Subgroups — #14907353

Order 2816.

548 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	Q₈×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	#13070448	C₂	1	6	yes	maximal, complemented
3	#13070419	C₂	1	3	yes	maximal, complemented
4	C₄×#66876	C₂	1	3	yes	maximal, complemented
5	C₈×C₁₁×#41	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₁×#41	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₁₁	C₄	1	3	yes	—
10	C₈×D₄×C₁₁	C₂²	1	3	yes	—
11	C₁₁×#431	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
13	C₂×#66876	C₂²	1	6	yes	—
14	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
16	#10622099	C₂²	1	3	yes	—
17	#10622099	C₂²	1	1	yes	characteristic
18	C₄×C₁₁⋊C₁₆	C₂²	1	6	yes	—
19	C₂×C₁₁×#41	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
22	C₄×Q₈×C₁₁	C₈	1	1	yes	characteristic
23	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×D₄×C₁₁	C₈	1	3	yes	—
25	C₄×D₄×C₁₁	C₂×C₄	1	3	yes	—
26	C₁₁×#117	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	3	yes	—
30	C₁₁×C₂²⋊C₈	C₂×C₄	1	6	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	1	yes	characteristic
35	#66876	—	2	12	no	—
36	C₂×C₁₁⋊C₁₆	C₂³	1	6	yes	—
37	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
38	#33485	—	11	1	no	maximal, Sylow
39	C₁₁×#41	—	2	4	no	—
40	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
41	C₂×D₄×C₁₁	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₁₁	—	2	6	no	—
46	C₂×C₈×C₁₁	—	2	6	no	—
47	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
51	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
52	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₁₁×C₂²⋊C₄	C₂×C₈	1	6	yes	—
54	C₄²×C₁₁	C₂×C₈	1	3	yes	—
55	C₄²×C₁₁	C₂×C₈	1	3	yes	—
56	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
58	C₁₁⋊C₁₆	—	2	6	no	—
59	C₁₁⋊C₁₆	#41	1	4	yes	complemented
60	C₈×#41	D₁₁	1	1	yes	characteristic
61	Q₈×C₁₆	—	11	2	no	—
62	#2169	—	11	6	no	—
63	#2163	—	11	3	no	—
64	C₄×M₃₂	—	11	3	no	—
65	C₂³×C₁₁	C₂²×C₈	1	3	yes	—
66	Q₈×C₁₁	M₃₂	1	4	yes	complemented
67	D₄×C₁₁	—	2	6	no	—
68	C₂×C₄×C₁₁	—	2	3	no	—
69	C₂×C₄×C₁₁	—	2	3	no	—
70	C₂×C₄×C₁₁	—	2	6	no	—
71	C₂×C₄×C₁₁	—	4	3	no	—
72	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₁₁	—	2	3	no	—
79	C₈×C₁₁	—	2	3	no	—
80	C₈×C₁₁	#131	1	2	yes	—
81	C₈×C₁₁	C₂×#41	1	2	yes	—
82	C₄×#41	Dic₁₁	1	1	yes	characteristic
83	C₂×M₃₂	—	11	6	no	—
84	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
85	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
86	C₈×D₄	Dic₁₁	1	3	yes	—
87	C₈×D₄	C₂×D₁₁	1	3	yes	—
88	#431	Dic₁₁	1	3	yes	—
89	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
90	C₄⋊C₁₆	—	11	6	no	—
91	C₂²⋊C₁₆	—	11	6	no	—
92	#346	—	11	1	no	—
93	#346	—	11	3	no	—
94	C₄×C₁₆	—	11	6	no	—
95	C₂²×C₁₁	—	2	3	no	—
96	C₂²×C₁₁	—	2	3	no	—
97	C₂²×C₁₁	—	2	3	no	—
98	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
99	C₄×C₁₁	—	2	1	no	—
100	C₄×C₁₁	—	2	1	no	—
101	C₄×C₁₁	—	2	3	no	—
102	C₄×C₁₁	—	2	3	no	—
103	C₄×C₁₁	C₂×M₃₂	1	6	yes	—
104	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
105	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
106	C₂×#41	C₁₁⋊C₈	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
108	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
109	C₄×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
110	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
111	C₄×D₄	C₁₁⋊C₈	1	3	yes	—
112	C₄×D₄	C₂×Dic₁₁	1	3	yes	—
113	#117	C₁₁⋊C₈	1	3	yes	—
114	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
115	M₃₂	—	22	12	no	—
116	C₂×C₁₆	—	11	2	no	—
117	C₂×C₁₆	—	11	6	no	—
118	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
119	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
120	C₂²⋊C₈	C₂×Dic₁₁	1	6	yes	—
121	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
122	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
123	C₄×C₈	C₂²×D₁₁	1	3	yes	—
124	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
125	C₂×C₁₁	—	4	3	no	—
126	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
127	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
128	C₂×C₁₁	#3407	1	1	yes	characteristic
129	#41	—	2	4	no	—
130	C₂×Q₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
131	C₂×D₄	C₂×C₁₁⋊C₈	1	3	yes	—
132	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
133	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
134	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
138	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
139	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
140	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
141	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
142	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
143	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	6	yes	—
144	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
145	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
146	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
147	C₄²	C₂²×Dic₁₁	1	3	yes	—
148	C₁₆	—	11	4	no	—
149	C₁₆	—	22	6	no	—
150	C₁₁	#33485	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂²×C₁₁⋊C₈	1	3	yes	—
152	Q₈	#66876	1	4	yes	complemented
153	D₄	—	2	6	no	—
154	C₂×C₄	—	2	3	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
161	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
162	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
163	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#67001	1	2	yes	—
167	C₈	#41×D₁₁	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	—	2	3	no	—
171	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	3	no	—
176	C₄	C₂×#66876	1	6	yes	—
177	C₄	C₂×#67001	1	1	yes	characteristic
178	C₄	#41×Dic₁₁	1	1	yes	characteristic
179	C₂	—	4	3	no	—
180	C₂	#41×C₁₁⋊C₈	1	1	yes	characteristic
181	C₂	C₂²×#66876	1	1	yes	characteristic
182	C₂	#13081719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₈×C₁₁×#41	C₂
prime radical	C₂×C₁₁×#41	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₈×#41	D₁₁
hypocenter	C₁₁	#33485
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₁₁⋊C₁₆	C₂	semidirect
2. #13070448	C₂	semidirect
3. #13070419	C₂	semidirect
4. C₄×#66876	C₂	semidirect
59. C₁₁⋊C₁₆	#41	semidirect
66. Q₈×C₁₁	M₃₂	semidirect
150. C₁₁	#33485	semidirect
152. Q₈	#66876	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₁₁⋊C₁₆	2
#13070448	2
#13070419	2
C₄×#66876	2
C₈×C₁₁×#41	2
#33485	11