Finite Group Databasebeta

Subgroups — #13070068

Order 1408.

190 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10622745	C₂	1	1	yes	maximal, characteristic
2	#10622618	C₂	1	1	yes	maximal, characteristic
3	#10622615	C₂	1	1	yes	maximal, characteristic
4	#10622612	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10622196	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622171	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4339	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
10	C₁₁×#117	C₂²	1	1	yes	characteristic
11	C₁₁×#117	C₂²	1	1	yes	characteristic
12	C₂×#4318	C₂²	1	1	yes	characteristic
13	C₂×#4318	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
15	#66888	—	2	1	no	—
16	#66887	—	2	1	no	—
17	#66874	—	2	1	no	—
18	#66873	—	2	1	no	—
19	#66869	—	2	2	no	—
20	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
21	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₁	—	2	1	no	—
23	#4339	—	2	1	no	—
24	#4339	—	2	1	no	—
25	#4339	—	2	1	no	—
26	#4339	—	2	1	no	—
27	C₂×D₄₄	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₁	—	2	1	no	—
29	C₂×Dic₂₂	D₄	1	1	yes	characteristic
30	C₁₁×M₁₆	—	2	1	no	—
31	C₁₁×M₁₆	—	2	1	no	—
32	C₂×C₈×C₁₁	—	2	1	no	—
33	C₁₁×C₄⋊C₄	—	2	1	no	—
34	C₁₁×C₄⋊C₄	—	2	1	no	—
35	C₁₁×C₂²⋊C₄	—	2	1	no	—
36	C₁₁×C₂²⋊C₄	—	2	1	no	—
37	C₄²×C₁₁	—	2	1	no	—
38	C₄²×C₁₁	D₄	1	2	yes	—
39	#4318	—	2	1	no	—
40	#4318	—	2	1	no	—
41	#4318	—	2	1	no	—
42	#4318	—	2	1	no	—
43	C₂×C₁₁⋊C₈	—	2	1	no	—
44	C₂×C₁₁⋊C₈	—	2	1	no	—
45	C₂×C₁₁⋊C₈	—	2	1	no	—
46	C₂×C₁₁⋊C₈	—	2	1	no	—
47	C₂×C₁₁⋊C₈	—	2	1	no	—
48	#2080	—	11	1	no	maximal, Sylow
49	C₂³×C₁₁	#41	1	1	yes	characteristic
50	C₂²×D₁₁	—	2	1	no	—
51	C₂×C₄×C₁₁	—	2	2	no	—
52	C₂×C₄×C₁₁	—	4	1	no	—
53	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
58	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
59	C₂²⋊D₁₁	—	4	1	no	—
60	C₂²⋊D₁₁	—	4	1	no	—
61	C₂×Dic₁₁	—	2	1	no	—
62	D₄₄	—	2	1	no	—
63	D₄₄	—	2	1	no	—
64	C₄×D₁₁	—	4	1	no	—
65	C₄×D₁₁	—	4	1	no	—
66	Dic₂₂	—	2	1	no	—
67	Dic₂₂	—	2	1	no	—
68	C₈×C₁₁	—	4	1	no	—
69	C₁₁⋊C₈	—	4	1	no	—
70	C₁₁⋊C₈	—	4	1	no	—
71	C₁₁⋊C₈	—	4	1	no	—
72	#424	—	11	1	no	—
73	#421	—	11	1	no	—
74	#416	—	11	1	no	—
75	#413	—	11	1	no	—
76	#344	—	11	1	no	—
77	#341	—	11	1	no	—
78	H(3,ℤ/4)	D₁₁	1	1	yes	characteristic, complemented
79	C₂²×C₁₁	—	4	1	no	—
80	C₂²×C₁₁	#123	1	1	yes	characteristic
81	C₂²×C₁₁	#123	1	1	yes	characteristic
82	C₂²×C₁₁	#126	1	1	yes	characteristic
83	C₂×D₁₁	—	4	1	no	—
84	C₂×D₁₁	—	8	1	no	—
85	C₄×C₁₁	—	4	2	no	—
86	C₄×C₁₁	—	8	1	no	—
87	C₄×C₁₁	#121	1	1	yes	characteristic
88	C₄×C₁₁	#123	1	1	yes	characteristic
89	C₄×C₁₁	#124	1	1	yes	characteristic
90	C₄×C₁₁	#126	1	1	yes	characteristic
91	Dic₁₁	—	4	1	no	—
92	C₂×#41	—	11	1	no	—
93	C₂×M₁₆	—	11	1	no	—
94	C₂×M₁₆	—	11	1	no	—
