—
49	C₂²×C₄×C₇	—	2	4	no	—
50	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₇×Q₁₆	—	2	32	no	—
55	C₂×C₇×C₈	—	2	8	no	—
56	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
57	C₇×C₄⋊C₄	—	2	16	no	—
58	C₇×C₄⋊C₄	—	2	8	no	—
59	C₇×C₄⋊C₄	—	4	8	no	—
60	C₇×C₄⋊C₄	—	4	8	no	—
61	C₄²×C₇	—	2	4	no	—
62	C₄²×C₇	—	4	2	no	—
63	C₄²×C₇	—	4	8	no	—
64	C₄²×C₇	Q₁₆	1	8	yes	complemented
65	C₄²×C₇	C₂×D₄	1	4	yes	—
66	C₂²×Q₁₆	C₂²×C₇	1	4	yes	—
67	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
68	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
69	C₄×Q₁₆	—	2	16	no	—
70	C₂×#107	C₂²×C₇	1	4	yes	—
71	C₂×#103	C₂²×C₇	1	8	yes	—
72	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
73	#391	C₂²×C₇	1	4	yes	—
74	#389	—	2	4	no	—
75	#384	C₂²×C₇	1	2	yes	—
76	C₄×C₄⋊C₄	—	2	4	no	—
77	C₄³	C₂²×C₇	1	1	yes	characteristic
78	#336	C₂²×C₇	1	4	yes	—
79	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
80	C₇×Q₈	—	2	16	no	—
81	C₇×Q₈	—	4	8	no	—
82	C₂×C₄×C₇	—	2	4	no	—
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	8	no	—
86	C₂×C₄×C₇	—	4	4	no	—
87	C₂×C₄×C₇	—	4	4	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	—	4	8	no	—
90	C₂×C₄×C₇	#127	1	2	yes	—
91	C₂×C₄×C₇	#127	1	2	yes	—
92	C₂×C₄×C₇	C₂×Q₁₆	1	8	yes	—
93	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₇	C₂×#41	1	4	yes	—
96	C₇×C₈	—	2	8	no	—
97	C₂²×Q₈	C₂³×C₇	1	4	yes	—
98	C₂×Q₁₆	—	2	16	no	—
99	C₂×Q₁₆	—	2	8	no	—
100	C₂²×C₈	C₂³×C₇	1	4	yes	—
101	C₄⋊Q₈	—	2	8	no	—
102	C₄×Q₈	—	2	16	no	—
103	C₂×C₄⋊C₄	—	2	4	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	—	2	4	no	—
106	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
107	C₂×C₄²	—	2	4	no	—
108	C₂×C₄²	C₇×D₄	1	2	yes	—
109	C₂×C₄²	C₇×D₄	1	2	yes	—
110	C₂×C₄²	C₂³×C₇	1	2	yes	—
111	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
112	#107	—	2	8	no	—
113	#103	—	2	16	no	—
114	C₄×C₈	C₇×D₄	1	8	yes	—
115	#95	—	2	8	no	—
116	C₂²×C₇	#525	1	2	yes	—
117	C₂²×C₇	C₂×#127	1	1	yes	characteristic
118	C₂²×C₇	C₂²×Q₁₆	1	2	yes	—
119	C₂²×C₇	#575	1	2	yes	—
120	C₄×C₇	—	2	8	no	—
121	C₄×C₇	—	2	2	no	—
122	C₄×C₇	—	4	8	no	—
123	C₄×C₇	—	8	4	no	—
124	C₄×C₇	#494	1	4	yes	—
125	C₄×C₇	#525	1	4	yes	—
126	C₂×Q₈	—	2	4	no	—
127	C₂×Q₈	—	2	8	no	—
128	C₂×Q₈	—	2	8	no	—
129	C₂×Q₈	C₇×#41	1	8	yes	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
134	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
136	Q₁₆	—	2	32	no	—
137	C₂×C₈	—	2	8	no	—
138	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
139	C₄⋊C₄	—	2	16	no	—
140	C₄⋊C₄	—	2	8	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₄²	—	4	8	no	—
146	C₄²	C₇×Q₁₆	1	8	yes	complemented
147	C₄²	C₂×C₇×D₄	1	4	yes	—
148	C₂×C₇	#2397	1	1	yes	characteristic
149	C₂×C₇	#3104	1	4	yes	—
150	C₂×C₇	C₂×#494	1	1	yes	characteristic
151	C₂×C₇	#3150	1	1	yes	characteristic
152	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	4	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	8	no	—
163	C₂×C₄	C₇×#127	1	2	yes	—
164	C₂×C₄	C₇×#127	1	2	yes	—
165	C₂×C₄	C₂×C₇×Q₁₆	1	8	yes	—
166	C₂×C₄	C₂×C₇×Q₁₆	1	4	yes	—
167	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
168	C₂×C₄	C₂×C₇×#41	1	4	yes	—
169	C₈	—	2	8	no	—
170	C₇	#23801	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²	C₇×#525	1	2	yes	—
172	C₂²	C₂×C₇×#127	1	1	yes	characteristic
173	C₂²	C₂²×C₇×Q₁₆	1	2	yes	—
174	C₂²	C₇×#575	1	2	yes	—
175	C₄	—	2	8	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	8	no	—
178	C₄	—	8	4	no	—
179	C₄	C₇×#494	1	4	yes	—
180	C₄	C₇×#525	1	4	yes	—
181	C₂	C₇×#2397	1	1	yes	characteristic
182	C₂	C₇×#3104	1	4	yes	—
183	C₂	C₂×C₇×#494	1	1	yes	characteristic
184	C₂	C₇×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
19. #23801	C₇	direct (N × H)
64. C₄²×C₇	Q₁₆	semidirect
146. C₄²	C₇×Q₁₆	semidirect
170. C₇	#23801	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×C₄×C₇×Q₁₆	2
C₄×C₇×C₄⋊Q₈	2
C₇×#2095	2
C₇×#2016	2
C₇×#1744	2
#23801	7

Subgroups — C7×#23801 #13201017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23801 #13201017