Subgroups — #14665391

Order 2816.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070419	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#13066686	C₂	1	2	yes	maximal, complemented
4	#13066680	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₁₁×#431	C₄	1	1	yes	characteristic
7	C₄×C₁₁×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
10	C₁₁×#322	C₄	1	2	yes	—
11	C₈²×C₁₁	C₂²	1	2	yes	—
12	C₂²⋊(C₁₁⋊C₁₆)	—	2	1	no	—
13	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
14	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
17	#10622099	—	2	1	no	—
18	C₂×C₁₁×M₁₆	—	2	1	no	—
19	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₁₁×C₄⋊C₈	C₈	1	2	yes	—
23	C₁₁×C₂²⋊C₈	C₈	1	2	yes	—
24	C₁₁×#97	—	2	1	no	—
25	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₁⋊C₁₆	—	2	1	no	—
32	C₂×C₁₁⋊C₁₆	—	2	1	no	—
33	C₂×C₁₁⋊C₁₆	—	2	2	no	—
34	C₂×C₁₁⋊C₁₆	—	4	2	no	—
35	#12016	—	11	1	no	maximal, Sylow
36	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
39	C₁₁×M₁₆	—	2	4	no	—
40	C₂×C₈×C₁₁	—	2	1	no	—
41	C₂×C₈×C₁₁	—	2	1	no	—
42	C₂×C₈×C₁₁	—	2	2	no	—
43	C₂×C₈×C₁₁	—	4	1	no	—
44	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
52	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₁₁⋊C₁₆	—	4	1	no	—
54	C₁₁⋊C₁₆	—	4	2	no	—
55	C₁₁⋊C₁₆	—	4	1	no	—
56	#2163	—	11	1	no	—
57	C₂×C₄⋊C₁₆	—	11	1	no	—
58	C₈×M₁₆	D₁₁	1	1	yes	characteristic
59	#1370	—	11	2	no	—
60	#1367	—	11	2	no	—
61	C₂³×C₁₁	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₁	—	2	2	no	—
63	C₂×C₄×C₁₁	—	2	1	no	—
64	C₂×C₄×C₁₁	—	2	1	no	—
65	C₂×C₄×C₁₁	—	2	2	no	—
66	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₁₁	—	2	4	no	—
73	C₈×C₁₁	—	2	1	no	—
74	C₈×C₁₁	—	2	1	no	—
75	C₈×C₁₁	—	2	1	no	—
76	C₈×C₁₁	—	2	4	no	—
77	C₈×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₈×C₁₁	#137	1	1	yes	characteristic
79	C₂²×C₁₆	—	22	1	no	—
80	#431	Dic₁₁	1	1	yes	characteristic
81	C₄×M₁₆	Dic₁₁	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
83	C₄⋊C₁₆	—	11	1	no	—
84	C₄⋊C₁₆	—	11	2	no	—
85	C₄⋊C₁₆	—	11	1	no	—
86	C₄⋊C₁₆	—	22	1	no	—
87	C₂²⋊C₁₆	—	22	1	no	—
88	#346	—	22	1	no	—
89	#322	Dic₁₁	1	2	yes	—
90	C₈²	C₂×D₁₁	1	2	yes	—
91	C₂²×C₁₁	—	2	1	no	—
92	C₂²×C₁₁	—	2	1	no	—
93	C₂²×C₁₁	#364	1	2	yes	—
94	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₁₁	—	2	1	no	—
96	C₄×C₁₁	—	2	1	no	—
97	C₄×C₁₁	—	2	1	no	—
98	C₄×C₁₁	—	2	1	no	—
99	C₄×C₁₁	#428	1	1	yes	characteristic
100	C₄×C₁₁	#430	1	1	yes	characteristic
101	C₄×C₁₁	#504	1	2	yes	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
104	C₂²×C₈	D₄₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
106	C₂×C₁₆	—	22	1	no	—
107	C₂×C₁₆	—	22	1	no	—
108	C₂×C₁₆	—	22	2	no	—
109	C₂×C₁₆	—	44	2	no	—
110	C₄⋊C₈	C₁₁⋊C₈	1	2	yes	—
111	C₂²⋊C₈	C₁₁⋊C₈	1	2	yes	—
112	#97	—	2	1	no	—
113	C₄×C₈	Dic₂₂	1	1	yes	characteristic
114	C₄×C₈	D₄₄	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
116	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
117	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
119	C₂×C₁₁	—	2	2	no	—
120	C₂×C₁₁	#1565	1	1	yes	characteristic
121	C₂×C₁₁	#2151	1	1	yes	characteristic
122	C₂×C₁₁	C₂×#364	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	#4318	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	C₂×C₁₁⋊C₈	1	4	yes	—
132	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
134	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
135	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
136	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
137	C₄²	#4318	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
139	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
140	C₁₆	—	44	2	no	—
141	C₁₆	—	44	1	no	—
142	C₁₆	—	44	1	no	—
143	C₁₁	#12016	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
152	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
153	C₂×C₄	C₂×#4318	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	4	no	—
160	C₈	#66961	1	1	yes	characteristic
161	C₈	#66962	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#10622101	1	2	yes	—
165	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#10622734	1	1	yes	characteristic
171	C₄	#10622738	1	1	yes	characteristic
172	C₄	#10622753	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#13067368	1	1	yes	characteristic
175	C₂	#13070375	1	1	yes	characteristic
176	C₂	C₂×#10622101	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₈×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₈
prime radical	C₂³×C₁₁	C₄⋊C₈
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₈×M₁₆	D₁₁
hypocenter	C₁₁	#12016
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13066686	C₂	semidirect
4. #13066680	C₂	semidirect
143. C₁₁	#12016	semidirect

Maximal subgroups

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

subgroup	index
#13070419	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
#13066686	2
#13066680	2
C₈×C₁₁×M₁₆	2
#12016	11