Subgroups — #13073651

Order 1408.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66923	C₂	1	2	yes	maximal
2	C₄×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#10622589	C₂	1	2	yes	maximal
4	#10622586	C₂	1	2	yes	maximal
5	#10622548	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
7	#10622541	C₂	1	2	yes	maximal
8	#10622270	C₂	1	4	yes	maximal
9	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₁⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₁₁⋊C₄	C₂²	1	4	yes	—
20	C₂×Dic₁₁⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
22	#66923	—	2	8	no	—
23	C₁₁×#95	C₂²	1	2	yes	—
24	C₁₁⋊#95	C₂²	1	2	yes	—
25	C₁₁⋊#95	C₂²	1	4	yes	—
26	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₁	C₂³	1	4	yes	—
32	C₂²×Dic₁₁	C₂³	1	2	yes	—
33	C₂²×Dic₁₁	C₂³	1	2	yes	—
34	C₁₁×C₄⋊C₄	—	2	2	no	—
35	C₁₁×C₄⋊C₄	—	2	2	no	—
36	C₄²×C₁₁	—	2	4	no	—
37	C₄²×C₁₁	Q₈	1	4	yes	—
38	C₄⋊Dic₁₁	—	2	4	no	—
39	C₄⋊Dic₁₁	—	2	4	no	—
40	C₄⋊Dic₁₁	—	4	2	no	—
41	C₄⋊Dic₁₁	D₄	1	4	yes	—
42	C₄⋊Dic₁₁	Q₈	1	4	yes	—
43	Dic₁₁⋊C₄	—	2	8	no	—
44	Dic₁₁⋊C₄	—	2	8	no	—
45	Dic₁₁⋊C₄	—	4	4	no	—
46	C₄×Dic₁₁	—	2	4	no	—
47	#2552	—	11	1	no	maximal, Sylow
48	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₁₁	—	2	1	no	—
50	C₂×C₄×C₁₁	—	2	1	no	—
51	C₂×C₄×C₁₁	—	2	2	no	—
52	C₂×C₄×C₁₁	—	2	2	no	—
53	C₂×C₄×C₁₁	—	2	2	no	—
54	C₂×C₄×C₁₁	—	2	2	no	—
55	C₂×C₄×C₁₁	—	2	2	no	—
56	C₂×C₄×C₁₁	—	2	2	no	—
57	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₁₁	#41	1	2	yes	—
63	C₂×C₄×C₁₁	#41	1	2	yes	—
64	C₂×Dic₁₁	—	2	2	no	—
65	C₂×Dic₁₁	—	2	2	no	—
66	C₂×Dic₁₁	—	2	2	no	—
67	C₂×Dic₁₁	—	2	2	no	—
68	C₂×Dic₁₁	—	2	4	no	—
69	C₂×Dic₁₁	—	2	4	no	—
70	C₂×Dic₁₁	—	2	4	no	—
71	C₂×Dic₁₁	C₂×D₄	1	4	yes	—
72	C₂×Dic₁₁	C₂×Q₈	1	4	yes	—
73	C₂×#125	—	11	2	no	—
74	#398	—	11	4	no	—
75	#389	—	11	2	no	—
76	#389	—	11	2	no	—
77	#384	—	11	2	no	—
78	#384	—	11	1	no	—
79	C₄×C₄⋊C₄	—	11	1	no	—
80	C₄×C₄⋊C₄	D₁₁	1	1	yes	characteristic
81	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
85	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
86	C₂²×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
88	C₄×C₁₁	—	2	2	no	—
89	C₄×C₁₁	—	4	2	no	—
90	C₄×C₁₁	—	4	2	no	—
91	C₄×C₁₁	#125	1	4	yes	complemented
92	C₄×C₁₁	C₄⋊Q₈	1	4	yes	—
93	Dic₁₁	—	2	4	no	—
94	Dic₁₁	—	2	4	no	—
95	Dic₁₁	—	4	4	no	—
96	#125	—	22	8	no	—
97	C₂×C₄⋊C₄	—	11	1	no	—
98	C₂×C₄⋊C₄	—	11	1	no	—
99	C₂×C₄⋊C₄	—	11	2	no	—
100	C₂×C₄⋊C₄	—	11	4	no	—
101	C₂×C₄⋊C₄	—	11	4	no	—
102	C₂×C₄⋊C₄	—	11	2	no	—
103	C₂×C₄⋊C₄	—	11	4	no	—
104	C₂×C₄⋊C₄	—	11	2	no	—
105	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
107	C₂×C₄²	—	11	2	no	—
108	C₂×C₄²	C₂×D₁₁	1	2	yes	—
