Subgroups — #13265407

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	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#11727498	C₂	1	1	yes	maximal, characteristic
3	C₈×#1136	C₂	1	1	yes	maximal, characteristic
4	#11725361	C₂	1	2	yes	maximal, complemented
5	#11725360	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×C₁₆	—	2	1	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	#89697	C₄	1	1	yes	characteristic
9	C₄×#1136	C₄	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	—	2	1	no	—
11	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₂²⋊(C₇⋊C₁₆)	—	2	1	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₄×C₇⋊C₁₆	—	2	1	no	—
16	#89249	C₄	1	2	yes	—
17	C₈×C₇⋊C₈	C₂²	1	2	yes	—
18	#12015	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#1136	C₈	1	1	yes	characteristic
23	C₂×#1136	C₈	1	1	yes	characteristic
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₂×C₇×C₁₆	—	4	2	no	—
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	D₄	1	1	yes	characteristic
28	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
31	C₂×C₇⋊C₁₆	—	2	1	no	—
32	C₂×C₇⋊C₁₆	—	2	1	no	—
33	C₄⋊(C₇⋊C₈)	—	2	1	no	—
34	#6377	C₈	1	2	yes	—
35	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
37	#2162	—	7	1	no	—
38	C₂×C₄⋊C₁₆	D₇	1	1	yes	characteristic
39	C₈×M₁₆	—	7	1	no	—
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	C₂×C₇×C₈	—	2	1	no	—
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	—	4	1	no	—
48	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₇	M₁₆	1	1	yes	characteristic
55	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
56	#1136	—	2	2	no	—
57	#1136	C₁₆	1	4	yes	complemented
58	C₂×C₇⋊C₈	—	2	2	no	—
59	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
60	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
61	C₇×C₁₆	—	4	2	no	—
62	C₇⋊C₁₆	—	4	1	no	—
63	C₇⋊C₁₆	—	4	1	no	—
64	C₂²×C₁₆	—	2	1	no	—
65	#431	—	7	1	no	—
66	C₄×M₁₆	—	7	1	no	—
67	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
68	C₄⋊C₁₆	—	2	1	no	—
69	C₄⋊C₁₆	—	7	1	no	—
70	C₄⋊C₁₆	—	7	1	no	—
71	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
72	C₂²⋊C₁₆	—	14	1	no	—
73	C₄×C₁₆	—	14	1	no	—
74	#322	—	7	2	no	—
75	C₈²	—	7	2	no	—
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	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₇×C₈	#137	1	1	yes	characteristic
93	C₇⋊C₈	—	2	2	no	—
94	C₇⋊C₈	—	4	2	no	—
95	C₇⋊C₈	C₂×C₁₆	1	4	yes	complemented
96	C₂×M₁₆	—	7	1	no	—
97	C₂×M₁₆	—	7	1	no	—
98	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
100	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	—	14	1	no	—
104	C₂×C₁₆	—	14	1	no	—
105	C₄⋊C₈	—	14	1	no	—
106	C₂²⋊C₈	—	14	1	no	—
107	#97	—	7	2	no	—
108	C₄×C₈	—	7	2	no	—
109	C₄×C₈	—	7	2	no	—
110	C₄×C₈	Dic₁₄	1	1	yes	characteristic
111	C₄×C₈	C₄×D₇	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇	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	C₈×D₇	1	1	yes	characteristic
127	C₂²×C₄	#1131	1	1	yes	characteristic
128	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
129	M₁₆	—	7	4	no	—
130	M₁₆	—	14	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	7	2	no	—
135	C₂×C₈	—	7	2	no	—
136	C₂×C₈	—	14	2	no	—
137	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
138	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
141	C₄²	C₈×D₇	1	1	yes	characteristic
142	C₄²	#1131	1	1	yes	characteristic
143	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
144	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
145	C₁₆	—	4	2	no	—
146	C₁₆	—	28	1	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	Dic₇⋊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	D₇×C₁₆	1	2	yes	—
157	C₂×C₄	#6372	1	2	yes	—
158	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
159	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
160	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
162	C₂×C₄	C₂×#1131	1	1	yes	characteristic
163	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	7	4	no	—
168	C₈	—	14	2	no	—
169	C₈	—	28	2	no	—
170	C₈	#6495	1	1	yes	characteristic
171	C₈	#6505	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	Dic₇⋊C₁₆	1	2	yes	—
176	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	Dic₇⋊C₁₆	1	2	yes	—
181	C₄	C₂×D₇×C₁₆	1	1	yes	characteristic
182	C₄	C₂×#6372	1	1	yes	characteristic
183	C₄	#89736	1	1	yes	characteristic
184	C₄	#89896	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#11728174	1	1	yes	characteristic
187	C₂	C₂×Dic₇⋊C₁₆	1	1	yes	characteristic
188	C₂	#11729177	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×C₄⋊C₁₆	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇
hypocenter	C₇	#12015
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #11725361	C₂	semidirect
5. #11725360	C₂	semidirect
57. #1136	C₁₆	semidirect
95. C₇⋊C₈	C₂×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
C₂×C₇×C₄⋊C₁₆	2
#11727498	2
C₈×#1136	2
#11725361	2
#11725360	2
#12015	7