Subgroups — #14670212

Order 2816.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622128	C₂	1	2	yes	maximal
2	C₂×#10622127	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#13068780	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#107	C₂²	1	2	yes	—
7	C₂×C₁₁×#106	—	2	1	no	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₁₁×#382	—	2	1	no	—
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂×#66881	—	2	1	no	—
12	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	#10622549	—	2	1	no	—
15	#10622128	—	2	4	no	—
16	#10622127	—	2	4	no	—
17	C₄×C₁₁⋊C₁₆	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₁	—	2	2	no	—
23	C₁₁×#107	—	2	2	no	—
24	C₁₁×#107	—	2	2	no	—
25	C₁₁×#106	—	4	2	no	—
26	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₁	Q₈	1	2	yes	—
28	C₁₁×#95	—	4	2	no	—
29	C₁₁⋊#95	—	4	2	no	—
30	#66882	—	2	2	no	—
31	#66882	—	2	2	no	—
32	#66881	—	4	2	no	—
33	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
36	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
37	C₂×C₁₁⋊C₁₆	Q₈	1	2	yes	—
38	C₂×C₁₁⋊C₁₆	Q₈	1	2	yes	—
39	#12907	—	11	1	no	maximal, Sylow
40	C₂²×C₄×C₁₁	—	4	2	no	—
41	C₂²×C₄×C₁₁	D₈	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₁	—	4	2	no	—
44	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₁	D₈	1	2	yes	—
46	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
53	C₁₁×C₄⋊C₄	—	4	2	no	—
54	C₁₁×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
56	C₄²×C₁₁	Q₁₆	1	2	yes	—
57	C₄⋊Dic₁₁	—	4	2	no	—
58	C₄⋊Dic₁₁	—	4	2	no	—
59	C₁₁⋊C₁₆	C₄⋊C₄	1	4	yes	—
60	C₁₁⋊C₁₆	C₄⋊C₄	1	4	yes	—
61	C₂×#367	—	11	2	no	—
62	C₂×#366	—	11	2	no	—
63	C₂×C₄×C₁₆	—	11	1	no	—
64	#1846	—	11	1	no	—
65	#1846	D₁₁	1	1	yes	characteristic
66	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₁	—	2	4	no	—
68	C₂×C₄×C₁₁	—	8	2	no	—
69	C₂×C₄×C₁₁	—	8	4	no	—
70	C₂×C₄×C₁₁	#107	1	4	yes	—
71	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×Dic₁₁	—	8	2	no	—
78	C₂×Dic₁₁	—	8	4	no	—
79	C₈×C₁₁	#107	1	4	yes	—
80	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₁₆	—	11	1	no	—
85	C₂²×C₁₆	—	11	1	no	—
86	C₂×#107	—	11	2	no	—
87	C₂×#107	C₂×D₁₁	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×#106	—	22	1	no	—
90	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
91	#382	—	2	1	no	—
92	#382	—	22	1	no	—
93	#367	—	22	4	no	—
94	#366	—	22	4	no	—
95	C₄×C₁₆	—	11	4	no	—
96	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
97	C₂²×C₁₁	#493	1	1	yes	characteristic
98	C₂²×C₁₁	#500	1	1	yes	characteristic
99	C₂²×C₁₁	#508	1	4	yes	—
100	C₄×C₁₁	—	2	4	no	—
101	C₄×C₁₁	—	16	2	no	—
102	C₄×C₁₁	#384	1	1	yes	characteristic
103	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₁	#494	1	1	yes	characteristic
105	C₄×C₁₁	#500	1	1	yes	characteristic
106	Dic₁₁	—	16	2	no	—
107	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	22	2	no	—
111	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
112	C₂×C₁₆	—	11	2	no	—
113	C₂×C₁₆	—	11	2	no	—
114	C₂×C₁₆	—	11	2	no	—
115	C₂×C₁₆	—	11	2	no	—
116	C₂×C₁₆	—	11	2	no	—
117	C₂×C₁₆	—	11	2	no	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	—	22	2	no	—
121	#107	—	22	2	no	—
122	#106	—	4	2	no	—
123	#106	—	44	2	no	—
124	C₄×C₈	Dic₂₂	1	2	yes	—
125	C₄×C₈	C₄×D₁₁	1	2	yes	—
126	#95	—	4	2	no	—
127	#95	—	44	2	no	—
128	C₂×C₁₁	#1845	1	1	yes	characteristic
129	C₂×C₁₁	#2158	1	2	yes	—
130	C₂×C₁₁	#2249	1	2	yes	—
131	C₂×C₁₁	#2253	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	44	2	no	—
134	C₂²×C₄	#4323	1	2	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
136	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
137	C₂×C₈	#4323	1	2	yes	—
138	C₂×C₈	#4326	1	2	yes	—
139	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
141	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
142	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	44	2	no	—
148	C₄⋊C₄	—	44	2	no	—
149	C₄²	Dic₁₁⋊C₄	1	2	yes	—
150	C₄²	#4326	1	2	yes	—
151	C₁₆	—	11	4	no	—
152	C₁₆	—	11	4	no	—
153	C₁₁	#12907	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂×#4323	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	88	4	no	—
159	C₂×C₄	—	88	2	no	—
160	C₂×C₄	#66871	1	4	yes	—
161	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₂×#4323	1	1	yes	characteristic
163	C₂×C₄	#66993	1	1	yes	characteristic
164	C₂×C₄	C₂×#4326	1	1	yes	characteristic
165	C₂×C₄	C₂×#4326	1	1	yes	characteristic
166	C₂×C₄	#66997	1	1	yes	characteristic
167	C₈	#66871	1	4	yes	—
168	C₈	#66941	1	1	yes	characteristic
169	C₈	#66941	1	1	yes	characteristic
170	C₈	C₄⋊C₄×D₁₁	1	1	yes	characteristic
171	C₈	C₄⋊C₄×D₁₁	1	1	yes	characteristic
172	C₂²	C₂×#66871	1	1	yes	characteristic
173	C₂²	#10622686	1	1	yes	characteristic
174	C₂²	#10622691	1	1	yes	characteristic
175	C₂²	#10622807	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	176	2	no	—
179	C₄	C₂×#66871	1	1	yes	characteristic
180	C₄	#10622589	1	1	yes	characteristic
181	C₄	#10622691	1	1	yes	characteristic
182	C₄	#10622705	1	1	yes	characteristic
183	C₂	#13068775	1	1	yes	characteristic
184	C₂	#13070398	1	2	yes	—
185	C₂	#13070875	1	2	yes	—
186	C₂	#13070892	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1846	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1846	D₁₁
hypocenter	C₁₁	#12907
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
17. C₄×C₁₁⋊C₁₆	C₄	semidirect
153. C₁₁	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622128	2
C₂×#10622127	2
C₂×C₄×C₁₁⋊C₁₆	2
#13068780	2
C₁₁×#1846	2
#12907	11