Subgroups — #13265409

Order 1792.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 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	#11727498	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	C₄×C₇⋊C₁₆	—	2	1	no	—
18	#12015	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	C₈	1	1	yes	characteristic
20	C₂×C₇×M₁₆	C₈	1	1	yes	characteristic
21	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₇×C₄⋊C₈	—	2	1	no	—
25	C₇×C₂²⋊C₈	—	2	1	no	—
26	C₇×#97	C₈	1	2	yes	—
27	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₇×C₈	D₄	1	1	yes	characteristic
31	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₇⋊C₁₆	—	2	1	no	—
34	C₂×C₇⋊C₁₆	—	2	1	no	—
35	C₂×C₇⋊C₁₆	—	2	2	no	—
36	C₂×C₇⋊C₁₆	—	4	2	no	—
37	#2162	—	7	1	no	—
38	C₂×C₄⋊C₁₆	—	7	1	no	—
39	C₈×M₁₆	D₇	1	1	yes	characteristic
40	#1370	—	7	2	no	—
41	#1367	—	7	2	no	—
42	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
45	C₇×M₁₆	—	2	2	no	—
46	C₇×M₁₆	C₁₆	1	4	yes	complemented
47	C₂×C₇×C₈	—	2	1	no	—
48	C₂×C₇×C₈	—	2	1	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	4	1	no	—
51	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₇	M₁₆	1	1	yes	characteristic
60	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₇⋊C₁₆	—	4	1	no	—
62	C₇⋊C₁₆	—	4	2	no	—
63	C₇⋊C₁₆	—	4	1	no	—
64	C₂²×C₁₆	—	14	1	no	—
65	#431	Dic₇	1	1	yes	characteristic
66	C₄×M₁₆	Dic₇	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
68	C₄⋊C₁₆	—	7	1	no	—
69	C₄⋊C₁₆	—	7	2	no	—
70	C₄⋊C₁₆	—	7	1	no	—
71	C₄⋊C₁₆	—	14	1	no	—
72	C₂²⋊C₁₆	—	14	1	no	—
73	C₄×C₁₆	—	14	1	no	—
74	#322	Dic₇	1	2	yes	—
75	C₈²	C₂×D₇	1	2	yes	—
76	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇	—	2	1	no	—
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	4	1	no	—
80	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₇	M₃₂	1	2	yes	—
85	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
88	C₇×C₈	—	2	1	no	—
89	C₇×C₈	—	2	1	no	—
90	C₇×C₈	—	2	1	no	—
91	C₇×C₈	—	2	2	no	—
92	C₇×C₈	—	4	2	no	—
93	C₇×C₈	C₂×C₁₆	1	4	yes	complemented
94	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₇×C₈	#137	1	1	yes	characteristic
96	C₂×M₁₆	C₇⋊C₈	1	1	yes	characteristic
97	C₂×M₁₆	C₇⋊C₈	1	1	yes	characteristic
98	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
99	C₂²×C₈	D₂₈	1	1	yes	characteristic
100	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
101	C₂×C₁₆	—	14	1	no	—
102	C₂×C₁₆	—	14	1	no	—
103	C₂×C₁₆	—	14	2	no	—
104	C₂×C₁₆	—	28	2	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₂²⋊C₈	—	2	1	no	—
107	#97	C₇⋊C₈	1	2	yes	—
108	C₄×C₈	Dic₁₄	1	1	yes	characteristic
109	C₄×C₈	D₂₈	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₇	1	2	yes	—
111	C₄×C₈	C₂×Dic₇	1	2	yes	—
112	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
114	C₂²×C₇	—	2	1	no	—
115	C₂²×C₇	—	2	1	no	—
116	C₂²×C₇	C₄⋊C₁₆	1	2	yes	—
117	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₇	—	2	1	no	—
119	C₄×C₇	—	2	1	no	—
120	C₄×C₇	—	2	1	no	—
121	C₄×C₇	C₄⋊C₁₆	1	2	yes	—
122	C₄×C₇	#428	1	1	yes	characteristic
123	C₄×C₇	#430	1	1	yes	characteristic
124	C₄×C₇	C₂²×C₁₆	1	1	yes	characteristic
125	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	#1136	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	C₇⋊C₁₆	1	4	yes	complemented
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
136	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
137	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
140	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
141	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
142	C₄²	#1136	1	1	yes	characteristic
143	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
144	C₄²	C₂²×Dic₇	1	1	yes	characteristic
145	C₁₆	—	28	1	no	—
146	C₁₆	—	28	2	no	—
147	C₁₆	—	28	1	no	—
148	C₂×C₇	—	2	2	no	—
149	C₂×C₇	#1565	1	1	yes	characteristic
150	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
151	C₂×C₇	#2153	1	1	yes	characteristic
152	C₂³	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
157	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
158	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
159	C₂×C₄	C₂×C₇⋊C₁₆	1	2	yes	—
160	C₂×C₄	#6386	1	2	yes	—
161	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#1136	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	4	2	no	—
169	C₈	C₂×C₇⋊C₁₆	1	4	yes	complemented
170	C₈	#6471	1	1	yes	characteristic
171	C₈	#6472	1	1	yes	characteristic
172	C₇	#12015	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊(C₇⋊C₁₆)	1	2	yes	—
176	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊(C₇⋊C₁₆)	1	2	yes	—
181	C₄	C₂²×C₇⋊C₁₆	1	1	yes	characteristic
182	C₄	C₂×#6386	1	1	yes	characteristic
183	C₄	#89893	1	1	yes	characteristic
184	C₄	#89897	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×C₄⋊(C₇⋊C₁₆)	1	1	yes	characteristic
187	C₂	#11727604	1	1	yes	characteristic
188	C₂	#11727639	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₁₆
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	#12015
LCS	G ▷ 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
46. C₇×M₁₆	C₁₆	semidirect
93. C₇×C₈	C₂×C₁₆	semidirect
130. M₁₆	C₇⋊C₁₆	semidirect
169. C₈	C₂×C₇⋊C₁₆	semidirect
172. C₇	#12015	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
#11727498	2
C₂×C₄⋊(C₇⋊C₁₆)	2
#11725246	2
#11725245	2
#12015	7