Subgroups — #13265419

Order 1792.

232 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#11727497	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	C₂×#6386	—	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	#12017	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	C₈	1	2	yes	—
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	—	2	1	no	—
24	C₇×C₂²⋊C₈	—	2	1	no	—
25	C₇×#97	C₈	1	1	yes	characteristic
26	C₇×#97	C₈	1	1	yes	characteristic
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	#6386	—	4	2	no	—
34	C₂×C₇⋊C₁₆	—	2	1	no	—
35	C₂×C₇⋊C₁₆	—	2	1	no	—
36	C₂×C₇⋊C₁₆	—	2	2	no	—
37	#2163	—	7	1	no	—
38	#2150	—	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	4	no	—
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	—	2	1	no	—
48	C₂×C₇×C₈	—	2	2	no	—
49	C₂×C₇×C₈	—	4	1	no	—
50	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₇	M₁₆	1	1	yes	characteristic
59	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₇⋊C₁₆	—	4	1	no	—
61	C₇⋊C₁₆	—	4	2	no	—
62	C₇⋊C₁₆	—	4	1	no	—
63	C₂×M₃₂	—	14	1	no	—
64	#431	Dic₇	1	1	yes	characteristic
65	C₄×M₁₆	Dic₇	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
67	C₄⋊C₁₆	—	7	1	no	—
68	C₄⋊C₁₆	—	7	2	no	—
69	C₄⋊C₁₆	—	7	1	no	—
70	C₄⋊C₁₆	—	14	1	no	—
71	C₂²⋊C₁₆	—	14	1	no	—
72	#346	—	14	1	no	—
73	#322	Dic₇	1	2	yes	—
74	C₈²	C₂×D₇	1	2	yes	—
75	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	1	no	—
78	C₂×C₄×C₇	—	2	2	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×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
86	C₇×C₈	—	2	1	no	—
87	C₇×C₈	—	2	1	no	—
88	C₇×C₈	—	2	1	no	—
89	C₇×C₈	—	2	2	no	—
90	C₇×C₈	—	4	2	no	—
91	C₇×C₈	M₃₂	1	4	yes	complemented
92	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₇×C₈	#137	1	1	yes	characteristic
94	C₂×M₁₆	C₇⋊C₈	1	2	yes	—
95	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
96	C₂²×C₈	D₂₈	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
98	M₃₂	—	28	2	no	—
99	C₂×C₁₆	—	14	1	no	—
100	C₂×C₁₆	—	14	1	no	—
101	C₂×C₁₆	—	14	2	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₂²⋊C₈	—	2	1	no	—
104	#97	C₇⋊C₈	1	1	yes	characteristic
105	#97	C₇⋊C₈	1	1	yes	characteristic
106	C₄×C₈	Dic₁₄	1	1	yes	characteristic
107	C₄×C₈	D₂₈	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₇	1	2	yes	—
109	C₄×C₈	C₂×Dic₇	1	2	yes	—
110	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
111	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
112	C₂²×C₇	—	2	1	no	—
113	C₂²×C₇	—	2	1	no	—
114	C₂²×C₇	—	2	1	no	—
115	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	—	2	1	no	—
118	C₄×C₇	—	2	1	no	—
119	C₄×C₇	—	2	1	no	—
120	C₄×C₇	#428	1	1	yes	characteristic
121	C₄×C₇	#430	1	1	yes	characteristic
122	C₄×C₇	C₂×M₃₂	1	2	yes	—
123	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	#1136	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
132	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
136	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
137	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
138	C₄²	#1136	1	1	yes	characteristic
139	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
140	C₄²	C₂²×Dic₇	1	1	yes	characteristic
141	C₁₆	—	28	1	no	—
142	C₁₆	—	28	2	no	—
143	C₁₆	—	28	1	no	—
144	C₂×C₇	—	4	1	no	—
145	C₂×C₇	#1565	1	1	yes	characteristic
146	C₂×C₇	#2150	1	1	yes	characteristic
147	C₂×C₇	#2153	1	1	yes	characteristic
148	C₂³	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
154	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
155	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
157	C₂×C₄	C₂×#1136	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	2	no	—
164	C₈	#6386	1	4	yes	complemented
165	C₈	#6471	1	1	yes	characteristic
166	C₈	#6472	1	1	yes	characteristic
167	C₇	#12017	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	C₂×#6386	1	2	yes	—
177	C₄	#89893	1	1	yes	characteristic
178	C₄	#89897	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	#11727497	1	1	yes	characteristic
181	C₂	#11727604	1	1	yes	characteristic
182	C₂	#11727639	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	#12017
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
91. C₇×C₈	M₃₂	semidirect
164. C₈	#6386	semidirect
167. C₇	#12017	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
#11727497	2
#11725246	2
#11725245	2
#12017	7