Subgroups — C₂×#10623261 #13082151

Order 1408.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66999	C₂	1	4	yes	maximal, complemented
3	#10623261	C₂	1	16	yes	maximal, direct product factor
4	C₂×#66950	C₂	1	2	yes	maximal, complemented
5	C₂×#66946	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₄×D₁₁	C₂	1	2	yes	maximal, complemented
7	C₂×#66943	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₁	C₂²	1	2	yes	—
11	C₂²×D₄₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₁	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₁	C₂²	1	4	yes	—
14	C₂²×Dic₂₂	C₂²	1	1	yes	characteristic
15	C₁₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
16	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
19	#66999	C₂²	1	32	yes	complemented
20	C₂×(C₂×D₁₁)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₁₁)⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₁	C₂²	1	4	yes	—
25	C₂×Dic₁₁⋊C₄	C₂²	1	2	yes	—
26	C₂×Dic₁₁⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
28	#66950	C₂²	1	16	yes	complemented
29	#66946	C₂²	1	16	yes	—
30	C₄⋊C₄×D₁₁	C₂²	1	16	yes	complemented
31	#66943	C₂²	1	16	yes	complemented
32	C₄×D₄₄	C₂²	1	8	yes	—
33	C₄×Dic₂₂	C₂²	1	8	yes	complemented
34	C₂³×D₁₁	C₂³	1	2	yes	—
35	C₂×Q₈×C₁₁	—	4	4	no	—
36	C₂×Q₈×C₁₁	C₂³	1	8	yes	—
37	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₁	C₂³	1	2	yes	—
44	C₂×D₄₄	Q₈	1	8	yes	complemented
45	C₂×D₄₄	C₂³	1	4	yes	—
46	C₂×C₄×D₁₁	—	2	8	no	—
47	C₂×C₄×D₁₁	—	4	8	no	—
48	C₂×C₄×D₁₁	C₂³	1	8	yes	—
49	C₂×C₄×D₁₁	C₂³	1	16	yes	—
50	C₂×Dic₂₂	—	2	4	no	—
51	C₂×Dic₂₂	C₂³	1	4	yes	—
52	C₁₁×C₄⋊C₄	C₂³	1	8	yes	—
53	C₁₁×C₄⋊C₄	C₂³	1	8	yes	—
54	C₄²×C₁₁	C₂³	1	4	yes	—
55	(C₂×D₁₁)⋊C₄	C₂³	1	8	yes	—
56	(C₂×D₁₁)⋊C₄	C₂³	1	16	yes	—
57	C₄⋊Dic₁₁	C₂³	1	4	yes	—
58	C₄⋊Dic₁₁	C₂³	1	4	yes	—
59	C₄⋊Dic₁₁	C₂³	1	16	yes	—
60	Dic₁₁⋊C₄	C₂³	1	8	yes	—
61	Dic₁₁⋊C₄	C₂³	1	16	yes	—
62	C₄×Dic₁₁	C₂³	1	8	yes	—
63	C₂×#554	—	11	1	no	maximal, Sylow
64	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
65	C₂²×D₁₁	—	2	4	no	—
66	C₂²×D₁₁	—	2	4	no	—
67	C₂²×D₁₁	C₂×Q₈	1	8	yes	—
68	C₂²×D₁₁	C₂⁴	1	4	yes	—
69	Q₈×C₁₁	—	4	8	no	—
70	C₂×C₄×C₁₁	—	2	8	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₁₁	#41	1	4	yes	—
74	C₂×C₄×C₁₁	C₂⁴	1	2	yes	—
75	C₂×C₄×C₁₁	C₂⁴	1	8	yes	—
76	C₂×C₄×C₁₁	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₁	C₂⁴	1	2	yes	—
78	C₂×Dic₁₁	—	2	4	no	—
79	C₂×Dic₁₁	—	2	4	no	—
