Subgroups — #14669086

Order 2816.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×#10622099	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#13066694	C₂	1	2	yes	maximal
5	C₁₁×#1726	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
7	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
8	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₁₁×#376	C₄	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	C₄	1	8	yes	complemented
16	#10622099	—	2	2	no	—
17	C₄×C₁₁⋊C₁₆	—	2	2	no	—
18	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₁₁×#97	—	2	2	no	—
26	C₁₁×#97	—	2	2	no	—
27	C₁₁×#97	C₈	1	4	yes	—
28	C₄×C₈×C₁₁	—	2	2	no	—
29	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₁	D₄	1	2	yes	—
31	C₄×C₈×C₁₁	Q₈	1	2	yes	—
32	C₁₁×#95	C₈	1	4	yes	—
33	C₂×C₁₁⋊C₁₆	—	2	2	no	—
34	C₂×C₁₁⋊C₁₆	—	2	2	no	—
35	C₂×C₁₁⋊C₁₆	—	2	2	no	—
36	C₂×C₁₁⋊C₁₆	—	2	2	no	—
37	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
38	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
39	#12654	—	11	1	no	maximal, Sylow
40	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	—	2	1	no	—
45	C₂×C₈×C₁₁	—	2	1	no	—
46	C₂×C₈×C₁₁	—	2	1	no	—
47	C₂×C₈×C₁₁	—	2	1	no	—
48	C₂×C₈×C₁₁	C₄²	1	2	yes	—
49	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₁	#41	1	2	yes	—
57	C₄²×C₁₁	—	2	1	no	—
58	C₄²×C₁₁	—	2	1	no	—
59	C₄²×C₁₁	C₄²	1	2	yes	—
60	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
61	C₄²×C₁₁	C₂×C₈	1	2	yes	—
62	C₄²×C₁₁	M₁₆	1	2	yes	—
63	C₁₁⋊C₁₆	—	2	4	no	—
64	C₁₁⋊C₁₆	—	2	4	no	—
65	C₂×C₄⋊C₁₆	—	11	2	no	—
66	C₂×#346	—	11	1	no	—
67	C₂×C₄×C₁₆	—	11	1	no	—
68	#1726	D₁₁	1	1	yes	characteristic
69	#1374	—	11	2	no	—
70	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	—	2	4	no	—
72	C₂×C₄×C₁₁	—	2	4	no	—
73	C₂×C₄×C₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	4	no	—
75	C₂×C₄×C₁₁	C₄×C₈	1	4	yes	—
76	C₂×C₄×C₁₁	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
83	C₈×C₁₁	—	2	2	no	—
84	C₈×C₁₁	—	2	2	no	—
85	C₈×C₁₁	C₄⋊C₈	1	4	yes	—
86	C₈×C₁₁	C₄×D₄	1	2	yes	—
87	C₈×C₁₁	C₄×Q₈	1	2	yes	—
88	C₂²×C₁₆	—	11	2	no	—
89	C₂²×C₁₆	—	11	2	no	—
90	C₂×#97	Dic₁₁	1	1	yes	characteristic
91	C₂×#97	Dic₁₁	1	1	yes	characteristic
92	C₂×#97	Dic₁₁	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
96	#376	Dic₁₁	1	1	yes	characteristic
97	C₄⋊C₁₆	—	11	8	no	—
98	#346	—	22	2	no	—
99	C₄×C₁₆	—	22	2	no	—
100	C₂²×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
101	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₁	#431	1	1	yes	characteristic
103	C₂²×C₁₁	#504	1	4	yes	—
104	C₄×C₁₁	—	2	4	no	—
105	C₄×C₁₁	—	2	4	no	—
106	C₄×C₁₁	—	4	2	no	—
107	C₄×C₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₁₁	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₁₁	#431	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
115	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
118	C₂×C₁₆	—	11	4	no	—
119	C₂×C₁₆	—	11	4	no	—
120	C₂×C₁₆	—	22	2	no	—
121	C₂×C₁₆	—	22	2	no	—
122	C₂×C₁₆	—	22	2	no	—
123	C₂×C₁₆	—	22	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	#97	C₁₁⋊C₈	1	4	yes	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	Dic₂₂	1	2	yes	—
129	C₄×C₈	C₄×D₁₁	1	4	yes	—
130	C₄×C₈	D₄₄	1	2	yes	—
131	#95	C₁₁⋊C₈	1	4	yes	—
132	C₂×C₁₁	C₄×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₁₁	#2108	1	2	yes	—
134	C₂×C₁₁	#2151	1	2	yes	—
135	C₂×C₁₁	#2164	1	2	yes	—
136	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
137	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
145	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
146	C₂×C₈	#4318	1	2	yes	—
147	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
149	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
151	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
152	C₂×C₈	#4339	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
156	C₄²	#4318	1	2	yes	—
157	C₄²	C₄×Dic₁₁	1	2	yes	—
158	C₄²	C₄⋊Dic₁₁	1	2	yes	—
159	C₁₆	—	22	4	no	—
160	C₁₆	—	22	4	no	—
161	C₁₁	#12654	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₄×C₁₁⋊C₈	1	4	yes	—
168	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	4	yes	—
169	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
170	C₂×C₄	C₂×#4318	1	1	yes	characteristic
171	C₂×C₄	C₂×#4318	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
174	C₂×C₄	#66979	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	C₄⋊(C₁₁⋊C₈)	1	4	yes	—
178	C₈	C₄×Dic₂₂	1	2	yes	—
179	C₈	C₄×D₄₄	1	2	yes	—
180	C₂²	C₂×C₄×C₁₁⋊C₈	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
182	C₂²	#10622538	1	1	yes	characteristic
183	C₂²	#10622753	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×#4318	1	1	yes	characteristic
188	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
189	C₄	#10622538	1	1	yes	characteristic
190	C₄	C₄×C₄⋊Dic₁₁	1	1	yes	characteristic
191	C₂	C₄×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
192	C₂	#13070213	1	2	yes	—
193	C₂	#13070375	1	2	yes	—
194	C₂	#13070423	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂×C₄×C₈
Fitting	C₁₁×#1726	C₂
socle, prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1726	D₁₁
hypocenter	C₁₁	#12654
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
15. C₄⋊(C₁₁⋊C₁₆)	C₄	semidirect
161. C₁₁	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×#10622099	2
C₂×C₄×C₁₁⋊C₁₆	2
#13066694	2
C₁₁×#1726	2
#12654	11