Subgroups — #14670199

Order 2816.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622151	C₂	1	4	yes	maximal
2	#13068788	C₂	1	1	yes	maximal, characteristic
3	#13068773	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×#66883	—	2	1	no	—
8	C₂×#66883	C₂²	1	2	yes	—
9	C₂×#66881	—	2	1	no	—
10	C₂×#66881	C₂²	1	2	yes	—
11	#10622548	—	2	1	no	—
12	#10622537	—	2	1	no	—
13	C₁₁×#346	C₄	1	2	yes	complemented
14	C₄×C₁₁×C₁₆	C₄	1	2	yes	complemented
15	#10622151	—	2	8	no	—
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₁	—	2	2	no	—
20	C₂×C₄⋊Dic₁₁	—	4	2	no	—
21	C₂×#4318	—	2	2	no	—
22	C₁₁×M₃₂	D₄	1	4	yes	—
23	C₁₁×M₃₂	Q₈	1	4	yes	—
24	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
29	#66883	—	2	4	no	—
30	#66883	—	4	2	no	—
31	#66881	—	2	4	no	—
32	#66881	—	4	2	no	—
33	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
34	#12903	—	11	1	no	maximal, Sylow
35	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
36	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₁	—	4	2	no	—
38	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
46	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₁	SD₁₆	1	2	yes	—
49	C₄⋊Dic₁₁	—	4	2	no	—
50	C₄⋊Dic₁₁	—	4	2	no	—
51	C₄⋊Dic₁₁	—	8	4	no	—
52	#4318	—	4	2	no	—
53	#4318	—	4	2	no	—
54	C₂×C₁₁⋊C₈	—	4	2	no	—
55	C₁₁×C₁₆	C₄⋊C₄	1	8	yes	complemented
56	C₂×#365	—	11	4	no	—
57	C₄×M₃₂	D₁₁	1	1	yes	characteristic
58	#1848	—	11	1	no	—
59	#1844	—	11	1	no	—
60	C₂³×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₁	—	2	4	no	—
62	C₂×C₄×C₁₁	#106	1	2	yes	—
63	C₂×C₄×C₁₁	#106	1	2	yes	—
64	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×Dic₁₁	—	8	4	no	—
71	C₂×Dic₁₁	—	8	2	no	—
72	C₈×C₁₁	#106	1	2	yes	—
73	C₈×C₁₁	#106	1	2	yes	—
74	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
75	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
76	C₁₁⋊C₈	—	8	2	no	—
77	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
78	C₂×#108	—	11	2	no	—
79	C₂×#108	—	22	1	no	—
80	C₂×#106	—	11	2	no	—
81	C₂×#106	—	22	1	no	—
82	#423	—	22	1	no	—
83	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
84	#384	—	22	1	no	—
85	#365	—	22	8	no	—
86	#346	Dic₁₁	1	2	yes	complemented
87	C₄×C₁₆	Dic₁₁	1	2	yes	complemented
88	C₂²×C₁₁	—	2	2	no	—
89	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
90	C₂²×C₁₁	#492	1	1	yes	characteristic
91	C₂²×C₁₁	#498	1	1	yes	characteristic
92	C₄×C₁₁	—	2	2	no	—
93	C₄×C₁₁	#365	1	4	yes	complemented
94	C₄×C₁₁	#384	1	1	yes	characteristic
95	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
96	C₄×C₁₁	#492	1	1	yes	characteristic
97	C₄×C₁₁	#498	1	1	yes	characteristic
98	Dic₁₁	—	16	2	no	—
99	C₂×M₁₆	—	22	2	no	—
100	C₂²×C₈	D₄₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	22	2	no	—
103	C₂×C₄⋊C₄	—	44	2	no	—
104	C₂×C₄²	D₄₄	1	1	yes	characteristic
105	M₃₂	Dic₂₂	1	4	yes	—
106	M₃₂	D₄₄	1	4	yes	—
107	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	—
108	#108	—	22	4	no	—
109	#108	—	44	2	no	—
110	#106	—	22	4	no	—
111	#106	—	44	2	no	—
112	C₄⋊C₈	—	44	2	no	—
113	C₄×C₈	Dic₂₂	1	1	yes	characteristic
114	C₄×C₈	Dic₂₂	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
117	C₂×C₁₁	—	2	1	no	—
118	C₂×C₁₁	—	2	1	no	—
119	C₂×C₁₁	#1844	1	1	yes	characteristic
120	C₂×C₁₁	C₂×#365	1	2	yes	—
121	C₂²×C₄	—	44	2	no	—
122	C₂²×C₄	#4314	1	2	yes	—
123	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
124	M₁₆	—	44	2	no	—
125	M₁₆	—	44	2	no	—
126	C₂×C₈	—	44	2	no	—
127	C₂×C₈	#4314	1	2	yes	—
128	C₂×C₈	#4314	1	2	yes	—
129	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
132	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
133	C₂×C₈	C₂×D₄₄	1	2	yes	—
134	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
135	C₄⋊C₄	—	44	2	no	—
136	C₄⋊C₄	—	44	2	no	—
137	C₄⋊C₄	—	88	4	no	—
138	C₄²	#4314	1	2	yes	—
139	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
140	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
141	C₁₆	C₄⋊Dic₁₁	1	8	yes	complemented
142	C₁₁	#12903	1	1	yes	characteristic, complemented, Sylow
143	C₂³	C₂×#4314	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	88	4	no	—
146	C₂×C₄	—	88	2	no	—
147	C₂×C₄	#66881	1	2	yes	—
148	C₂×C₄	#66881	1	2	yes	—
149	C₂×C₄	#66922	1	1	yes	characteristic
150	C₂×C₄	#66927	1	1	yes	characteristic
151	C₂×C₄	C₂×#4314	1	1	yes	characteristic
152	C₂×C₄	C₂×#4314	1	1	yes	characteristic
153	C₂×C₄	C₂×#4314	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
155	C₈	—	88	2	no	—
156	C₈	#66881	1	2	yes	—
157	C₈	#66881	1	2	yes	—
158	C₈	#66922	1	2	yes	—
159	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#10622293	1	1	yes	characteristic
162	C₂²	#10622306	1	1	yes	characteristic
163	C₂²	C₂×#66881	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	176	2	no	—
166	C₄	#10622151	1	4	yes	complemented
167	C₄	#10622293	1	1	yes	characteristic
168	C₄	#10622306	1	1	yes	characteristic
169	C₄	#10622548	1	1	yes	characteristic
170	C₄	C₂×#66881	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#13068773	1	1	yes	characteristic
174	C₂	C₂×#10622151	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#66881
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₄×C₁₁×M₃₂	C₂
socle	C₂²×C₁₁	C₂×#106
prime radical	C₂³×C₁₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#12903
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₁₁×#346	C₄	semidirect
14. C₄×C₁₁×C₁₆	C₄	semidirect
55. C₁₁×C₁₆	C₄⋊C₄	semidirect
86. #346	Dic₁₁	semidirect
87. C₄×C₁₆	Dic₁₁	semidirect
93. C₄×C₁₁	#365	semidirect
141. C₁₆	C₄⋊Dic₁₁	semidirect
142. C₁₁	#12903	semidirect
166. C₄	#10622151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622151	2
#13068788	2
#13068773	2
C₄×C₁₁×M₃₂	2
#12903	11