80	C₂×Dic₁₁	—	2	4	no	—
81	C₂×Dic₁₁	C₂⁴	1	4	yes	—
82	C₂×Dic₁₁	C₂⁴	1	4	yes	—
83	C₂×Dic₁₁	C₂⁴	1	4	yes	—
84	D₄₄	C₂×Q₈	1	16	yes	complemented
85	C₄×D₁₁	—	2	16	no	—
86	C₄×D₁₁	—	4	16	no	—
87	Dic₂₂	—	2	8	no	—
88	#554	—	11	16	no	—
89	C₂×C₄⋊Q₈	D₁₁	1	1	yes	characteristic, complemented
90	C₂×#125	—	11	2	no	—
91	C₂×C₂²⋊Q₈	—	11	4	no	—
92	C₂×C₂²⋊Q₈	—	11	2	no	—
93	C₂×C₄×Q₈	—	11	1	no	—
94	C₂×C₄×D₄	—	11	2	no	—
95	C₂×C₄×D₄	—	11	1	no	—
96	C₂²×C₄⋊C₄	—	11	2	no	—
97	C₂²×C₁₁	C₂²×Q₈	1	2	yes	—
98	C₂²×C₁₁	C₂×#41	1	2	yes	—
99	C₂²×C₁₁	H(5,𝔽₂)	1	2	yes	—
100	C₂²×C₁₁	C₂⁵	1	1	yes	characteristic
101	C₂×D₁₁	—	2	16	no	—
102	C₂×D₁₁	—	2	4	no	—
103	C₂×D₁₁	—	2	4	no	—
104	C₂×D₁₁	C₂²×Q₈	1	8	yes	—
105	C₄×C₁₁	—	2	2	no	—
106	C₄×C₁₁	—	2	8	no	—
107	C₄×C₁₁	C₂²×Q₈	1	4	yes	—
108	C₄×C₁₁	C₂×#41	1	4	yes	—
109	Dic₁₁	—	2	4	no	—
110	Dic₁₁	—	2	4	no	—
111	Dic₁₁	—	2	4	no	—
112	C₂²×Q₈	—	11	1	no	—
113	C₂²×Q₈	C₂×D₁₁	1	2	yes	—
114	C₂²×D₄	—	11	1	no	—
115	C₂³×C₄	—	11	2	no	—
116	C₂³×C₄	—	11	4	no	—
117	C₄⋊Q₈	C₂×D₁₁	1	8	yes	complemented
118	#125	—	11	16	no	—
119	C₂²⋊Q₈	—	11	32	no	—
120	C₂²⋊Q₈	—	11	16	no	—
121	C₄×Q₈	—	11	8	no	—
122	C₄×D₄	—	11	16	no	—
123	C₄×D₄	—	11	8	no	—
124	C₂×C₄⋊C₄	—	11	1	no	—
125	C₂×C₄⋊C₄	—	11	4	no	—
126	C₂×C₄⋊C₄	—	11	1	no	—
127	C₂×C₄⋊C₄	—	11	2	no	—
128	C₂×C₄⋊C₄	—	11	4	no	—
129	C₂×C₄⋊C₄	—	11	16	no	—
130	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
131	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
132	C₂×C₂²⋊C₄	—	11	4	no	—
133	C₂×C₂²⋊C₄	—	11	2	no	—
134	C₂×C₄²	—	11	2	no	—
135	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
136	C₂×C₁₁	#554	1	4	yes	complemented
137	C₂×C₁₁	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₁₁	C₂²×#41	1	1	yes	characteristic
139	C₂×C₁₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₁₁	—	2	8	no	—
141	C₂⁴	—	11	2	no	—
142	C₂×Q₈	—	4	4	no	—
143	C₂×Q₈	—	11	4	no	—
144	C₂×Q₈	—	22	4	no	—
145	C₂×Q₈	C₂²×D₁₁	1	8	yes	—
146	C₂×D₄	—	11	8	no	—
147	C₂×D₄	—	11	4	no	—
148	C₂²×C₄	—	11	2	no	—
149	C₂²×C₄	—	11	2	no	—
150	C₂²×C₄	—	11	2	no	—
151	C₂²×C₄	—	11	8	no	—
152	C₂²×C₄	—	11	16	no	—
153	C₂²×C₄	—	22	8	no	—
154	C₂²×C₄	—	44	8	no	—
155	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
156	C₂²×C₄	C₂²×D₁₁	1	4	yes	—
157	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
159	C₄⋊C₄	—	11	4	no	—
160	C₄⋊C₄	—	11	16	no	—
161	C₄⋊C₄	—	11	4	no	—
162	C₄⋊C₄	—	11	8	no	—
163	C₄⋊C₄	—	11	16	no	—
164	C₄⋊C₄	C₂²×D₁₁	1	8	yes	—
