Subgroups — #14670209

Order 2816.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 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	4	yes	maximal
2	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13068776	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#107	—	2	1	no	—
6	C₂×C₁₁×#107	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₁₁×#384	—	2	1	no	—
9	C₂×#66882	—	2	1	no	—
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
12	#10622548	—	2	1	no	—
13	#10622128	—	2	8	no	—
14	C₄×C₁₁⋊C₁₆	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₁×C₄⋊C₄	—	4	2	no	—
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₁	—	2	2	no	—
21	C₂×C₄⋊Dic₁₁	—	4	2	no	—
22	C₁₁×#107	—	2	4	no	—
23	C₁₁×#107	—	4	2	no	—
24	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₁	Q₈	1	2	yes	—
26	#66882	—	2	4	no	—
27	#66882	—	4	2	no	—
28	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₁⋊C₁₆	D₄	1	4	yes	—
30	C₂×C₁₁⋊C₁₆	Q₈	1	4	yes	—
31	#12906	—	11	1	no	maximal, Sylow
32	C₂²×C₄×C₁₁	—	4	2	no	—
33	C₂²×C₄×C₁₁	D₈	1	2	yes	—
34	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₁₁	—	4	2	no	—
36	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₁₁	D₈	1	2	yes	—
38	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
39	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
43	C₁₁×C₄⋊C₄	—	4	2	no	—
44	C₁₁×C₄⋊C₄	—	4	2	no	—
45	C₁₁×C₄⋊C₄	—	8	4	no	—
46	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₁	Q₁₆	1	2	yes	—
48	C₄⋊Dic₁₁	—	4	2	no	—
49	C₄⋊Dic₁₁	—	4	2	no	—
50	C₄⋊Dic₁₁	—	8	4	no	—
51	C₁₁⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
52	C₂×#367	—	11	4	no	—
53	C₂×C₄×C₁₆	—	11	1	no	—
54	#1845	—	11	1	no	—
55	#1845	D₁₁	1	1	yes	characteristic
56	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₁	—	8	2	no	—
58	C₂×C₄×C₁₁	—	8	4	no	—
59	C₂×C₄×C₁₁	#107	1	4	yes	—
60	C₂×C₄×C₁₁	SD₃₂	1	4	yes	—
61	C₂×C₄×C₁₁	SD₃₂	1	4	yes	—
62	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
64	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₁₁	—	8	2	no	—
69	C₂×Dic₁₁	—	8	4	no	—
70	C₈×C₁₁	#107	1	4	yes	—
71	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
72	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
73	C₂²×C₁₆	—	11	2	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	11	2	no	—
76	C₂×#107	—	22	1	no	—
77	C₂×#107	C₂×D₁₁	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
79	#384	—	2	1	no	—
80	#384	—	22	1	no	—
81	#367	—	22	8	no	—
82	C₄×C₁₆	—	11	4	no	—
83	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
84	C₂²×C₁₁	#493	1	1	yes	characteristic
85	C₂²×C₁₁	#500	1	1	yes	characteristic
86	C₂²×C₁₁	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₁₁	C₂×SD₃₂	1	2	yes	—
88	C₄×C₁₁	—	16	2	no	—
89	C₄×C₁₁	#367	1	8	yes	complemented
90	C₄×C₁₁	#384	1	1	yes	characteristic
91	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₁	#494	1	1	yes	characteristic
93	C₄×C₁₁	#500	1	1	yes	characteristic
94	Dic₁₁	—	16	2	no	—
95	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	22	2	no	—
100	C₂×C₄⋊C₄	—	44	2	no	—
101	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
102	C₂×C₁₆	—	11	4	no	—
103	C₂×C₁₆	—	11	4	no	—
104	C₂×C₁₆	—	11	4	no	—
105	#107	—	2	4	no	—
106	#107	—	4	2	no	—
107	#107	—	22	4	no	—
108	#107	—	44	2	no	—
109	C₄×C₈	Dic₂₂	1	2	yes	—
110	C₄×C₈	C₄×D₁₁	1	2	yes	—
111	C₂×C₁₁	#1845	1	1	yes	characteristic
112	C₂×C₁₁	C₂×#367	1	2	yes	—
113	C₂×C₁₁	#2248	1	1	yes	characteristic
114	C₂×C₁₁	#2248	1	1	yes	characteristic
115	C₂×C₁₁	#2254	1	2	yes	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	—	44	2	no	—
118	C₂²×C₄	#4323	1	2	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
120	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
121	C₂×C₈	#4323	1	2	yes	—
122	C₂×C₈	#4326	1	2	yes	—
123	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
124	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
125	C₂×C₈	D₄×D₁₁	1	2	yes	—
126	C₂×C₈	Q₈×D₁₁	1	2	yes	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄⋊C₄	—	44	2	no	—
131	C₄⋊C₄	—	44	2	no	—
132	C₄⋊C₄	—	88	4	no	—
133	C₄²	Dic₁₁⋊C₄	1	2	yes	—
134	C₄²	#4326	1	2	yes	—
135	C₁₆	—	11	8	no	—
136	C₁₁	#12906	1	1	yes	characteristic, complemented, Sylow
137	C₂³	C₂×#4323	1	1	yes	characteristic
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	88	4	no	—
141	C₂×C₄	—	88	2	no	—
142	C₂×C₄	#66871	1	4	yes	—
143	C₂×C₄	#66891	1	4	yes	—
144	C₂×C₄	#66892	1	4	yes	—
145	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
146	C₂×C₄	C₂×#4323	1	1	yes	characteristic
147	C₂×C₄	#66993	1	1	yes	characteristic
148	C₂×C₄	C₂×#4326	1	1	yes	characteristic
149	C₂×C₄	C₂×#4326	1	1	yes	characteristic
150	C₂×C₄	#66997	1	1	yes	characteristic
151	C₈	#66871	1	4	yes	—
152	C₈	#66941	1	2	yes	—
153	C₈	C₄⋊C₄×D₁₁	1	2	yes	—
154	C₂²	C₂×#66871	1	1	yes	characteristic
155	C₂²	#10622686	1	1	yes	characteristic
156	C₂²	#10622691	1	1	yes	characteristic
157	C₂²	C₂×#66891	1	2	yes	—
158	C₂²	C₂×#66892	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	176	2	no	—
161	C₄	#10622128	1	8	yes	complemented
162	C₄	C₂×#66871	1	1	yes	characteristic
163	C₄	#10622589	1	1	yes	characteristic
164	C₄	#10622691	1	1	yes	characteristic
165	C₄	#10622705	1	1	yes	characteristic
166	C₂	#13068775	1	1	yes	characteristic
167	C₂	C₂×#10622128	1	2	yes	—
168	C₂	#13070871	1	1	yes	characteristic
169	C₂	#13070873	1	1	yes	characteristic
170	C₂	#13070896	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1845	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1845	D₁₁
hypocenter	C₁₁	#12906
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
14. C₄×C₁₁⋊C₁₆	C₄	semidirect
51. C₁₁⋊C₁₆	C₄⋊C₄	semidirect
89. C₄×C₁₁	#367	semidirect
136. C₁₁	#12906	semidirect
161. C₄	#10622128	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×C₄×C₁₁⋊C₁₆	2
#13068776	2
C₁₁×#1845	2
#12906	11