Subgroups — C₂²×#89256 #13506422

Order 1792.

942 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×#89256	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₄	1	6	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	6	yes	—
9	C₂²×C₇⋊C₁₆	C₄	1	16	yes	complemented
10	C₂²×C₇⋊C₁₆	C₂²	1	12	yes	—
11	#89256	C₂²	1	16	yes	direct product factor
12	C₂²×#346	—	7	1	no	maximal, Sylow
13	C₂³×C₄×C₇	C₈	1	2	yes	—
14	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₈	1	8	yes	—
16	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
17	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	6	yes	—
19	C₂²×C₇×C₈	C₂³	1	3	yes	—
20	C₂×C₄²×C₇	C₈	1	6	yes	—
21	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
22	C₄×C₇×C₈	C₂×C₄	1	12	yes	—
23	C₄×C₇×C₈	C₂³	1	4	yes	—
24	C₂×C₇⋊C₁₆	C₂×C₄	1	48	yes	complemented
25	C₂×C₇⋊C₁₆	C₂³	1	8	yes	—
26	C₂³×C₁₆	—	7	2	no	—
27	C₂²×C₄×C₈	D₇	1	1	yes	characteristic
28	C₂×#346	—	7	12	no	—
29	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₇	—	2	8	no	—
31	C₂²×C₄×C₇	C₄²	1	4	yes	—
32	C₂²×C₄×C₇	C₂×C₈	1	12	yes	—
33	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
36	C₂×C₇×C₈	C₄²	1	12	yes	—
37	C₂×C₇×C₈	C₂×C₈	1	24	yes	—
38	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
39	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
40	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
41	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
42	C₄²×C₇	C₂×C₈	1	12	yes	—
43	C₄²×C₇	C₂²×C₄	1	4	yes	—
44	C₇⋊C₁₆	C₂²×C₄	1	16	yes	complemented
45	C₂³×C₈	Dic₇	1	1	yes	characteristic
46	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
47	C₂²×C₄²	Dic₇	1	1	yes	characteristic
48	C₂²×C₁₆	—	7	16	no	—
49	C₂²×C₁₆	—	7	12	no	—
50	C₂×C₄×C₈	Dic₇	1	6	yes	—
51	C₂×C₄×C₈	C₂×D₇	1	6	yes	—
52	#346	—	7	16	no	—
53	C₂³×C₇	C₄×C₈	1	4	yes	—
54	C₂³×C₇	M₃₂	1	8	yes	—
55	C₂³×C₇	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₇	—	2	24	no	—
57	C₂×C₄×C₇	C₄×C₈	1	12	yes	—
58	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
59	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
60	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
61	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
63	C₇×C₈	C₂×C₄²	1	6	yes	—
64	C₇×C₈	C₂²×C₈	1	8	yes	—
65	C₇×C₈	C₂³×C₄	1	2	yes	—
66	C₂³×C₄	C₇⋊C₈	1	2	yes	—
67	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
68	C₂²×C₈	C₇⋊C₈	1	8	yes	—
69	C₂²×C₈	C₄×D₇	1	8	yes	—
70	C₂²×C₈	C₂×Dic₇	1	6	yes	—
71	C₂²×C₈	C₂×Dic₇	1	3	yes	—
72	C₂²×C₈	C₂²×D₇	1	3	yes	—
73	C₂×C₄²	C₇⋊C₈	1	6	yes	—
74	C₂×C₄²	C₂×Dic₇	1	6	yes	—
75	C₂×C₁₆	—	7	48	no	—
76	C₂×C₁₆	—	7	8	no	—
77	C₄×C₈	C₂×Dic₇	1	12	yes	—
78	C₄×C₈	C₂²×D₇	1	4	yes	—
79	C₂²×C₇	#346	1	16	yes	complemented
80	C₂²×C₇	C₂×C₄×C₈	1	6	yes	—
81	C₂²×C₇	C₂×M₃₂	1	12	yes	—
82	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₇	—	2	8	no	—
84	C₄×C₇	C₂×C₄×C₈	1	6	yes	—
85	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
87	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₂×C₇⋊C₈	1	12	yes	—
90	C₂²×C₄	C₂×C₇⋊C₈	1	4	yes	—
91	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
92	C₂²×C₄	C₄×Dic₇	1	4	yes	—
93	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
94	C₂×C₈	C₂×C₇⋊C₈	1	24	yes	—
95	C₂×C₈	C₄×Dic₇	1	12	yes	—
96	C₂×C₈	C₂×C₄×D₇	1	12	yes	—
97	C₂×C₈	C₂²×Dic₇	1	4	yes	—
98	C₂×C₈	C₂²×Dic₇	1	3	yes	—
99	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
100	C₄²	C₂×C₇⋊C₈	1	12	yes	—
101	C₄²	C₂²×Dic₇	1	4	yes	—
102	C₁₆	—	7	16	no	—
103	C₂×C₇	C₂×#346	1	12	yes	complemented
104	C₂×C₇	C₂²×C₄×C₈	1	1	yes	characteristic
105	C₂×C₇	C₂²×M₃₂	1	2	yes	—
106	C₂³	C₄×C₇⋊C₈	1	4	yes	—
107	C₂³	#6386	1	8	yes	—
108	C₂³	C₂²×C₇⋊C₈	1	3	yes	—
109	C₂×C₄	—	2	24	no	—
110	C₂×C₄	C₄×C₇⋊C₈	1	12	yes	—
111	C₂×C₄	C₂²×C₇⋊C₈	1	8	yes	—
112	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
113	C₂×C₄	C₂²×C₇⋊C₈	1	6	yes	—
114	C₂×C₄	C₂×C₄×Dic₇	1	6	yes	—
115	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
116	C₈	C₂²×C₇⋊C₈	1	8	yes	—
117	C₈	C₂×C₄×Dic₇	1	6	yes	—
118	C₈	C₂²×C₄×D₇	1	2	yes	—
119	C₇	C₂²×#346	1	1	yes	characteristic, complemented, Sylow
120	C₂²	#89256	1	16	yes	direct product factor
121	C₂²	C₂×C₄×C₇⋊C₈	1	6	yes	—
122	C₂²	C₂×#6386	1	12	yes	—
123	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
124	C₄	—	2	8	no	—
125	C₄	C₂×C₄×C₇⋊C₈	1	6	yes	—
126	C₄	C₂³×C₇⋊C₈	1	1	yes	characteristic
127	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
128	C₂	C₂×#89256	1	12	yes	direct product factor
129	C₂	C₂²×C₄×C₇⋊C₈	1	1	yes	characteristic
130	C₂	C₂²×#6386	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₇
commutator	C₂×C₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂²×C₄×C₇×C₈	C₂
socle, prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇
hypocenter	C₇	C₂²×#346
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#89256	C₂	direct (N × H)
9. C₂²×C₇⋊C₁₆	C₄	semidirect
11. #89256	C₂²	direct (N × H)
24. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
44. C₇⋊C₁₆	C₂²×C₄	semidirect
79. C₂²×C₇	#346	semidirect
103. C₂×C₇	C₂×#346	semidirect
119. C₇	C₂²×#346	semidirect
120. C₂²	#89256	direct (N × H)
128. C₂	C₂×#89256	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂³×C₇⋊C₁₆	2
C₂×#89256	2
C₂²×#346	7

Subgroups — C22×#89256 #13506422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89256 #13506422