95	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
96	C₂²×C₈	—	11	1	no	—
97	#117	C₂×D₁₁	1	1	yes	characteristic
98	#117	C₂×D₁₁	1	1	yes	characteristic
99	C₄⋊C₈	—	22	2	no	—
100	#104	—	22	2	no	—
101	#103	—	22	1	no	—
102	#102	—	22	1	no	—
103	#101	—	22	1	no	—
104	#100	—	22	1	no	—
105	C₂×C₁₁	—	2	1	no	—
106	C₂×C₁₁	—	2	1	no	—
107	C₂×C₁₁	—	2	1	no	—
108	C₂×C₁₁	#397	1	1	yes	characteristic
109	D₁₁	—	8	1	no	—
110	#41	—	22	1	no	—
111	#41	—	22	1	no	—
112	#41	—	22	1	no	—
113	#41	—	22	1	no	—
114	C₂×Q₈	—	11	1	no	—
115	C₂×D₄	—	11	1	no	—
116	C₂×D₄	—	22	1	no	—
117	C₂²×C₄	—	22	1	no	—
118	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
119	M₁₆	—	2	1	no	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	22	1	no	—
122	M₁₆	—	22	1	no	—
123	M₁₆	—	22	1	no	—
124	M₁₆	—	22	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	22	1	no	—
127	C₂×C₈	—	22	1	no	—
128	C₂×C₈	—	22	1	no	—
129	C₂×C₈	—	22	1	no	—
130	C₂×C₈	—	22	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	D₄₄	1	2	yes	—
137	C₁₁	#2080	1	1	yes	characteristic, complemented, Sylow
138	C₂³	—	22	1	no	—
139	C₂³	#4341	1	1	yes	characteristic
140	Q₈	—	22	1	no	—
141	Q₈	—	22	1	no	—
142	D₄	—	22	1	no	—
143	D₄	—	22	1	no	—
144	D₄	—	44	1	no	—
145	D₄	—	44	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	22	1	no	—
149	C₂×C₄	—	44	1	no	—
150	C₂×C₄	—	44	1	no	—
151	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
152	C₂×C₄	#4339	1	1	yes	characteristic
153	C₂×C₄	#4339	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
155	C₂×C₄	#4341	1	1	yes	characteristic
156	C₂×C₄	#4343	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	44	1	no	—
159	C₈	—	44	1	no	—
160	C₈	—	44	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	44	1	no	—
163	C₂²	—	88	1	no	—
164	C₂²	#66936	1	1	yes	characteristic
165	C₂²	#66939	1	1	yes	characteristic
166	C₂²	#66951	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	8	1	no	—
169	C₄	—	44	1	no	—
170	C₄	#66928	1	1	yes	characteristic
171	C₄	#66936	1	1	yes	characteristic
172	C₄	#66948	1	1	yes	characteristic
173	C₄	#66951	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	88	1	no	—
178	C₂	#10622287	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#66928
commutator, cosocle	C₂²×C₄×C₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×H(3,ℤ/4)	C₂
socle	C₂×C₁₁	#397
prime radical	C₂×#4339	C₂²
hypercenter	H(3,ℤ/4)	D₁₁
hypocenter	C₁₁	#2080
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
4. #10622612	C₂	semidirect
5. C₁₁×H(3,ℤ/4)	C₂	semidirect
6. #10622196	C₂	semidirect
7. #10622171	C₂	semidirect
78. H(3,ℤ/4)	D₁₁	semidirect
137. C₁₁	#2080	semidirect

Maximal subgroups

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

subgroup	index
#10622745	2
#10622618	2
#10622615	2
#10622612	2
C₁₁×H(3,ℤ/4)	2
#10622196	2
#10622171	2
#2080	11