Subgroups — C₇×#12033 #13189249

Order 1792.

222 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₇×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic
4	C₇×#1372	C₂	1	4	yes	maximal, complemented
5	C₇×#423	—	2	1	no	—
6	C₂×C₇×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₇×C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×#335	—	2	1	no	—
12	C₇×#335	—	2	1	no	—
13	C₇×#335	C₂²	1	2	yes	—
14	C₇×#335	C₂²	1	2	yes	—
15	#12033	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×M₁₆	—	4	1	no	—
17	C₂²×C₇×C₈	—	4	1	no	—
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
22	C₇×C₄⋊C₈	—	2	2	no	—
23	C₇×C₄⋊C₈	—	2	2	no	—
24	C₇×C₄⋊C₈	—	4	1	no	—
25	C₇×C₄⋊C₈	—	4	1	no	—
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	D₄	1	1	yes	characteristic
28	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
30	#2162	C₂×C₇	1	1	yes	characteristic
31	#1567	C₂×C₇	1	1	yes	characteristic
32	C₂×#335	C₂×C₇	1	1	yes	characteristic
33	#1372	C₂×C₇	1	4	yes	complemented
34	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
37	C₇×M₁₆	—	8	2	no	—
38	C₂×C₇×C₈	—	2	2	no	—
39	C₂×C₇×C₈	—	4	1	no	—
40	C₂×C₇×C₈	—	4	2	no	—
41	C₂×C₇×C₈	—	4	2	no	—
42	C₂×C₇×C₈	—	8	2	no	—
43	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇	D₈	1	1	yes	characteristic
49	C₄²×C₇	Q₁₆	1	1	yes	characteristic
50	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
51	C₇×C₁₆	—	2	4	no	—
52	#423	—	2	1	no	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
55	C₄⋊C₁₆	C₄×C₇	1	2	yes	—
56	C₂²⋊C₁₆	C₄×C₇	1	2	yes	—
57	C₄×C₁₆	C₂²×C₇	1	2	yes	—
58	#335	—	2	1	no	—
59	#335	—	2	1	no	—
60	#335	C₂²×C₇	1	2	yes	—
61	#335	C₂²×C₇	1	2	yes	—
62	C₂³×C₇	#107	1	1	yes	characteristic
63	C₂×C₄×C₇	—	2	1	no	—
64	C₂×C₄×C₇	—	2	1	no	—
65	C₂×C₄×C₇	—	4	1	no	—
66	C₂×C₄×C₇	#107	1	1	yes	characteristic
67	C₂×C₄×C₇	#107	1	1	yes	characteristic
68	C₂×C₄×C₇	#107	1	1	yes	characteristic
69	C₂×C₄×C₇	SD₃₂	1	2	yes	—
70	C₂×C₄×C₇	SD₃₂	1	2	yes	—
71	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₇×C₈	—	2	1	no	—
75	C₇×C₈	—	2	1	no	—
76	C₇×C₈	—	2	1	no	—
77	C₇×C₈	—	8	2	no	—
78	C₇×C₈	—	8	2	no	—
79	C₇×C₈	#131	1	2	yes	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂²×C₈	—	4	1	no	—
82	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
83	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
84	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
85	C₂×C₁₆	C₂×C₄×C₇	1	4	yes	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
91	C₄×C₈	C₇×D₄	1	1	yes	characteristic
92	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
93	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
94	C₂²×C₇	—	2	1	no	—
95	C₂²×C₇	—	2	1	no	—
96	C₂²×C₇	#367	1	2	yes	—
97	C₂²×C₇	C₂×#107	1	1	yes	characteristic
98	C₄×C₇	—	2	1	no	—
99	C₄×C₇	—	2	1	no	—
100	C₄×C₇	—	2	1	no	—
101	C₄×C₇	#367	1	2	yes	—
102	C₄×C₇	#424	1	1	yes	characteristic
103	C₄×C₇	#430	1	1	yes	characteristic
104	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
105	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
108	C₂²×C₄	C₇×Q₁₆	1	1	yes	characteristic
109	M₁₆	—	8	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	8	2	no	—
115	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
118	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
119	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
120	C₄²	C₇×D₈	1	1	yes	characteristic
121	C₄²	C₇×Q₁₆	1	1	yes	characteristic
122	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	C₂×C₇	—	2	2	no	—
125	C₂×C₇	#1571	1	1	yes	characteristic
126	C₂×C₇	C₂×#367	1	1	yes	characteristic
127	C₂×C₇	#2161	1	1	yes	characteristic
128	C₂³	C₇×#107	1	1	yes	characteristic
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	C₇×#107	1	1	yes	characteristic
133	C₂×C₄	C₇×#107	1	1	yes	characteristic
134	C₂×C₄	C₇×#107	1	1	yes	characteristic
135	C₂×C₄	C₇×SD₃₂	1	2	yes	—
136	C₂×C₄	C₇×SD₃₂	1	2	yes	—
137	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
139	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	2	no	—
144	C₈	—	8	2	no	—
145	C₈	C₇×#131	1	2	yes	—
146	C₇	#12033	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₇×#367	1	2	yes	—
150	C₂²	C₂×C₇×#107	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	C₇×#367	1	2	yes	—
155	C₄	C₇×#424	1	1	yes	characteristic
156	C₄	C₇×#430	1	1	yes	characteristic
157	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
158	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	C₇×#1571	1	1	yes	characteristic
161	C₂	C₂×C₇×#367	1	1	yes	characteristic
162	C₂	C₇×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#107
prime radical	C₂³×C₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1372	C₂	semidirect
15. #12033	C₇	direct (N × H)
33. #1372	C₂×C₇	semidirect
146. C₇	#12033	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2162	2
C₇×#1567	2
C₂×C₇×#335	2
C₇×#1372	2
#12033	7

Subgroups — C7×#12033 #13189249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12033 #13189249