Subgroups — #14640600

Order 2816.

306 conjugacy classes of subgroups.

The subgroup lattice has 162 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	1	yes	maximal, characteristic
2	C₂×C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₁×C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₁⋊C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₁₁⋊C₈	C₄	1	2	yes	—
8	C₄×C₁₁×C₁₆	—	2	1	no	—
9	C₄×C₁₁×C₁₆	—	2	1	no	—
10	C₄×C₁₁⋊C₁₆	—	2	1	no	—
11	C₄×C₁₁⋊C₁₆	—	2	1	no	—
12	C₈×C₁₁⋊C₈	—	2	1	no	—
13	C₈×C₁₁⋊C₈	—	2	1	no	—
14	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₁⋊C₈	C₈	1	4	yes	—
18	C₂×C₁₁×C₁₆	—	2	2	no	—
19	C₂×C₁₁×C₁₆	—	2	2	no	—
20	C₂×C₁₁×C₁₆	C₈	1	4	yes	complemented
21	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
25	C₂×C₁₁⋊C₁₆	—	2	2	no	—
26	C₂×C₁₁⋊C₁₆	—	2	2	no	—
27	C₂×C₁₁⋊C₁₆	C₈	1	4	yes	complemented
28	C₄×C₁₁⋊C₈	—	2	2	no	—
29	C₄×C₁₁⋊C₈	—	2	2	no	—
30	#7136	—	11	1	no	maximal, Sylow
31	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
40	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
41	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₁	M₁₆	1	2	yes	—
44	C₂×C₁₁⋊C₈	—	2	4	no	—
45	C₂×C₁₁⋊C₈	—	2	4	no	—
46	C₂×C₁₁⋊C₈	C₁₆	1	8	yes	complemented
47	C₁₁×C₁₆	—	2	4	no	—
48	C₁₁⋊C₁₆	—	2	4	no	—
49	C₂×C₄×C₁₆	—	11	1	no	—
50	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic
51	C₂×C₈²	—	11	1	no	—
52	C₂³×C₁₁	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₁	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₁	#97	1	1	yes	characteristic
57	C₂×C₄×C₁₁	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₁	#97	1	1	yes	characteristic
59	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₁	C₂×C₁₆	1	4	yes	—
62	C₂×C₄×C₁₁	M₃₂	1	4	yes	—
63	C₈×C₁₁	C₂²⋊C₈	1	2	yes	—
64	C₈×C₁₁	C₂²⋊C₈	1	2	yes	—
65	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
66	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
67	C₁₁⋊C₈	—	2	8	no	—
68	C₂²×C₁₆	—	11	2	no	—
69	C₂²×C₁₆	Dic₁₁	1	2	yes	—
70	C₂×C₄×C₈	—	11	2	no	—
71	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
72	C₄×C₁₆	—	2	1	no	—
73	C₄×C₁₆	—	2	1	no	—
74	C₄×C₁₆	—	22	1	no	—
75	C₄×C₁₆	—	22	1	no	—
76	C₈²	—	22	1	no	—
77	C₈²	—	22	1	no	—
78	C₂²×C₁₁	C₈²	1	1	yes	characteristic
79	C₂²×C₁₁	#322	1	1	yes	characteristic
80	C₂²×C₁₁	#336	1	1	yes	characteristic
81	C₂²×C₁₁	C₄×C₁₆	1	2	yes	—
82	C₂²×C₁₁	#346	1	2	yes	—
83	C₄×C₁₁	#322	1	1	yes	characteristic
84	C₄×C₁₁	#322	1	1	yes	characteristic
85	C₄×C₁₁	#336	1	1	yes	characteristic
86	C₄×C₁₁	#336	1	1	yes	characteristic
87	C₄×C₁₁	C₂²⋊C₁₆	1	4	yes	—
88	C₄×C₁₁	C₄⋊C₁₆	1	4	yes	—
89	C₂²×C₈	—	11	4	no	—
90	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	11	4	no	—
96	C₂×C₁₆	—	22	2	no	—
97	C₂×C₁₆	—	22	2	no	—
98	C₂×C₁₆	C₁₁⋊C₈	1	4	yes	complemented
