Subgroups — #14670221

Order 2816.

314 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622131	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13068788	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#108	—	2	1	no	—
6	C₂×C₁₁×#108	C₂²	1	2	yes	—
7	C₁₁×#423	—	2	1	no	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×#66883	—	2	1	no	—
10	C₂×#66883	C₂²	1	2	yes	—
11	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
12	#10622537	—	2	1	no	—
13	#10622131	—	2	8	no	—
14	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	—
15	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	—
16	C₂×C₁₁×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	C₂×#4318	—	2	2	no	—
21	C₁₁×#108	—	2	4	no	—
22	C₁₁×#108	—	4	2	no	—
23	C₁₁×C₄⋊C₈	—	4	2	no	—
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
28	#66883	—	2	4	no	—
29	#66883	—	4	2	no	—
30	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₁₁⋊C₁₆	D₄	1	4	yes	—
32	C₂×C₁₁⋊C₁₆	Q₈	1	4	yes	—
33	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
34	#12909	—	11	1	no	maximal, Sylow
35	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
36	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
37	C₁₁×M₁₆	—	4	2	no	—
38	C₁₁×M₁₆	—	4	2	no	—
39	C₂×C₈×C₁₁	—	4	2	no	—
40	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₁	D₈	1	2	yes	—
43	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
44	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
48	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₁	D₈	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×#368	—	11	4	no	—
56	C₂×C₄×C₁₆	—	11	1	no	—
57	#1848	—	11	1	no	—
58	#1848	D₁₁	1	1	yes	characteristic
59	C₂³×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₁	—	2	4	no	—
61	C₂×C₄×C₁₁	#107	1	2	yes	—
62	C₂×C₄×C₁₁	#107	1	2	yes	—
63	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
65	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₁₁	—	8	2	no	—
70	C₈×C₁₁	#107	1	2	yes	—
71	C₈×C₁₁	#107	1	2	yes	—
72	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
74	C₁₁⋊C₈	—	8	2	no	—
75	C₂²×C₁₆	—	11	2	no	—
76	C₂×#108	—	2	1	no	—
77	C₂×#108	—	11	2	no	—
78	C₂×#108	—	22	1	no	—
79	C₂×#108	C₂×D₁₁	1	2	yes	—
80	#423	—	2	1	no	—
81	#423	—	22	1	no	—
82	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
83	#368	—	22	8	no	—
84	C₄×C₁₆	—	11	2	no	—
85	C₄×C₁₆	—	11	2	no	—
86	C₂²×C₁₁	—	2	2	no	—
87	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
88	C₂²×C₁₁	#494	1	1	yes	characteristic
89	C₂²×C₁₁	#500	1	1	yes	characteristic
90	C₄×C₁₁	—	2	2	no	—
91	C₄×C₁₁	#368	1	4	yes	complemented
92	C₄×C₁₁	#384	1	1	yes	characteristic
93	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
94	C₄×C₁₁	#493	1	1	yes	characteristic
95	C₄×C₁₁	#500	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	22	2	no	—
98	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
101	C₂×C₁₆	—	11	4	no	—
102	C₂×C₁₆	—	11	4	no	—
103	C₂×C₁₆	—	11	4	no	—
104	#108	—	2	4	no	—
105	#108	—	4	2	no	—
106	#108	—	22	4	no	—
107	#108	—	44	2	no	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	—	44	2	no	—
110	C₄×C₈	Dic₂₂	1	1	yes	characteristic
111	C₄×C₈	Dic₂₂	1	1	yes	characteristic
112	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
113	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
114	C₂×C₁₁	—	2	1	no	—
115	C₂×C₁₁	—	2	1	no	—
116	C₂×C₁₁	#1845	1	1	yes	characteristic
117	C₂×C₁₁	C₂×#368	1	2	yes	—
118	C₂²×C₄	#4326	1	2	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	M₁₆	—	4	2	no	—
122	M₁₆	—	44	2	no	—
123	M₁₆	—	44	2	no	—
124	C₂×C₈	—	4	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	#4323	1	2	yes	—
129	C₂×C₈	#4326	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	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
135	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
136	C₄²	#4323	1	2	yes	—
137	C₁₆	—	11	8	no	—
138	C₁₁	#12909	1	1	yes	characteristic, complemented, Sylow
139	C₂³	C₂×#4326	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	#66871	1	2	yes	—
142	C₂×C₄	#66871	1	2	yes	—
143	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂×#4323	1	1	yes	characteristic
145	C₂×C₄	C₂×#4323	1	1	yes	characteristic
146	C₂×C₄	#66993	1	1	yes	characteristic
147	C₂×C₄	C₂×#4326	1	1	yes	characteristic
148	C₂×C₄	#66997	1	1	yes	characteristic
149	C₈	—	8	2	no	—
150	C₈	—	88	2	no	—
151	C₈	#66871	1	2	yes	—
152	C₈	#66871	1	2	yes	—
153	C₈	#66941	1	2	yes	—
154	C₈	C₄⋊C₄×D₁₁	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#66871	1	1	yes	characteristic
157	C₂²	#10622691	1	1	yes	characteristic
158	C₂²	#10622705	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	#10622131	1	4	yes	complemented
161	C₄	C₂×#66871	1	1	yes	characteristic
162	C₄	#10622589	1	1	yes	characteristic
163	C₄	#10622686	1	1	yes	characteristic
164	C₄	#10622691	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#13068775	1	1	yes	characteristic
168	C₂	C₂×#10622131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1848	C₂
socle	C₂²×C₁₁	C₂×#107
prime radical	C₂³×C₁₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1848	D₁₁
hypocenter	C₁₁	#12909
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
91. C₄×C₁₁	#368	semidirect
138. C₁₁	#12909	semidirect
160. C₄	#10622131	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622131	2
C₂×C₄×C₁₁⋊C₁₆	2
#13068788	2
C₁₁×#1848	2
#12909	11