Subgroups — #13073642

Order 1408.

556 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×#66941	C₂	1	2	yes	maximal
2	C₂×#66922	C₂	1	2	yes	maximal
3	C₄×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#10622598	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	#10622269	C₂	1	4	yes	maximal
9	C₂²×Dic₂₂	C₂²	1	4	yes	—
10	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₁	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₁⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
22	#66941	—	2	8	no	—
23	#66922	—	2	8	no	—
24	C₁₁×#95	C₂²	1	2	yes	—
25	C₁₁⋊#95	C₂²	1	2	yes	—
26	C₁₁⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₁	C₂³	1	4	yes	—
33	C₂²×Dic₁₁	C₂³	1	2	yes	—
34	C₂²×Dic₁₁	C₂³	1	2	yes	—
35	C₂×Dic₂₂	—	2	8	no	—
36	C₂×Dic₂₂	—	2	8	no	—
37	C₂×Dic₂₂	—	4	4	no	—
38	C₁₁×C₄⋊C₄	D₄	1	4	yes	—
39	C₁₁×C₄⋊C₄	Q₈	1	4	yes	complemented
40	C₄²×C₁₁	—	2	4	no	—
41	C₄²×C₁₁	Q₈	1	4	yes	complemented
42	C₄⋊Dic₁₁	—	2	8	no	—
43	C₄⋊Dic₁₁	—	2	4	no	—
44	C₄⋊Dic₁₁	—	2	4	no	—
45	C₄⋊Dic₁₁	—	4	2	no	—
46	C₄⋊Dic₁₁	D₄	1	4	yes	—
47	C₄⋊Dic₁₁	Q₈	1	4	yes	complemented
48	Dic₁₁⋊C₄	—	2	8	no	—
49	C₄×Dic₁₁	—	2	4	no	—
50	#2551	—	11	1	no	maximal, Sylow
51	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₁	—	2	2	no	—
53	C₂×C₄×C₁₁	—	2	2	no	—
54	C₂×C₄×C₁₁	—	2	1	no	—
55	C₂×C₄×C₁₁	—	2	2	no	—
56	C₂×C₄×C₁₁	—	2	1	no	—
57	C₂×C₄×C₁₁	—	2	2	no	—
58	C₂×C₄×C₁₁	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
67	C₂×Dic₁₁	—	2	2	no	—
68	C₂×Dic₁₁	—	2	2	no	—
69	C₂×Dic₁₁	—	2	2	no	—
70	C₂×Dic₁₁	—	2	2	no	—
71	C₂×Dic₁₁	—	2	4	no	—
72	C₂×Dic₁₁	—	2	4	no	—
73	C₂×Dic₁₁	—	2	4	no	—
74	C₂×Dic₁₁	C₂×D₄	1	4	yes	—
75	C₂×Dic₁₁	C₂×Q₈	1	4	yes	—
76	Dic₂₂	—	4	16	no	—
77	C₂×C₄⋊Q₈	—	11	2	no	—
78	C₂×C₄⋊Q₈	—	11	2	no	—
79	#395	—	11	4	no	—
80	#389	—	11	2	no	—
81	#389	—	11	2	no	—
82	#384	—	11	1	no	—
83	C₄×C₄⋊C₄	—	11	1	no	—
84	C₄×C₄⋊C₄	D₁₁	1	1	yes	characteristic
85	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
88	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂²×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
92	C₄×C₁₁	—	2	2	no	—
93	C₄×C₁₁	—	2	4	no	—
94	C₄×C₁₁	—	2	4	no	—
95	C₄×C₁₁	C₄⋊Q₈	1	4	yes	complemented
96	C₄×C₁₁	C₄⋊Q₈	1	4	yes	complemented
97	Dic₁₁	—	2	4	no	—
98	Dic₁₁	—	2	4	no	—
99	Dic₁₁	—	4	4	no	—
100	C₂²×Q₈	—	11	4	no	—
101	C₄⋊Q₈	—	22	8	no	—
102	C₄⋊Q₈	—	22	8	no	—
103	C₂×C₄⋊C₄	—	11	1	no	—
104	C₂×C₄⋊C₄	—	11	1	no	—
105	C₂×C₄⋊C₄	—	11	2	no	—
106	C₂×C₄⋊C₄	—	11	2	no	—
107	C₂×C₄⋊C₄	—	11	4	no	—
108	C₂×C₄⋊C₄	—	11	4	no	—
109	C₂×C₄⋊C₄	—	11	2	no	—
110	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
112	C₂×C₄²	—	11	2	no	—
113	C₂×C₄²	C₂×D₁₁	1	2	yes	—
114	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
115	#95	—	11	2	no	—
116	#95	—	11	4	no	—
117	#95	C₂×D₁₁	1	2	yes	—
118	C₂×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₁₁	#544	1	1	yes	characteristic