99	C₄×C₈	—	22	2	no	—
100	C₄×C₈	—	22	2	no	—
101	C₄×C₈	Dic₂₂	1	1	yes	characteristic
102	C₄×C₈	D₄₄	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
105	C₂×C₁₁	#1273	1	1	yes	characteristic
106	C₂×C₁₁	C₈×C₁₆	1	1	yes	characteristic
107	C₂×C₁₁	#1311	1	1	yes	characteristic
108	C₂×C₁₁	#1312	1	2	yes	—
109	C₂×C₁₁	#1374	1	2	yes	—
110	C₂²×C₄	C₈×D₁₁	1	2	yes	—
111	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
112	C₂×C₈	—	11	8	no	—
113	C₂×C₈	—	22	4	no	—
114	C₂×C₈	—	22	4	no	—
115	C₂×C₈	C₈×D₁₁	1	2	yes	—
116	C₂×C₈	#4313	1	2	yes	—
117	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
118	C₂×C₈	#4318	1	2	yes	—
119	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
121	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
123	C₄²	#4313	1	2	yes	—
124	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
125	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
126	C₁₆	—	2	4	no	—
127	C₁₆	—	22	4	no	—
128	C₁₁	#7136	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₈×Dic₁₁	1	1	yes	characteristic
130	C₂×C₄	C₁₆×D₁₁	1	4	yes	—
131	C₂×C₄	#66862	1	4	yes	—
132	C₂×C₄	C₄×C₁₁⋊C₈	1	1	yes	characteristic
133	C₂×C₄	#66867	1	1	yes	characteristic
134	C₂×C₄	C₈×Dic₁₁	1	1	yes	characteristic
135	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
136	C₂×C₄	#66879	1	1	yes	characteristic
137	C₂×C₄	#66879	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
139	C₂×C₄	C₁₁⋊#95	1	1	yes	characteristic
140	C₈	—	22	8	no	—
141	C₈	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
142	C₈	Dic₁₁⋊C₈	1	2	yes	—
143	C₈	(C₂×D₁₁)⋊C₈	1	2	yes	—
144	C₈	C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
145	C₂²	C₈×C₁₁⋊C₈	1	1	yes	characteristic
146	C₂²	#10622092	1	1	yes	characteristic
147	C₂²	C₁₆×Dic₁₁	1	2	yes	—
148	C₂²	#10622140	1	2	yes	—
149	C₂²	#10622188	1	1	yes	characteristic
150	C₄	#10622092	1	1	yes	characteristic
151	C₄	#10622093	1	1	yes	characteristic
152	C₄	Dic₁₁⋊C₁₆	1	4	yes	—
153	C₄	(C₂×D₁₁)⋊C₁₆	1	4	yes	—
154	C₄	#10622162	1	1	yes	characteristic
155	C₄	#10622188	1	1	yes	characteristic
156	C₂	C₁₁⋊#1273	1	1	yes	characteristic
157	C₂	C₁₆×C₁₁⋊C₈	1	1	yes	characteristic
158	C₂	#13066532	1	1	yes	characteristic
159	C₂	#13066535	1	2	yes	—
160	C₂	#13066693	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄×C₈	C₂×D₁₁
commutator	C₂×C₁₁	C₈×C₁₆
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₄×C₈
cosocle	C₂×C₄×C₈×C₁₁	C₂²
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#7136
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
20. C₂×C₁₁×C₁₆	C₈	semidirect
27. C₂×C₁₁⋊C₁₆	C₈	semidirect
46. C₂×C₁₁⋊C₈	C₁₆	semidirect
98. C₂×C₁₆	C₁₁⋊C₈	semidirect
128. C₁₁	#7136	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×C₈×C₁₁⋊C₈	2
C₂×C₄×C₁₁×C₁₆	2
#7136	11