165	C₄⋊C₄	C₂²×D₁₁	1	8	yes	—
166	C₂²⋊C₄	—	11	16	no	—
167	C₂²⋊C₄	—	11	8	no	—
168	C₄²	—	11	8	no	—
169	C₄²	C₂²×D₁₁	1	4	yes	—
170	C₁₁	C₂×#554	1	1	yes	characteristic, complemented, Sylow
171	C₂³	—	11	8	no	—
172	C₂³	—	11	4	no	—
173	C₂³	—	22	4	no	—
174	C₂³	—	22	4	no	—
175	C₂³	C₂³×D₁₁	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	22	8	no	—
178	D₄	—	11	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	11	4	no	—
182	C₂×C₄	—	11	4	no	—
183	C₂×C₄	—	11	4	no	—
184	C₂×C₄	—	22	4	no	—
185	C₂×C₄	—	22	4	no	—
186	C₂×C₄	—	22	4	no	—
187	C₂×C₄	—	22	16	no	—
188	C₂×C₄	—	44	16	no	—
189	C₂×C₄	Q₈×D₁₁	1	4	yes	—
190	C₂×C₄	#4343	1	4	yes	—
191	C₂×C₄	C₂³×D₁₁	1	2	yes	—
192	C₂×C₄	C₂³×D₁₁	1	8	yes	—
193	C₂×C₄	C₂³×D₁₁	1	2	yes	—
194	C₂×C₄	C₂³×D₁₁	1	2	yes	—
195	C₂²	—	11	8	no	—
196	C₂²	—	22	16	no	—
197	C₂²	—	22	4	no	—
198	C₂²	—	22	4	no	—
199	C₂²	#67037	1	2	yes	—
200	C₂²	C₂×Q₈×D₁₁	1	2	yes	—
201	C₂²	C₂×#4343	1	2	yes	—
202	C₂²	C₂⁴×D₁₁	1	1	yes	characteristic
203	C₄	—	2	8	no	—
204	C₄	—	2	2	no	—
205	C₄	—	22	4	no	—
206	C₄	—	22	4	no	—
207	C₄	—	22	4	no	—
208	C₄	C₂×Q₈×D₁₁	1	4	yes	—
209	C₄	C₂×#4343	1	4	yes	—
210	C₂	—	22	8	no	—
211	C₂	#10623261	1	4	yes	direct product factor
212	C₂	C₂×#67037	1	1	yes	characteristic
213	C₂	C₂²×Q₈×D₁₁	1	1	yes	characteristic
214	C₂	C₂²×#4343	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁
Fitting	C₂×C₁₁×C₄⋊Q₈	C₂
socle	C₂³×C₁₁	C₂⁴
prime radical	C₂²×D₄₄	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₁
hypocenter	C₁₁	C₂×#554
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×C₄⋊Q₈	C₂	semidirect
2. C₂×#66999	C₂	semidirect
3. #10623261	C₂	direct (N × H)
4. C₂×#66950	C₂	semidirect
6. C₂×C₄⋊C₄×D₁₁	C₂	semidirect
7. C₂×#66943	C₂	semidirect
9. C₂×C₄×Dic₂₂	C₂	semidirect
15. C₁₁×C₄⋊Q₈	C₂²	semidirect
19. #66999	C₂²	semidirect
28. #66950	C₂²	semidirect
30. C₄⋊C₄×D₁₁	C₂²	semidirect
31. #66943	C₂²	semidirect
33. C₄×Dic₂₂	C₂²	semidirect
44. C₂×D₄₄	Q₈	semidirect
84. D₄₄	C₂×Q₈	semidirect
89. C₂×C₄⋊Q₈	D₁₁	semidirect
117. C₄⋊Q₈	C₂×D₁₁	semidirect
136. C₂×C₁₁	#554	semidirect
170. C₁₁	C₂×#554	semidirect
211. C₂	#10623261	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×C₄⋊Q₈	2
C₂×#66999	2
#10623261	2
C₂×#66950	2
C₂×#66946	2
C₂×C₄⋊C₄×D₁₁	2
C₂×#66943	2
C₂×C₄×D₄₄	2
C₂×C₄×Dic₂₂	2
C₂×#554	11

Subgroups — C2×#10623261 #13082151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10623261 #13082151