Subgroups — #14670198

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×#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×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#106	—	2	1	no	—
6	C₂×C₁₁×#106	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₁₁×#384	—	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×C₁₁×C₄⋊C₄	—	4	2	no	—
20	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
21	C₂×#4318	—	2	2	no	—
22	C₁₁×#106	—	2	4	no	—
23	C₁₁×#106	—	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	#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	#12903	—	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	C₁₁×C₄⋊C₄	—	8	4	no	—
49	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₁	SD₁₆	1	2	yes	—
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	—	11	1	no	—
58	#1848	—	11	1	no	—
59	#1844	D₁₁	1	1	yes	characteristic
60	C₂³×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₁	—	2	4	no	—
62	C₂×C₄×C₁₁	—	8	4	no	—
63	C₂×C₄×C₁₁	—	8	2	no	—
64	C₂×C₄×C₁₁	#106	1	2	yes	—
65	C₂×C₄×C₁₁	#106	1	2	yes	—
66	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₄⋊Q₈	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	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
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	—	11	2	no	—
78	C₂×#108	—	11	2	no	—
79	C₂×#108	—	22	1	no	—
80	C₂×#106	—	2	1	no	—
81	C₂×#106	C₂×D₁₁	1	2	yes	—
82	#423	—	22	1	no	—
83	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
84	#384	—	2	1	no	—
85	#365	—	22	8	no	—
86	#346	—	11	2	no	—
87	C₄×C₁₆	—	11	2	no	—
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	—	16	2	no	—
94	C₄×C₁₁	#365	1	4	yes	complemented
95	C₄×C₁₁	#384	1	1	yes	characteristic
96	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
97	C₄×C₁₁	#492	1	1	yes	characteristic
98	C₄×C₁₁	#498	1	1	yes	characteristic
99	C₂×M₁₆	—	22	2	no	—
100	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
105	M₃₂	—	11	4	no	—
106	M₃₂	—	11	4	no	—
107	C₂×C₁₆	—	11	4	no	—
108	#108	—	22	4	no	—
109	#108	—	44	2	no	—
110	#106	—	2	4	no	—
111	#106	—	4	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	C₄×D₁₁	1	1	yes	characteristic
116	C₄×C₈	C₄×D₁₁	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	—	4	2	no	—
122	C₂²×C₄	#4324	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
124	M₁₆	—	44	2	no	—
125	M₁₆	—	44	2	no	—
126	C₂×C₈	—	44	2	no	—
127	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
129	C₂×C₈	#4324	1	2	yes	—
130	C₂×C₈	#4325	1	2	yes	—
131	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
133	C₂×C₈	D₄×D₁₁	1	2	yes	—
134	C₂×C₈	Q₈×D₁₁	1	2	yes	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	8	4	no	—
138	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
139	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
140	C₄²	#4325	1	2	yes	—
141	C₁₆	—	11	8	no	—
142	C₁₁	#12903	1	1	yes	characteristic, complemented, Sylow
143	C₂³	C₂×#4324	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	#66872	1	2	yes	—
148	C₂×C₄	#66872	1	2	yes	—
149	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
150	C₂×C₄	C₂×#4324	1	1	yes	characteristic
151	C₂×C₄	#66993	1	1	yes	characteristic
152	C₂×C₄	C₂×#4325	1	1	yes	characteristic
153	C₂×C₄	C₂×#4325	1	1	yes	characteristic
154	C₂×C₄	#66997	1	1	yes	characteristic
155	C₈	—	88	2	no	—
156	C₈	#66872	1	2	yes	—
157	C₈	#66872	1	2	yes	—
158	C₈	#66941	1	2	yes	—
159	C₈	C₄⋊C₄×D₁₁	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#66872	1	1	yes	characteristic
162	C₂²	#10622689	1	1	yes	characteristic
163	C₂²	#10622692	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	16	2	no	—
166	C₄	#10622125	1	4	yes	complemented
167	C₄	C₂×#66872	1	1	yes	characteristic
168	C₄	#10622589	1	1	yes	characteristic
169	C₄	#10622692	1	1	yes	characteristic
170	C₄	#10622698	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#13068772	1	1	yes	characteristic
174	C₂	C₂×#10622125	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#66872
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1844	C₂
socle	C₂²×C₁₁	C₂×#106
prime radical	C₂³×C₁₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1844	D₁₁
hypocenter	C₁₁	#12903
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

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
55. C₁₁⋊C₁₆	C₄⋊C₄	semidirect
94. C₄×C₁₁	#365	semidirect
142. C₁₁	#12903	semidirect
166. C₄	#10622125	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×#1844	2
#12903	11