Subgroups — C₇×#23836 #13201052

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#125	C₂	1	2	yes	maximal
2	C₇×#2085	C₂	1	4	yes	maximal
3	C₇×#2059	C₂	1	4	yes	maximal
4	C₄×C₇×#107	C₂	1	2	yes	maximal
5	C₄×C₇×#106	C₂	1	2	yes	maximal
6	C₇×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×#125	C₂²	1	2	yes	—
8	C₂×C₇×#107	C₂²	1	4	yes	—
9	C₂×C₇×#106	C₂²	1	4	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#389	C₂²	1	4	yes	—
13	C₇×#387	—	2	4	no	—
14	C₇×#382	C₂²	1	2	yes	—
15	C₄×C₇×C₄⋊C₄	—	2	4	no	—
16	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
17	C₄³×C₇	C₂²	1	1	yes	characteristic
18	C₇×#340	C₂²	1	2	yes	—
19	C₇×#340	C₂²	1	2	yes	—
20	C₇×#340	C₂²	1	2	yes	—
21	C₇×#340	C₂²	1	2	yes	—
22	C₇×#336	C₂²	1	4	yes	—
23	#23836	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₇×C₈	C₂³	1	2	yes	—
25	C₂²×C₇×C₈	C₂³	1	2	yes	—
26	C₇×#125	—	2	8	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×C₇×C₄⋊C₄	C₂³	1	4	yes	—
31	C₂×C₄²×C₇	—	2	4	no	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	D₄	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₇×#107	—	2	8	no	—
37	C₇×#106	—	2	8	no	—
38	C₄×C₇×C₈	Q₈	1	4	yes	—
39	C₄×C₇×C₈	Q₈	1	4	yes	—
40	C₇×#95	—	2	4	no	—
41	C₇×#95	—	2	4	no	—
42	C₇×#95	—	2	4	no	—
43	C₄×#125	C₂×C₇	1	2	yes	—
44	#2085	C₂×C₇	1	4	yes	—
45	#2059	C₂×C₇	1	4	yes	—
46	C₄×#107	C₂×C₇	1	2	yes	—
47	C₄×#106	C₂×C₇	1	2	yes	—
48	#1744	C₂×C₇	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	4	no	—
50	C₂²×C₄×C₇	—	2	4	no	—
51	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
55	C₂×C₇×C₈	—	2	4	no	—
56	C₂×C₇×C₈	—	2	4	no	—
57	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
58	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
59	C₇×C₄⋊C₄	—	2	16	no	—
60	C₇×C₄⋊C₄	—	2	4	no	—
61	C₇×C₄⋊C₄	—	2	4	no	—
62	C₇×C₄⋊C₄	—	2	4	no	—
63	C₇×C₄⋊C₄	—	4	8	no	—
64	C₇×C₄⋊C₄	#41	1	8	yes	—
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	—	2	4	no	—
67	C₄²×C₇	—	4	2	no	—
68	C₄²×C₇	—	4	8	no	—
69	C₄²×C₇	C₂×Q₈	1	2	yes	—
70	C₄²×C₇	C₂×Q₈	1	2	yes	—
71	C₂×#125	C₂²×C₇	1	2	yes	—
72	C₂×#107	C₂²×C₇	1	4	yes	—
73	C₂×#106	C₂²×C₇	1	4	yes	—
74	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
76	#389	C₂²×C₇	1	4	yes	—
77	#387	—	2	4	no	—
78	#382	C₂²×C₇	1	2	yes	—
79	C₄×C₄⋊C₄	—	2	4	no	—
80	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
81	C₄³	C₂²×C₇	1	1	yes	characteristic
82	#340	C₂²×C₇	1	2	yes	—
83	#340	C₂²×C₇	1	2	yes	—
84	#340	C₂²×C₇	1	2	yes	—
85	#340	C₂²×C₇	1	2	yes	—
86	#336	C₂²×C₇	1	4	yes	—
87	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	—	2	8	no	—
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	4	4	no	—
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	—	4	4	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	4	no	—
98	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
99	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₇	#135	1	4	yes	—
101	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×#41	1	4	yes	—
104	C₇×C₈	—	2	4	no	—
105	C₇×C₈	—	2	4	no	—
106	C₂²×C₈	C₂³×C₇	1	2	yes	—
107	C₂²×C₈	C₂³×C₇	1	2	yes	—
108	#125	—	2	8	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
112	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
113	C₂×C₄²	—	2	4	no	—
114	C₂×C₄²	C₇×D₄	1	2	yes	—
115	C₂×C₄²	C₇×D₄	1	2	yes	—
116	C₂×C₄²	C₂³×C₇	1	2	yes	—
117	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
118	#107	—	2	8	no	—
119	#106	—	2	8	no	—
120	C₄×C₈	C₇×Q₈	1	4	yes	—
121	C₄×C₈	C₇×Q₈	1	4	yes	—
122	#95	—	2	4	no	—
123	#95	—	2	4	no	—
124	#95	—	2	4	no	—
125	C₂²×C₇	#525	1	2	yes	—
126	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂²×C₇	C₂×#135	1	2	yes	—
128	C₂²×C₇	#575	1	2	yes	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	4	4	no	—
131	C₄×C₇	—	4	8	no	—
132	C₄×C₇	—	8	4	no	—
133	C₄×C₇	#499	1	4	yes	—
134	C₄×C₇	#533	1	2	yes	—
135	C₄×C₇	#533	1	2	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
139	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
141	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
145	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
146	C₄⋊C₄	—	2	16	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	C₇×#41	1	8	yes	—
152	C₄²	—	2	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	8	no	—
156	C₄²	C₂×C₇×Q₈	1	2	yes	—
157	C₄²	C₂×C₇×Q₈	1	2	yes	—
158	C₂×C₇	#2399	1	1	yes	characteristic
159	C₂×C₇	#3102	1	4	yes	—
160	C₂×C₇	C₂×#499	1	1	yes	characteristic
161	C₂×C₇	#3162	1	1	yes	characteristic
162	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
174	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
175	C₂×C₄	C₇×#135	1	4	yes	—
176	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
178	C₂×C₄	C₂×C₇×#41	1	4	yes	—
179	C₈	—	2	4	no	—
180	C₈	—	2	4	no	—
181	C₇	#23836	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	C₇×#525	1	2	yes	—
183	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
184	C₂²	C₂×C₇×#135	1	2	yes	—
185	C₂²	C₇×#575	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	4	no	—
188	C₄	—	4	8	no	—
189	C₄	—	8	4	no	—
190	C₄	C₇×#499	1	4	yes	—
191	C₄	C₇×#533	1	2	yes	—
192	C₄	C₇×#533	1	2	yes	—
193	C₂	C₇×#2399	1	1	yes	characteristic
194	C₂	C₇×#3102	1	4	yes	—
195	C₂	C₂×C₇×#499	1	1	yes	characteristic
196	C₂	C₇×#3162	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
23. #23836	C₇	direct (N × H)
181. C₇	#23836	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#125	2
C₇×#2085	2
C₇×#2059	2
C₄×C₇×#107	2
C₄×C₇×#106	2
C₇×#1744	2
#23836	7

Subgroups — C7×#23836 #13201052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23836 #13201052