Subgroups — C₇×#1770 #11730239

Order 896.

272 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#97	C₂	1	2	yes	maximal
3	C₂×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₇×#336	C₂	1	2	yes	maximal
6	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	2	yes	—
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₄²×C₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₈	C₄	1	4	yes	complemented
13	C₇×#97	—	2	4	no	—
14	C₄×C₇×C₈	C₄	1	4	yes	complemented
15	C₇×#95	C₄	1	2	yes	—
16	#1770	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₄×C₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₇×C₈	—	2	2	no	—
22	C₂×C₇×C₈	—	2	1	no	—
23	C₂×C₇×C₈	—	2	1	no	—
24	C₂×C₇×C₈	—	2	2	no	—
25	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₇×C₈	C₂×C₄	1	4	yes	—
27	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₇×C₈	D₄	1	2	yes	—
29	C₂×C₇×C₈	Q₈	1	2	yes	—
30	C₇×C₄⋊C₄	—	2	4	no	—
31	C₄²×C₇	—	2	2	no	—
32	C₄²×C₇	—	2	2	no	—
33	C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₄²×C₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
36	C₂×#97	C₂×C₇	1	2	yes	—
37	C₂×C₄×C₈	C₂×C₇	1	1	yes	characteristic
38	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
39	#336	C₂×C₇	1	2	yes	—
40	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	—	2	4	no	—
43	C₂×C₄×C₇	—	2	4	no	—
44	C₂×C₄×C₇	—	2	4	no	—
45	C₂×C₄×C₇	C₄²	1	2	yes	—
46	C₂×C₄×C₇	C₄²	1	2	yes	—
47	C₂×C₄×C₇	M₁₆	1	4	yes	—
48	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇	#41	1	1	yes	characteristic
53	C₂×C₄×C₇	#41	1	1	yes	characteristic
54	C₇×C₈	—	2	2	no	—
55	C₇×C₈	—	2	4	no	—
56	C₇×C₈	C₄⋊C₄	1	4	yes	complemented
57	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₇	1	2	yes	—
59	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
60	C₂×C₄⋊C₄	C₄×C₇	1	2	yes	—
61	C₂×C₄²	C₂²×C₇	1	2	yes	—
62	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
63	C₄⋊C₈	C₄×C₇	1	4	yes	complemented
64	#97	—	2	4	no	—
65	C₄×C₈	C₄×C₇	1	4	yes	complemented
66	#95	C₄×C₇	1	2	yes	—
67	C₂²×C₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₇	C₄×D₄	1	1	yes	characteristic
69	C₂²×C₇	C₄×Q₈	1	1	yes	characteristic
70	C₂²×C₇	C₂×M₁₆	1	2	yes	—
71	C₂²×C₇	#131	1	2	yes	—
72	C₄×C₇	—	2	2	no	—
73	C₄×C₇	—	4	4	no	—
74	C₄×C₇	#97	1	4	yes	complemented
75	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₇	#117	1	1	yes	characteristic
77	C₄×C₇	C₄×D₄	1	1	yes	characteristic
78	C₄×C₇	C₄×Q₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×C₄×C₇	1	4	yes	—
80	C₂²×C₄	C₂×C₄×C₇	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₄×C₇	1	1	yes	characteristic
82	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	C₂×C₄×C₇	1	2	yes	—
88	C₂×C₈	C₂×C₄×C₇	1	4	yes	—
89	C₂×C₈	C₂×C₄×C₇	1	2	yes	—
90	C₂×C₈	C₇×D₄	1	2	yes	—
91	C₂×C₈	C₇×Q₈	1	2	yes	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	2	2	no	—
95	C₄²	C₂×C₄×C₇	1	2	yes	—
96	C₄²	C₂×C₄×C₇	1	2	yes	—
97	C₂×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₇	C₂×#97	1	1	yes	characteristic
99	C₂×C₇	#405	1	1	yes	characteristic
100	C₂×C₇	#436	1	2	yes	—
101	C₂×C₇	#446	1	2	yes	—
102	C₂³	C₂²×C₄×C₇	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₄²×C₇	1	2	yes	—
108	C₂×C₄	C₄²×C₇	1	2	yes	—
109	C₂×C₄	C₇×M₁₆	1	4	yes	—
110	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
111	C₂×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
113	C₂×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
114	C₂×C₄	C₇×#41	1	1	yes	characteristic
115	C₂×C₄	C₇×#41	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	2	4	no	—
118	C₈	C₇×C₄⋊C₄	1	4	yes	complemented
119	C₇	#1770	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²	C₂×C₄²×C₇	1	1	yes	characteristic
121	C₂²	C₄×C₇×D₄	1	1	yes	characteristic
122	C₂²	C₄×C₇×Q₈	1	1	yes	characteristic
123	C₂²	C₂×C₇×M₁₆	1	2	yes	—
124	C₂²	C₇×#131	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	C₇×#97	1	4	yes	complemented
128	C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
129	C₄	C₇×#117	1	1	yes	characteristic
130	C₄	C₄×C₇×D₄	1	1	yes	characteristic
131	C₄	C₄×C₇×Q₈	1	1	yes	characteristic
132	C₂	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
133	C₂	C₂×C₇×#97	1	1	yes	characteristic
134	C₂	C₇×#405	1	1	yes	characteristic
135	C₂	C₇×#436	1	2	yes	—
136	C₂	C₇×#446	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
12. C₇×C₄⋊C₈	C₄	semidirect
14. C₄×C₇×C₈	C₄	semidirect
16. #1770	C₇	direct (N × H)
56. C₇×C₈	C₄⋊C₄	semidirect
63. C₄⋊C₈	C₄×C₇	semidirect
65. C₄×C₈	C₄×C₇	semidirect
74. C₄×C₇	#97	semidirect
118. C₈	C₇×C₄⋊C₄	semidirect
119. C₇	#1770	direct (N × H)
127. C₄	C₇×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₈	2
C₂×C₇×#97	2
C₂×C₄×C₇×C₈	2
C₄×C₇×C₄⋊C₄	2
C₇×#336	2
#1770	7

Subgroups — C7×#1770 #11730239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1770 #11730239