Subgroups — #13265441

Order 1792.

242 conjugacy classes of subgroups.

The subgroup lattice has 148 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	#11727485	C₂	1	2	yes	maximal
3	#11725307	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	#89696	—	2	2	no	—
7	C₄×C₇×C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₇×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₇×C₁₆	C₂²	1	2	yes	—
10	#89253	—	2	2	no	—
11	#89253	C₂²	1	4	yes	—
12	#12023	—	7	1	no	maximal, Sylow
13	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
16	C₂×#1136	—	4	2	no	—
17	C₂×C₇×C₁₆	—	2	4	no	—
18	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
20	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₇×C₈	D₄	1	1	yes	characteristic
22	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
24	C₄⋊(C₇⋊C₈)	—	2	4	no	—
25	C₄⋊(C₇⋊C₈)	—	4	2	no	—
26	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
27	#1567	—	7	2	no	—
28	#1372	—	7	4	no	—
29	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
32	C₂×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₇	D₈	1	1	yes	characteristic
40	C₄²×C₇	Q₁₆	1	1	yes	characteristic
41	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
42	#1136	—	8	4	no	—
43	C₂×C₇⋊C₈	—	4	4	no	—
44	C₇×C₁₆	—	2	2	no	—
45	C₇×C₁₆	M₁₆	1	4	yes	complemented
46	C₂²×C₁₆	Dic₇	1	2	yes	—
47	#423	—	14	2	no	—
48	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
49	C₄×C₁₆	Dic₇	1	1	yes	characteristic
50	C₄×C₁₆	Dic₇	1	1	yes	characteristic
51	C₄×C₁₆	C₂×D₇	1	2	yes	—
52	#335	—	7	4	no	—
53	#335	—	14	2	no	—
54	C₂³×C₇	#107	1	1	yes	characteristic
55	C₂×C₄×C₇	—	2	2	no	—
56	C₂×C₄×C₇	—	2	1	no	—
57	C₂×C₄×C₇	—	2	1	no	—
58	C₂×C₄×C₇	#107	1	1	yes	characteristic
59	C₂×C₄×C₇	#107	1	1	yes	characteristic
60	C₂×C₄×C₇	#107	1	1	yes	characteristic
61	C₂×C₄×C₇	SD₃₂	1	4	yes	—
62	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₇×C₈	—	2	1	no	—
66	C₇×C₈	—	2	1	no	—
67	C₇×C₈	#108	1	2	yes	—
68	C₇×C₈	C₂×M₁₆	1	2	yes	—
69	C₇⋊C₈	—	8	4	no	—
70	C₂×M₁₆	—	28	2	no	—
71	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
73	C₂×C₄²	D₂₈	1	1	yes	characteristic
74	C₂×C₁₆	—	2	4	no	—
75	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
76	C₂×C₁₆	C₂×Dic₇	1	2	yes	—
77	C₄⋊C₈	—	14	4	no	—
78	C₄⋊C₈	—	28	2	no	—
79	C₄×C₈	Dic₁₄	1	1	yes	characteristic
80	C₄×C₈	D₂₈	1	1	yes	characteristic
81	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
82	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
83	C₂²×C₇	—	2	1	no	—
84	C₂²×C₇	—	2	1	no	—
85	C₂²×C₇	#367	1	2	yes	—
86	C₂²×C₇	C₂×#107	1	1	yes	characteristic
87	C₄×C₇	—	2	1	no	—
88	C₄×C₇	—	2	1	no	—
89	C₄×C₇	#367	1	2	yes	—
90	C₄×C₇	#368	1	2	yes	—
91	C₄×C₇	#423	1	1	yes	characteristic
92	C₄×C₇	C₂×#108	1	1	yes	characteristic
93	C₄×C₇	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₄	D₅₆	1	1	yes	characteristic
95	C₂²×C₄	Dic₂₈	1	1	yes	characteristic
96	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
97	M₁₆	—	56	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	28	4	no	—
101	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
102	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
103	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
104	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
105	C₄²	D₅₆	1	1	yes	characteristic
106	C₄²	Dic₂₈	1	1	yes	characteristic
107	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
108	C₄²	C₂×D₂₈	1	1	yes	characteristic
109	C₁₆	—	2	2	no	—
110	C₁₆	#1136	1	4	yes	complemented
111	C₂×C₇	—	2	2	no	—
112	C₂×C₇	#1567	1	1	yes	characteristic
113	C₂×C₇	C₂×#367	1	1	yes	characteristic
114	C₂×C₇	C₂×#368	1	1	yes	characteristic
115	C₂³	#6392	1	1	yes	characteristic
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	#6374	1	4	yes	—
120	C₂×C₄	#6392	1	1	yes	characteristic
121	C₂×C₄	#6392	1	1	yes	characteristic
122	C₂×C₄	#6392	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
124	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	56	4	no	—
129	C₈	#6393	1	2	yes	—
130	C₈	C₂×#1136	1	2	yes	—
131	C₇	#12023	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#89300	1	2	yes	—
135	C₂²	C₂×#6392	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	#89300	1	2	yes	—
139	C₄	#89301	1	2	yes	—
140	C₄	C₂×#6374	1	2	yes	—
141	C₄	#89696	1	1	yes	characteristic
142	C₄	C₂×#6393	1	1	yes	characteristic
143	C₂	—	2	2	no	—
144	C₂	#11727485	1	1	yes	characteristic
145	C₂	C₂×#89300	1	1	yes	characteristic
146	C₂	C₂×#89301	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂×#107
prime radical	C₂³×C₇	#107
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#12023
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #11725307	C₂	semidirect
45. C₇×C₁₆	M₁₆	semidirect
110. C₁₆	#1136	semidirect
131. C₇	#12023	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
#11727485	2
#11725307	2
#12023	7