Subgroups — C₇×#13638 #13190854

Order 1792.

182 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#368	C₂	1	1	yes	maximal, characteristic
3	C₇×#2158	C₂	1	1	yes	maximal, characteristic
4	C₇×#1426	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₇×#108	—	2	1	no	—
7	C₇×#427	—	2	1	no	—
8	C₇×M₆₄	C₄	1	2	yes	complemented
9	C₇×M₆₄	C₂²	1	2	yes	—
10	C₂×C₇×C₃₂	C₄	1	2	yes	complemented
11	C₇×#368	—	2	1	no	—
12	C₇×#368	C₂²	1	2	yes	—
13	C₇×#367	C₂²	1	2	yes	—
14	C₇×#366	—	2	1	no	—
15	#13638	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×M₁₆	—	4	1	no	—
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₇×#117	—	4	1	no	—
19	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
24	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
25	C₇×#108	—	2	2	no	—
26	C₇×#108	—	4	1	no	—
27	C₇×#107	—	2	2	no	—
28	C₇×#106	—	4	1	no	—
29	C₇×C₃₂	C₂×C₄	1	4	yes	—
30	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic
31	C₂×#368	C₂×C₇	1	1	yes	characteristic
32	#2158	C₂×C₇	1	1	yes	characteristic
33	#1426	C₂×C₇	1	4	yes	complemented
34	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
35	C₇×M₁₆	—	4	2	no	—
36	C₇×M₁₆	—	8	1	no	—
37	C₂×C₇×C₈	—	8	1	no	—
38	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₇×C₈	D₈	1	1	yes	characteristic
41	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
42	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
43	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
44	C₇×C₄⋊C₄	—	4	2	no	—
45	C₇×C₂²⋊C₄	—	8	1	no	—
46	C₄²×C₇	—	8	1	no	—
47	C₇×C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₇×C₁₆	C₄⋊C₄	1	1	yes	characteristic
49	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
50	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
52	C₂×#108	—	2	1	no	—
53	#427	—	2	1	no	—
54	M₆₄	C₄×C₇	1	2	yes	complemented
55	M₆₄	C₂²×C₇	1	2	yes	—
56	C₂×C₃₂	C₄×C₇	1	2	yes	complemented
57	#368	—	2	1	no	—
58	#368	C₂²×C₇	1	2	yes	—
59	#367	C₂²×C₇	1	2	yes	—
60	#366	—	2	1	no	—
61	C₂³×C₇	SD₃₂	1	1	yes	characteristic
62	C₂×C₄×C₇	—	8	2	no	—
63	C₂×C₄×C₇	#107	1	1	yes	characteristic
64	C₂×C₄×C₇	#107	1	1	yes	characteristic
65	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
66	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
68	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
69	C₇×C₈	—	8	2	no	—
70	C₇×C₈	#107	1	1	yes	characteristic
71	C₇×C₈	#107	1	1	yes	characteristic
72	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×M₁₆	—	4	1	no	—
75	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
76	#117	—	4	1	no	—
77	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
78	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
79	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
80	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
81	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
82	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
83	#108	—	2	2	no	—
84	#108	—	4	1	no	—
85	#107	—	2	2	no	—
86	#106	—	4	1	no	—
87	C₃₂	C₂×C₄×C₇	1	4	yes	—
88	C₂²×C₇	—	4	1	no	—
89	C₂²×C₇	#367	1	1	yes	characteristic
90	C₂²×C₇	#367	1	1	yes	characteristic
91	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
92	C₄×C₇	—	16	2	no	—
93	C₄×C₇	#367	1	1	yes	characteristic
94	C₄×C₇	#367	1	1	yes	characteristic
95	C₄×C₇	C₂×#107	1	1	yes	characteristic
96	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
98	M₁₆	—	4	2	no	—
99	M₁₆	—	8	1	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₇×D₈	1	1	yes	characteristic
104	C₂×C₈	C₇×Q₁₆	1	1	yes	characteristic
105	C₂×C₈	C₇×Q₁₆	1	1	yes	characteristic
106	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	8	1	no	—
109	C₄²	—	8	1	no	—
110	C₁₆	C₇×C₄⋊C₄	1	1	yes	characteristic
111	C₁₆	C₇×C₄⋊C₄	1	1	yes	characteristic
112	C₁₆	C₂²×C₄×C₇	1	1	yes	characteristic
113	C₁₆	C₂×C₇×Q₈	1	1	yes	characteristic
114	C₂×C₇	—	2	1	no	—
115	C₂×C₇	—	2	1	no	—
116	C₂×C₇	—	2	1	no	—
117	C₂×C₇	C₂×#367	1	1	yes	characteristic
118	C₂³	C₇×SD₃₂	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	C₇×#107	1	1	yes	characteristic
121	C₂×C₄	C₇×#107	1	1	yes	characteristic
122	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
123	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
124	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
125	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
126	C₈	—	8	2	no	—
127	C₈	C₇×#107	1	1	yes	characteristic
128	C₈	C₇×#107	1	1	yes	characteristic
129	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
130	C₈	C₂×C₇×Q₁₆	1	1	yes	characteristic
131	C₇	#13638	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²	—	4	1	no	—
133	C₂²	C₇×#367	1	1	yes	characteristic
134	C₂²	C₇×#367	1	1	yes	characteristic
135	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
136	C₄	—	16	2	no	—
137	C₄	C₇×#367	1	1	yes	characteristic
138	C₄	C₇×#367	1	1	yes	characteristic
139	C₄	C₂×C₇×#107	1	1	yes	characteristic
140	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	C₂×C₇×#367	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#367
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₂×#367
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1426	C₂	semidirect
8. C₇×M₆₄	C₄	semidirect
10. C₂×C₇×C₃₂	C₄	semidirect
15. #13638	C₇	direct (N × H)
33. #1426	C₂×C₇	semidirect
54. M₆₄	C₄×C₇	semidirect
56. C₂×C₃₂	C₄×C₇	semidirect
131. C₇	#13638	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×M₆₄	2
C₂×C₇×#368	2
C₇×#2158	2
C₇×#1426	2
#13638	7

Subgroups — C7×#13638 #13190854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13638 #13190854