Subgroups — #14907422

Order 2816.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070525	C₂	1	6	yes	maximal, complemented
2	Q₈×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	#13070419	C₂	1	3	yes	maximal
4	#13070415	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	3	yes	—
8	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₁	C₂²	1	3	yes	—
11	C₂×C₁₁×C₄⋊C₈	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
13	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	3	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
17	#10622099	C₂²	1	6	yes	—
18	C₄×C₁₁⋊C₁₆	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₁	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₁	C₈	1	3	yes	—
23	C₄×Q₈×C₁₁	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₁	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₁×C₄⋊C₄	C₈	1	3	yes	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
28	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
29	C₁₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
30	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₁	C₂³	1	3	yes	—
34	C₄×C₈×C₁₁	C₂³	1	3	yes	—
35	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
36	C₂×C₁₁⋊C₁₆	C₂³	1	6	yes	—
37	#33498	—	11	1	no	maximal, Sylow
38	C₂×Q₈×C₁₁	—	2	4	no	—
39	C₂×Q₈×C₁₁	C₂×C₈	1	3	yes	—
40	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	—	2	2	no	—
45	C₂×C₈×C₁₁	—	4	3	no	—
46	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
50	C₁₁×C₄⋊C₄	C₂×C₈	1	6	yes	—
51	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
52	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₄²×C₁₁	C₂×C₈	1	3	yes	—
54	C₄²×C₁₁	C₂×C₈	1	3	yes	—
55	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
56	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
57	C₁₁⋊C₁₆	—	2	2	no	—
58	C₁₁⋊C₁₆	—	2	6	no	—
59	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
60	#2183	—	11	6	no	—
61	Q₈×C₁₆	—	11	2	no	—
62	#2163	—	11	3	no	—
63	#2162	—	11	3	no	—
64	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
65	Q₈×C₁₁	—	2	6	no	—
66	Q₈×C₁₁	M₃₂	1	4	yes	—
67	C₂×C₄×C₁₁	—	2	6	no	—
68	C₂×C₄×C₁₁	—	2	1	no	—
69	C₂×C₄×C₁₁	—	2	1	no	—
70	C₂×C₄×C₁₁	—	2	6	no	—
71	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₁₁	—	2	1	no	—
78	C₈×C₁₁	—	2	3	no	—
79	C₈×C₁₁	—	2	3	no	—
80	C₈×C₁₁	2₋¹⁺⁴	1	2	yes	—
81	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
82	C₈×Q₈	Dic₁₁	1	3	yes	—
83	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
84	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
85	C₈×Q₈	C₂×D₁₁	1	3	yes	—
86	C₂×C₄⋊C₈	Dic₁₁	1	3	yes	—
87	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
88	C₄⋊C₁₆	—	11	6	no	—
89	C₄⋊C₁₆	—	11	6	no	—
90	C₂²⋊C₁₆	—	11	1	no	—
91	C₂²⋊C₁₆	—	11	3	no	—
92	#346	—	11	6	no	—
93	C₄×C₁₆	—	11	6	no	—
94	C₂²×C₁₁	—	2	1	no	—
95	C₂²×C₁₁	—	2	1	no	—
96	C₂²×C₁₁	#504	1	2	yes	—
97	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
98	C₄×C₁₁	—	2	3	no	—
99	C₄×C₁₁	—	2	3	no	—
100	C₄×C₁₁	—	2	3	no	—
101	C₄×C₁₁	—	2	1	no	—
102	C₄×C₁₁	C₂×M₃₂	1	6	yes	—
103	C₄×C₁₁	#519	1	1	yes	characteristic
104	C₄×C₁₁	#568	1	1	yes	characteristic
105	C₂²×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
107	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
108	C₄×Q₈	C₁₁⋊C₈	1	3	yes	—
109	C₄×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
110	C₄×Q₈	C₂×Dic₁₁	1	3	yes	—
111	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₁₁⋊C₈	1	3	yes	—
113	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
114	C₂×C₁₆	—	11	2	no	—
115	C₂×C₁₆	—	11	6	no	—
116	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
117	C₄⋊C₈	C₂×Dic₁₁	1	6	yes	—
118	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
119	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
120	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
121	C₄×C₈	C₂²×D₁₁	1	3	yes	—
122	C₄×C₈	C₂²×D₁₁	1	3	yes	—
123	C₂×C₁₁	—	2	2	no	—
124	C₂×C₁₁	#2982	1	1	yes	characteristic
125	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
126	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	C₂×C₁₁⋊C₈	1	3	yes	—
129	C₂×Q₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
131	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
132	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	3	no	—
135	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
136	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
137	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
138	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
139	C₄⋊C₄	C₂×C₁₁⋊C₈	1	6	yes	—
140	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
141	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
142	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
143	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
144	C₄²	C₂²×Dic₁₁	1	3	yes	—
145	C₄²	C₂²×Dic₁₁	1	3	yes	—
146	C₁₆	—	22	2	no	—
147	C₁₆	—	22	6	no	—
148	C₁₁	#33498	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	#66876	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
157	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
158	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
159	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
160	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
161	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	3	no	—
164	C₈	—	2	3	no	—
165	C₈	#67040	1	2	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#10622753	1	2	yes	—
169	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
170	C₄	—	2	3	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	1	no	—
174	C₄	C₂×#66876	1	6	yes	—
175	C₄	#10623343	1	1	yes	characteristic
176	C₄	#10623350	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	#13077697	1	1	yes	characteristic
179	C₂	C₂²×#66876	1	1	yes	characteristic
180	C₂	C₂×#10622753	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₁₁	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	#33498
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #13070525	C₂	semidirect
2. Q₈×C₁₁⋊C₁₆	C₂	semidirect
148. C₁₁	#33498	semidirect

Maximal subgroups

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

subgroup	index
#13070525	2
Q₈×C₁₁⋊C₁₆	2
#13070419	2
#13070415	2
C₂×C₈×Q₈×C₁₁	2
#33498	11