Subgroups — #13270237

Order 1792.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#89300	C₂	1	2	yes	maximal
3	C₂×#89299	C₂	1	2	yes	maximal
4	#11727520	C₂	1	2	yes	maximal
5	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6392	C₂²	1	4	yes	—
9	C₂×#6391	—	2	2	no	—
10	#89708	—	2	2	no	—
11	C₄×C₇×C₁₆	C₄	1	4	yes	complemented
12	#89300	—	2	4	no	—
13	#89299	—	2	4	no	—
14	#12907	—	7	1	no	maximal, Sylow
15	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₇	—	2	4	no	—
19	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₇×C₁₆	D₄	1	2	yes	—
22	C₂×C₇×C₁₆	D₄	1	2	yes	—
23	C₂×C₇×C₁₆	Q₈	1	2	yes	—
24	C₂×C₇×C₁₆	Q₈	1	2	yes	—
25	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇×C₈	Q₈	1	2	yes	—
27	C₇⋊#95	—	4	4	no	—
28	#6392	—	2	4	no	—
29	#6392	—	2	4	no	—
30	#6391	—	4	4	no	—
31	C₂×#367	—	7	2	no	—
32	C₂×#366	—	7	2	no	—
33	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
34	#1846	—	7	2	no	—
35	C₂²×C₄×C₇	D₈	1	2	yes	—
36	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₇	—	4	4	no	—
38	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
39	C₂×C₇×C₈	D₈	1	2	yes	—
40	C₂×C₇×C₈	Q₁₆	1	2	yes	—
41	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇	C₄⋊C₄	1	2	yes	—
48	C₄²×C₇	Q₁₆	1	2	yes	—
49	C₄⋊Dic₇	—	4	4	no	—
50	C₄⋊Dic₇	—	4	4	no	—
51	C₇×C₁₆	C₄⋊C₄	1	4	yes	—
52	C₇×C₁₆	C₄⋊C₄	1	4	yes	—
53	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
54	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
55	C₂×#107	—	7	4	no	—
56	C₂×#106	—	14	2	no	—
57	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
58	#382	—	14	2	no	—
59	#367	—	14	4	no	—
60	#366	—	14	4	no	—
61	C₄×C₁₆	Dic₇	1	4	yes	complemented
62	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇	—	2	4	no	—
64	C₂×C₄×C₇	#107	1	4	yes	—
65	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇	#127	1	1	yes	characteristic
67	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₇	—	8	4	no	—
72	C₂×Dic₇	—	8	8	no	—
73	C₇×C₈	#107	1	4	yes	—
74	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
77	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₈	D₂₈	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	14	4	no	—
81	C₂×C₄²	D₂₈	1	1	yes	characteristic
82	C₂×C₁₆	Dic₁₄	1	2	yes	—
83	C₂×C₁₆	Dic₁₄	1	2	yes	—
84	C₂×C₁₆	D₂₈	1	2	yes	—
85	C₂×C₁₆	D₂₈	1	2	yes	—
86	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
87	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
88	#107	—	14	4	no	—
89	#107	—	14	4	no	—
90	#106	—	28	4	no	—
91	C₄×C₈	Dic₁₄	1	2	yes	—
92	C₄×C₈	C₂×Dic₇	1	2	yes	—
93	#95	—	28	4	no	—
94	C₂²×C₇	C₂×#107	1	1	yes	characteristic
95	C₂²×C₇	#493	1	1	yes	characteristic
96	C₂²×C₇	#500	1	1	yes	characteristic
97	C₂²×C₇	#508	1	4	yes	—
98	C₄×C₇	—	2	4	no	—
99	C₄×C₇	#384	1	1	yes	characteristic
100	C₄×C₇	C₂×#107	1	1	yes	characteristic
101	C₄×C₇	#494	1	1	yes	characteristic
102	C₄×C₇	#500	1	1	yes	characteristic
103	Dic₇	—	16	4	no	—
104	C₂²×C₄	—	28	4	no	—
105	C₂²×C₄	D₅₆	1	2	yes	—
106	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
107	C₂×C₈	D₅₆	1	2	yes	—
108	C₂×C₈	Dic₂₈	1	2	yes	—
109	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
110	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
111	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
114	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
115	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
116	C₄⋊C₄	—	28	4	no	—
117	C₄⋊C₄	—	28	4	no	—
118	C₄²	Dic₂₈	1	2	yes	—
119	C₄²	C₄⋊Dic₇	1	2	yes	—
120	C₁₆	C₄⋊Dic₇	1	4	yes	—
121	C₁₆	C₄⋊Dic₇	1	4	yes	—
122	C₂×C₇	#1845	1	1	yes	characteristic
123	C₂×C₇	#2158	1	2	yes	—
124	C₂×C₇	#2249	1	2	yes	—
125	C₂×C₇	#2253	1	2	yes	—
126	C₂³	C₂×D₅₆	1	1	yes	characteristic
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	56	8	no	—
129	C₂×C₄	—	56	4	no	—
130	C₂×C₄	#6392	1	4	yes	—
131	C₂×C₄	#6432	1	1	yes	characteristic
132	C₂×C₄	#6437	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
137	C₈	#6392	1	4	yes	—
138	C₈	#6432	1	1	yes	characteristic
139	C₈	#6432	1	1	yes	characteristic
140	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
141	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
142	C₇	#12907	1	1	yes	characteristic, complemented, Sylow
143	C₂²	#89454	1	1	yes	characteristic
144	C₂²	#89467	1	1	yes	characteristic
145	C₂²	#89676	1	4	yes	—
146	C₂²	C₂×#6392	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	112	4	no	—
149	C₄	#89454	1	1	yes	characteristic
150	C₄	#89471	1	1	yes	characteristic
151	C₄	#89707	1	1	yes	characteristic
152	C₄	C₂×#6392	1	1	yes	characteristic
153	C₂	#11726799	1	2	yes	—
154	C₂	#11726816	1	2	yes	—
155	C₂	#11727519	1	1	yes	characteristic
156	C₂	#11729158	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#12907
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₇×C₁₆	C₄	semidirect
61. C₄×C₁₆	Dic₇	semidirect
142. C₇	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×C₁₆	2
C₂×#89300	2
C₂×#89299	2
#11727520	2
#12907	7