Subgroups — #13265414

Order 1792.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#11727499	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#11725246	C₂	1	2	yes	maximal, complemented
5	#11725245	C₂	1	2	yes	maximal, complemented
6	C₇×#431	C₄	1	1	yes	characteristic
7	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇⋊C₁₆	—	2	1	no	—
10	C₇×#322	C₄	1	2	yes	—
11	C₇×C₈²	C₂²	1	2	yes	—
12	C₂²⋊(C₇⋊C₁₆)	—	2	1	no	—
13	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
14	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
17	#89256	—	2	1	no	—
18	#12016	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	—	2	1	no	—
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₇×C₄⋊C₈	C₈	1	2	yes	—
24	C₇×C₂²⋊C₈	C₈	1	2	yes	—
25	C₇×#97	—	2	1	no	—
26	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
31	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₇⋊C₁₆	—	2	1	no	—
33	C₂×C₇⋊C₁₆	—	2	1	no	—
34	C₂×C₇⋊C₁₆	—	2	2	no	—
35	C₂×C₇⋊C₁₆	—	4	2	no	—
36	#2163	—	7	1	no	—
37	C₂×C₄⋊C₁₆	—	7	1	no	—
38	C₈×M₁₆	D₇	1	1	yes	characteristic
39	#1370	—	7	2	no	—
40	#1367	—	7	2	no	—
41	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
44	C₇×M₁₆	—	2	4	no	—
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	—	2	1	no	—
48	C₂×C₇×C₈	—	4	1	no	—
49	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
52	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₇	M₁₆	1	1	yes	characteristic
57	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₇⋊C₁₆	—	4	1	no	—
59	C₇⋊C₁₆	—	4	2	no	—
60	C₇⋊C₁₆	—	4	1	no	—
61	C₂²×C₁₆	—	14	1	no	—
62	#431	Dic₇	1	1	yes	characteristic
63	C₄×M₁₆	Dic₇	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
65	C₄⋊C₁₆	—	7	1	no	—
66	C₄⋊C₁₆	—	7	2	no	—
67	C₄⋊C₁₆	—	7	1	no	—
68	C₄⋊C₁₆	—	14	1	no	—
69	C₂²⋊C₁₆	—	14	1	no	—
70	#346	—	14	1	no	—
71	#322	Dic₇	1	2	yes	—
72	C₈²	C₂×D₇	1	2	yes	—
73	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
84	C₇×C₈	—	2	4	no	—
85	C₇×C₈	—	2	1	no	—
86	C₇×C₈	—	2	1	no	—
87	C₇×C₈	—	2	1	no	—
88	C₇×C₈	—	2	4	no	—
89	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₇×C₈	#137	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
93	C₂²×C₈	D₂₈	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
95	C₂×C₁₆	—	14	1	no	—
96	C₂×C₁₆	—	14	1	no	—
97	C₂×C₁₆	—	14	2	no	—
98	C₂×C₁₆	—	28	2	no	—
99	C₄⋊C₈	C₇⋊C₈	1	2	yes	—
100	C₂²⋊C₈	C₇⋊C₈	1	2	yes	—
101	#97	—	2	1	no	—
102	C₄×C₈	Dic₁₄	1	1	yes	characteristic
103	C₄×C₈	D₂₈	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₇	1	2	yes	—
105	C₄×C₈	C₂×Dic₇	1	2	yes	—
106	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
108	C₂²×C₇	—	2	1	no	—
109	C₂²×C₇	—	2	1	no	—
110	C₂²×C₇	#364	1	2	yes	—
111	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₇	—	2	1	no	—
113	C₄×C₇	—	2	1	no	—
114	C₄×C₇	—	2	1	no	—
115	C₄×C₇	—	2	1	no	—
116	C₄×C₇	#428	1	1	yes	characteristic
117	C₄×C₇	#430	1	1	yes	characteristic
118	C₄×C₇	#504	1	2	yes	—
119	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	#1136	1	1	yes	characteristic
121	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
128	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
131	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
132	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
133	C₄²	#1136	1	1	yes	characteristic
134	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
135	C₄²	C₂²×Dic₇	1	1	yes	characteristic
136	C₁₆	—	28	1	no	—
137	C₁₆	—	28	2	no	—
138	C₁₆	—	28	1	no	—
139	C₂×C₇	—	2	2	no	—
140	C₂×C₇	#1565	1	1	yes	characteristic
141	C₂×C₇	#2151	1	1	yes	characteristic
142	C₂×C₇	C₂×#364	1	1	yes	characteristic
143	C₂³	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
149	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
152	C₂×C₄	C₂×#1136	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
154	C₈	—	2	4	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	4	no	—
159	C₈	#6471	1	1	yes	characteristic
160	C₈	#6472	1	1	yes	characteristic
161	C₇	#12016	1	1	yes	characteristic, complemented, Sylow
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#89258	1	2	yes	—
165	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#89893	1	1	yes	characteristic
171	C₄	#89897	1	1	yes	characteristic
172	C₄	#89912	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	C₂×#89258	1	1	yes	characteristic
175	C₂	#11727604	1	1	yes	characteristic
176	C₂	#11727636	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×M₁₆	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₈×M₁₆	D₇
hypocenter	C₇	#12016
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #11725246	C₂	semidirect
5. #11725245	C₂	semidirect
161. C₇	#12016	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₈×M₁₆	2
#11727499	2
C₂×C₄⋊(C₇⋊C₁₆)	2
#11725246	2
#11725245	2
#12016	7