Subgroups — #13077205

Order 1408.

492 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#66979	C₂	1	1	yes	maximal, characteristic
3	C₂²×#4318	C₂	1	1	yes	maximal, characteristic
4	#10622611	C₂	1	4	yes	maximal, complemented
5	#10622608	C₂	1	4	yes	maximal, complemented
6	#10622577	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
8	C₁₁×#117	—	2	1	no	—
9	C₁₁×#117	—	2	1	no	—
10	C₁₁×#117	C₂²	1	2	yes	—
11	C₁₁×#117	C₂²	1	2	yes	—
12	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
14	C₂×C₄²×C₁₁	—	2	1	no	—
15	C₂×C₂²⋊Dic₁₁	—	2	1	no	—
16	#66979	—	2	1	no	—
17	#66979	—	2	1	no	—
18	#66979	C₂²	1	2	yes	—
19	#66979	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₁	—	2	1	no	—
22	C₂×#4318	C₄	1	8	yes	complemented
23	C₂×#4318	C₂²	1	2	yes	—
24	C₂×#4318	C₂²	1	2	yes	—
25	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
26	#66872	C₂²	1	8	yes	complemented
27	#66871	C₂²	1	8	yes	complemented
28	C₂⁴×C₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₁	—	2	2	no	—
30	C₂²×C₄×C₁₁	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₁	Q₈	1	4	yes	—
35	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₁	—	2	2	no	—
39	C₁₁×C₄⋊C₄	—	2	2	no	—
40	C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
41	C₁₁×C₂²⋊C₄	—	2	2	no	—
42	C₁₁×C₂²⋊C₄	—	2	2	no	—
43	C₄²×C₁₁	—	2	2	no	—
44	C₄²×C₁₁	—	2	2	no	—
45	C₂²⋊Dic₁₁	—	2	2	no	—
46	C₂²⋊Dic₁₁	—	2	2	no	—
47	C₄⋊Dic₁₁	—	2	2	no	—
48	C₄⋊Dic₁₁	C₂³	1	4	yes	—
49	C₄×Dic₁₁	—	2	2	no	—
50	C₄×Dic₁₁	—	2	2	no	—
51	#4318	C₂×C₄	1	8	yes	complemented
52	#4318	C₂×C₄	1	8	yes	complemented
53	C₂×C₁₁⋊C₈	C₂×C₄	1	8	yes	complemented
54	C₂×C₁₁⋊C₈	C₂³	1	4	yes	—
55	#2911	—	11	1	no	maximal, Sylow
56	C₂³×C₁₁	—	4	2	no	—
57	C₂³×C₁₁	C₄⋊C₄	1	4	yes	—
58	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₁	—	2	4	no	—
62	C₂×C₄×C₁₁	—	4	4	no	—
63	C₂×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
66	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₁₁	—	2	4	no	—
77	C₂×Dic₁₁	—	4	4	no	—
78	C₁₁⋊C₈	C₂²×C₄	1	8	yes	—
79	C₂²×M₁₆	—	11	1	no	—
80	C₂×#117	—	11	1	no	—
81	C₂×#117	D₁₁	1	1	yes	characteristic
82	#428	—	11	4	no	—
83	#428	—	11	4	no	—
84	#427	—	11	4	no	—
85	C₂²×C₁₁	—	2	2	no	—
86	C₂²×C₁₁	—	2	2	no	—
87	C₂²×C₁₁	—	2	2	no	—
88	C₂²×C₁₁	—	4	4	no	—
89	C₂²×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₂²×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₁₁	—	4	4	no	—
94	C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
99	Dic₁₁	—	4	4	no	—
100	C₂³×C₄	C₂×D₁₁	1	1	yes	characteristic
101	C₂×M₁₆	—	11	2	no	—
102	C₂×M₁₆	—	11	2	no	—
103	C₂×M₁₆	—	11	8	no	—
104	C₂²×C₈	—	11	2	no	—
105	#117	—	2	1	no	—
106	#117	—	2	1	no	—
107	#117	—	11	2	no	—
108	#117	—	11	2	no	—
109	#117	—	22	1	no	—
110	#117	—	22	1	no	—
111	#117	C₂×D₁₁	1	2	yes	—
112	#117	C₂×D₁₁	1	2	yes	—
113	C₂×C₄⋊C₄	—	11	2	no	—
114	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
115	C₂×C₂²⋊C₄	—	2	1	no	—
116	C₂×C₂²⋊C₄	—	22	1	no	—
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	—	22	1	no	—
119	#107	—	11	8	no	—
120	#106	—	11	8	no	—
121	C₂×C₁₁	—	2	2	no	—
122	C₂×C₁₁	—	2	2	no	—
123	C₂×C₁₁	—	2	2	no	—
124	C₂×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₁₁	#575	1	2	yes	—
126	C₂⁴	C₂²⋊D₁₁	1	1	yes	characteristic
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	22	2	no	—
129	C₂²×C₄	Dic₂₂	1	4	yes	—
130	C₂²×C₄	C₄×D₁₁	1	4	yes	—
131	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
137	M₁₆	—	11	8	no	—
138	M₁₆	—	11	8	no	—
139	C₂×C₈	—	11	8	no	—
140	C₂×C₈	—	11	4	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	11	4	no	—
143	C₄⋊C₄	—	22	2	no	—
144	C₄⋊C₄	C₂²×D₁₁	1	4	yes	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	22	2	no	—
148	C₂²⋊C₄	—	22	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	22	2	no	—
152	C₄²	—	22	2	no	—
153	C₁₁	#2911	1	1	yes	characteristic, complemented, Sylow
154	C₂³	—	4	2	no	—
155	C₂³	Dic₁₁⋊C₄	1	4	yes	—
156	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
157	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
158	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	22	4	no	—
162	C₂×C₄	—	44	4	no	—
163	C₂×C₄	Dic₁₁⋊C₄	1	4	yes	—
164	C₂×C₄	Dic₁₁⋊C₄	1	4	yes	—
165	C₂×C₄	Dic₁₁⋊C₄	1	4	yes	—
166	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
167	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
168	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
169	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
170	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
176	C₈	—	11	8	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×Dic₁₁⋊C₄	1	2	yes	—
182	C₂²	C₂×Dic₁₁⋊C₄	1	2	yes	—
183	C₂²	C₂×Dic₁₁⋊C₄	1	2	yes	—
184	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	—	44	4	no	—
187	C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
188	C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
189	C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
190	C₄	C₂²×Dic₂₂	1	1	yes	characteristic
191	C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
196	C₂	#10623353	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₁
commutator	C₄×C₁₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₂×C₁₁×#117	C₂
socle	C₂²×C₁₁	C₂²×D₄
prime radical	C₂⁴×C₁₁	D₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	C₂×#117	D₁₁
hypocenter	C₁₁	#2911
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #10622611	C₂	semidirect
5. #10622608	C₂	semidirect
6. #10622577	C₂	semidirect
22. C₂×#4318	C₄	semidirect
26. #66872	C₂²	semidirect
27. #66871	C₂²	semidirect
51. #4318	C₂×C₄	semidirect
52. #4318	C₂×C₄	semidirect
53. C₂×C₁₁⋊C₈	C₂×C₄	semidirect
153. C₁₁	#2911	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#117	2
C₂×#66979	2
C₂²×#4318	2
#10622611	2
#10622608	2
#10622577	2
#2911	11