109	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
110	#95	—	11	2	no	—
111	#95	—	11	4	no	—
112	#95	C₂×D₁₁	1	2	yes	—
113	C₂×C₁₁	C₂×#125	1	1	yes	characteristic
114	C₂×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₁₁	#544	1	1	yes	characteristic
116	C₂×C₁₁	#547	1	1	yes	characteristic
117	C₂×C₁₁	#550	1	1	yes	characteristic
118	C₂×C₁₁	#554	1	1	yes	characteristic
119	C₂×C₁₁	#557	1	1	yes	characteristic
120	C₂²×C₄	—	11	2	no	—
121	C₂²×C₄	—	11	2	no	—
122	C₂²×C₄	—	11	4	no	—
123	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
124	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	11	4	no	—
131	C₄⋊C₄	—	11	4	no	—
132	C₄⋊C₄	—	22	8	no	—
133	C₄⋊C₄	—	22	8	no	—
134	C₄⋊C₄	—	22	4	no	—
135	C₄⋊C₄	—	22	4	no	—
136	C₄⋊C₄	—	44	4	no	—
137	C₄⋊C₄	—	44	2	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	22	4	no	—
140	C₄²	Dic₂₂	1	4	yes	—
141	C₁₁	#2552	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂³×D₁₁	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	11	4	no	—
152	C₂×C₄	—	11	4	no	—
153	C₂×C₄	—	22	2	no	—
154	C₂×C₄	—	22	2	no	—
155	C₂×C₄	—	22	2	no	—
156	C₂×C₄	—	22	2	no	—
157	C₂×C₄	—	22	4	no	—
158	C₂×C₄	—	22	4	no	—
159	C₂×C₄	—	22	4	no	—
160	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
161	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
162	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
163	C₂×C₄	#4339	1	2	yes	—
164	C₂×C₄	#4339	1	2	yes	—
165	C₂×C₄	D₄×D₁₁	1	2	yes	—
166	C₂×C₄	Q₈×D₁₁	1	2	yes	—
167	C₂²	C₂²×Dic₂₂	1	1	yes	characteristic
168	C₂²	C₂×#4339	1	1	yes	characteristic
169	C₂²	C₂×#4339	1	1	yes	characteristic
170	C₂²	C₂×D₄×D₁₁	1	1	yes	characteristic
171	C₂²	C₂×Q₈×D₁₁	1	1	yes	characteristic
172	C₂²	#67042	1	1	yes	characteristic
173	C₂²	#67043	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	22	4	no	—
178	C₄	—	22	4	no	—
179	C₄	—	44	4	no	—
180	C₄	#66923	1	4	yes	complemented
181	C₄	#66941	1	4	yes	—
182	C₂	C₂×#66923	1	1	yes	characteristic
183	C₂	C₂×#66941	1	1	yes	characteristic
184	C₂	#10623050	1	1	yes	characteristic
185	C₂	#10623082	1	1	yes	characteristic
186	C₂	#10623084	1	1	yes	characteristic
187	C₂	#10623098	1	1	yes	characteristic
188	C₂	#10623110	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁
commutator, socle, prime radical, cosocle	C₂³×C₁₁	C₂⁴
Fitting	C₄×C₁₁×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₁₁
hypocenter	C₁₁	#2552
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
91. C₄×C₁₁	#125	semidirect
141. C₁₁	#2552	semidirect
180. C₄	#66923	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66923	2
C₄×C₁₁×C₄⋊C₄	2
#10622589	2
#10622586	2
#10622548	2
C₄×C₄⋊Dic₁₁	2
#10622541	2
#10622270	2
#2552	11