Subgroups — #14670202

Order 2816.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622125	C₂	1	4	yes	maximal
2	C₄×#66876	C₂	1	1	yes	maximal, characteristic
3	#13068788	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#107	—	2	1	no	—
6	C₂×C₁₁×#106	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₁₁×#382	—	2	1	no	—
9	C₂×#66883	—	2	1	no	—
10	C₂×#66883	C₂²	1	2	yes	—
11	C₂×#66876	C₂²	1	2	yes	—
12	#10622537	—	2	1	no	—
13	#10622125	—	2	8	no	—
14	#10622099	C₄	1	2	yes	complemented
15	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	complemented
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	C₂×#4318	—	2	2	no	—
21	C₁₁×#107	—	4	2	no	—
22	C₁₁×#106	—	2	4	no	—
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₁₁×#95	—	4	2	no	—
28	#66883	—	2	4	no	—
29	#66883	—	4	2	no	—
30	#66876	D₄	1	4	yes	—
31	#66876	Q₈	1	4	yes	—
32	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
33	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
34	#12904	—	11	1	no	maximal, Sylow
35	C₂²×C₄×C₁₁	—	4	2	no	—
36	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
37	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
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	C₁₁×C₄⋊C₄	—	4	2	no	—
47	C₁₁×C₄⋊C₄	—	4	2	no	—
48	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₁	SD₁₆	1	2	yes	—
51	#4318	—	4	2	no	—
52	#4318	—	4	2	no	—
53	C₂×C₁₁⋊C₈	—	4	2	no	—
54	C₁₁⋊C₁₆	C₄⋊C₄	1	8	yes	—
55	C₂×#365	—	11	4	no	—
56	C₄×M₃₂	—	11	1	no	—
57	#1848	—	11	1	no	—
58	#1846	D₁₁	1	1	yes	characteristic
59	C₂³×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₁	—	4	2	no	—
61	C₂×C₄×C₁₁	—	8	4	no	—
62	C₂×C₄×C₁₁	—	8	2	no	—
63	C₂×C₄×C₁₁	#106	1	2	yes	—
64	C₂×C₄×C₁₁	#106	1	2	yes	—
65	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₄⋊Q₈	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×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₈×C₁₁	#106	1	2	yes	—
72	C₈×C₁₁	#106	1	2	yes	—
73	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
74	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
75	C₁₁⋊C₈	—	8	2	no	—
76	C₂×M₃₂	—	11	2	no	—
77	C₂×#108	—	11	2	no	—
78	C₂×#108	—	22	1	no	—
79	C₂×#107	—	2	1	no	—
80	C₂×#106	C₂×D₁₁	1	2	yes	—
81	#423	—	22	1	no	—
82	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
83	#382	—	2	1	no	—
84	#365	—	22	8	no	—
85	#346	—	11	2	no	—
86	C₄×C₁₆	—	11	2	no	—
87	C₂²×C₁₁	—	2	2	no	—
88	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
89	C₂²×C₁₁	#492	1	1	yes	characteristic
90	C₂²×C₁₁	#498	1	1	yes	characteristic
91	C₄×C₁₁	—	2	2	no	—
92	C₄×C₁₁	—	2	2	no	—
93	C₄×C₁₁	—	16	2	no	—
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	C₂×M₁₆	—	22	2	no	—
99	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
103	M₃₂	—	11	4	no	—
104	M₃₂	—	11	4	no	—
105	C₂×C₁₆	—	11	4	no	—
106	#108	—	22	4	no	—
107	#108	—	44	2	no	—
108	#107	—	4	2	no	—
109	#106	—	2	4	no	—
110	C₄⋊C₈	—	44	2	no	—
111	C₄×C₈	Dic₂₂	1	1	yes	characteristic
112	C₄×C₈	Dic₂₂	1	1	yes	characteristic
113	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
114	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
115	#95	—	4	2	no	—
116	C₂×C₁₁	—	2	1	no	—
117	C₂×C₁₁	—	2	1	no	—
118	C₂×C₁₁	#1844	1	1	yes	characteristic
119	C₂×C₁₁	#2155	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#4324	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
123	M₁₆	—	44	2	no	—
124	M₁₆	—	44	2	no	—
125	C₂×C₈	—	44	2	no	—
126	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#4324	1	2	yes	—
129	C₂×C₈	#4325	1	2	yes	—
130	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
132	C₂×C₈	D₄×D₁₁	1	2	yes	—
133	C₂×C₈	Q₈×D₁₁	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
137	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
138	C₄²	#4325	1	2	yes	—
139	C₁₆	—	11	8	no	—
140	C₁₁	#12904	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₂×#4324	1	1	yes	characteristic
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#66872	1	2	yes	—
146	C₂×C₄	#66872	1	2	yes	—
147	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₂×#4324	1	1	yes	characteristic
149	C₂×C₄	#66993	1	1	yes	characteristic
150	C₂×C₄	C₂×#4325	1	1	yes	characteristic
151	C₂×C₄	C₂×#4325	1	1	yes	characteristic
152	C₂×C₄	#66997	1	1	yes	characteristic
153	C₈	—	88	2	no	—
154	C₈	#66872	1	2	yes	—
155	C₈	#66872	1	2	yes	—
156	C₈	#66941	1	2	yes	—
157	C₈	C₄⋊C₄×D₁₁	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#66872	1	1	yes	characteristic
160	C₂²	#10622689	1	1	yes	characteristic
161	C₂²	#10622692	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	C₂×#66872	1	1	yes	characteristic
166	C₄	#10622589	1	1	yes	characteristic
167	C₄	#10622692	1	1	yes	characteristic
168	C₄	#10622698	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#13068772	1	1	yes	characteristic
172	C₂	#13070388	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#66872
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1846	C₂
socle	C₂²×C₁₁	C₂×#106
prime radical	C₂³×C₁₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1846	D₁₁
hypocenter	C₁₁	#12904
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
14. #10622099	C₄	semidirect
15. C₄×C₁₁⋊C₁₆	C₄	semidirect
140. C₁₁	#12904	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622125	2
C₄×#66876	2
#13068788	2
C₁₁×#1846	2
#12904	11