Subgroups — C₇×#12011 #13189227

Order 1792.

220 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#364	C₂	1	2	yes	maximal
2	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇×#1373	C₂	1	4	yes	maximal, complemented
4	C₂×C₇×M₃₂	—	2	2	no	—
5	C₂×C₇×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₇×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#364	—	2	2	no	—
9	C₇×#364	C₂²	1	4	yes	—
10	C₇×#346	C₄	1	2	yes	—
11	C₄×C₇×C₁₆	C₂²	1	2	yes	—
12	#12011	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
15	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
16	C₇×M₃₂	—	2	2	no	—
17	C₇×M₃₂	—	2	4	no	—
18	C₇×M₃₂	—	4	2	no	—
19	C₇×M₃₂	C₈	1	4	yes	—
20	C₂×C₇×C₁₆	—	4	2	no	—
21	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	D₄	1	1	yes	characteristic
25	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂×#364	C₂×C₇	1	2	yes	—
28	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic
29	#1373	C₂×C₇	1	4	yes	complemented
30	C₂²×C₄×C₇	—	2	1	no	—
31	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
35	C₂×C₇×C₈	M₁₆	1	2	yes	—
36	C₂×C₇×C₈	D₈	1	2	yes	—
37	C₂×C₇×C₈	Q₁₆	1	2	yes	—
38	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇	—	2	1	no	—
41	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
43	C₇×C₁₆	—	2	2	no	—
44	C₇×C₁₆	—	4	4	no	—
45	C₇×C₁₆	C₂×C₈	1	4	yes	—
46	C₂×M₃₂	—	2	2	no	—
47	C₂×M₃₂	C₄×C₇	1	1	yes	characteristic
48	C₂×M₃₂	C₄×C₇	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
50	#364	—	2	2	no	—
51	#364	C₂²×C₇	1	4	yes	—
52	#346	C₄×C₇	1	2	yes	—
53	C₄×C₁₆	C₂²×C₇	1	2	yes	—
54	C₂³×C₇	#108	1	1	yes	characteristic
55	C₂×C₄×C₇	—	2	1	no	—
56	C₂×C₄×C₇	—	2	1	no	—
57	C₂×C₄×C₇	—	4	2	no	—
58	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₇	#107	1	2	yes	—
60	C₂×C₄×C₇	#108	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₇×C₈	C₄⋊C₈	1	2	yes	—
63	C₇×C₈	#107	1	2	yes	—
64	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
65	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
66	C₇×C₈	C₂×D₈	1	1	yes	characteristic
67	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
68	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
69	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
70	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
71	M₃₂	—	2	2	no	—
72	M₃₂	—	2	4	no	—
73	M₃₂	—	4	2	no	—
74	M₃₂	C₇×C₈	1	4	yes	—
75	C₂×C₁₆	—	4	2	no	—
76	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
77	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
78	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
79	C₄×C₈	C₇×D₄	1	1	yes	characteristic
80	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
81	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
82	C₂²×C₇	—	4	1	no	—
83	C₂²×C₇	#335	1	2	yes	—
84	C₂²×C₇	C₂×#108	1	1	yes	characteristic
85	C₄×C₇	—	2	2	no	—
86	C₄×C₇	—	4	1	no	—
87	C₄×C₇	#335	1	2	yes	—
88	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₇	C₂×#107	1	1	yes	characteristic
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
95	C₂×C₈	C₇×M₁₆	1	2	yes	—
96	C₂×C₈	C₇×D₈	1	2	yes	—
97	C₂×C₈	C₇×Q₁₆	1	2	yes	—
98	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
99	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
100	C₄²	—	2	1	no	—
101	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
102	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	4	4	no	—
105	C₁₆	C₂×C₇×C₈	1	4	yes	—
106	C₂×C₇	—	2	1	no	—
107	C₂×C₇	—	2	2	no	—
108	C₂×C₇	C₂×#335	1	1	yes	characteristic
109	C₂³	C₇×#108	1	1	yes	characteristic
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
114	C₂×C₄	C₇×#107	1	2	yes	—
115	C₂×C₄	C₇×#108	1	1	yes	characteristic
116	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
117	C₈	C₇×C₄⋊C₈	1	2	yes	—
118	C₈	C₇×#107	1	2	yes	—
119	C₈	C₂²×C₇×C₈	1	1	yes	characteristic
120	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
121	C₈	C₂×C₇×D₈	1	1	yes	characteristic
122	C₈	C₂×C₇×Q₁₆	1	1	yes	characteristic
123	C₇	#12011	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²	—	4	1	no	—
125	C₂²	C₇×#335	1	2	yes	—
126	C₂²	C₂×C₇×#108	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	C₇×#335	1	2	yes	—
130	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
131	C₄	C₂×C₇×#107	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×C₇×#335	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	C₂×D₈
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂×C₇	C₂×#335
prime radical	C₂³×C₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1373	C₂	semidirect
12. #12011	C₇	direct (N × H)
29. #1373	C₂×C₇	semidirect
123. C₇	#12011	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×#364	2
C₄×C₇×M₃₂	2
C₇×#1373	2
#12011	7

Subgroups — C7×#12011 #13189227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12011 #13189227