121	C₂×C₁₁	D₄²	1	1	yes	characteristic
122	C₂×C₁₁	D₄×Q₈	1	1	yes	characteristic
123	C₂×C₁₁	D₄×Q₈	1	1	yes	characteristic
124	C₂×C₁₁	Q₈²	1	1	yes	characteristic
125	C₂×Q₈	—	22	8	no	—
126	C₂×Q₈	—	22	8	no	—
127	C₂×Q₈	—	44	4	no	—
128	C₂²×C₄	—	11	2	no	—
129	C₂²×C₄	—	11	2	no	—
130	C₂²×C₄	—	11	4	no	—
131	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
133	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
134	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
136	C₄⋊C₄	—	11	4	no	—
137	C₄⋊C₄	—	11	4	no	—
138	C₄⋊C₄	—	22	8	no	—
139	C₄⋊C₄	—	22	8	no	—
140	C₄⋊C₄	—	22	4	no	—
141	C₄⋊C₄	—	22	4	no	—
142	C₄⋊C₄	—	44	2	no	—
143	C₄⋊C₄	Dic₂₂	1	4	yes	complemented
144	C₄⋊C₄	D₄₄	1	4	yes	—
145	C₄²	—	2	4	no	—
146	C₄²	—	22	4	no	—
147	C₄²	Dic₂₂	1	4	yes	complemented
148	C₁₁	#2551	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂³×D₁₁	1	1	yes	characteristic
150	Q₈	—	44	16	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	11	4	no	—
158	C₂×C₄	—	11	4	no	—
159	C₂×C₄	—	22	2	no	—
160	C₂×C₄	—	22	2	no	—
161	C₂×C₄	—	22	2	no	—
162	C₂×C₄	—	22	2	no	—
163	C₂×C₄	—	22	4	no	—
164	C₂×C₄	—	22	4	no	—
165	C₂×C₄	—	22	4	no	—
166	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
167	C₂×C₄	C₂×Dic₂₂	1	4	yes	—
168	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
169	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
170	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
171	C₂×C₄	C₂×D₄₄	1	4	yes	—
172	C₂×C₄	C₂×D₄₄	1	2	yes	—
173	C₂×C₄	D₄×D₁₁	1	2	yes	—
174	C₂×C₄	Q₈×D₁₁	1	2	yes	—
175	C₂²	C₂²×Dic₂₂	1	1	yes	characteristic
176	C₂²	C₂²×Dic₂₂	1	1	yes	characteristic
177	C₂²	C₂²×D₄₄	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₁₁	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₁₁	1	1	yes	characteristic
180	C₂²	#67042	1	1	yes	characteristic
181	C₂²	#67043	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	22	4	no	—
186	C₄	—	22	4	no	—
187	C₄	—	44	4	no	—
188	C₄	#66922	1	4	yes	complemented
189	C₄	#66941	1	4	yes	complemented
190	C₂	C₂×#66922	1	1	yes	characteristic
191	C₂	C₂×#66941	1	1	yes	characteristic
192	C₂	#10623050	1	1	yes	characteristic
193	C₂	D₄×Dic₂₂	1	1	yes	characteristic
194	C₂	D₄×D₄₄	1	1	yes	characteristic
195	C₂	Q₈×Dic₂₂	1	1	yes	characteristic
196	C₂	Q₈×D₄₄	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	#2551
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
39. C₁₁×C₄⋊C₄	Q₈	semidirect
41. C₄²×C₁₁	Q₈	semidirect
47. C₄⋊Dic₁₁	Q₈	semidirect
95. C₄×C₁₁	C₄⋊Q₈	semidirect
96. C₄×C₁₁	C₄⋊Q₈	semidirect
143. C₄⋊C₄	Dic₂₂	semidirect
147. C₄²	Dic₂₂	semidirect
148. C₁₁	#2551	semidirect
188. C₄	#66922	semidirect
189. C₄	#66941	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66941	2
C₂×#66922	2
C₄×C₁₁×C₄⋊C₄	2
#10622598	2
#10622548	2
C₄×C₄⋊Dic₁₁	2
#10622541	2
#10622269	2
#2551	11