Subgroups — C₁₁×#3486 #13066324

Order 1408.

1492 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₈×C₁₁	C₂	1	8	yes	maximal, complemented
4	C₁₁×#548	C₂	1	8	yes	maximal, complemented
5	C₁₁×#536	C₂	1	6	yes	maximal, complemented
6	C₁₁×#525	C₂	1	6	yes	maximal, complemented
7	C₁₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₁×2₋¹⁺⁴	—	2	8	no	—
9	C₂×C₁₁×#41	C₂²	1	6	yes	—
10	C₂×C₁₁×#41	C₂²	1	4	yes	complemented
11	C₂²×Q₈×C₁₁	—	2	8	no	—
12	C₂²×Q₈×C₁₁	C₂²	1	3	yes	—
13	C₂²×Q₈×C₁₁	C₂²	1	8	yes	—
14	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₁	C₂²	1	4	yes	complemented
16	C₂³×C₄×C₁₁	C₂²	1	3	yes	—
17	C₁₁×C₄⋊Q₈	C₂²	1	12	yes	complemented
18	C₁₁×#124	C₂²	1	12	yes	complemented
19	C₁₁×#123	C₂²	1	12	yes	—
20	C₁₁×C₂²⋊Q₈	C₂²	1	12	yes	—
21	C₁₁×C₂²⋊Q₈	C₂²	1	24	yes	complemented
22	C₁₁×#121	C₂²	1	12	yes	—
23	C₁₁×#120	C₂²	1	4	yes	—
24	C₄×Q₈×C₁₁	C₂²	1	8	yes	complemented
25	C₄×D₄×C₁₁	C₂²	1	24	yes	—
26	C₁₁×#117	C₂²	1	6	yes	—
27	C₂⁴×C₁₁	C₂³	1	1	yes	characteristic
28	C₁₁×#41	—	2	16	no	—
29	C₁₁×#41	—	2	24	no	—
30	C₂×Q₈×C₁₁	—	2	32	no	—
31	C₂×Q₈×C₁₁	—	2	16	no	—
32	C₂×Q₈×C₁₁	—	2	12	no	—
33	C₂×Q₈×C₁₁	D₄	1	8	yes	complemented
34	C₂×Q₈×C₁₁	C₂³	1	6	yes	—
35	C₂×Q₈×C₁₁	C₂³	1	12	yes	—
36	C₂×Q₈×C₁₁	C₂³	1	4	yes	—
37	C₂×D₄×C₁₁	C₂³	1	8	yes	—
38	C₂×D₄×C₁₁	C₂³	1	4	yes	—
39	C₂×D₄×C₁₁	C₂³	1	6	yes	—
40	C₂²×C₄×C₁₁	—	2	12	no	—
41	C₂²×C₄×C₁₁	C₂³	1	12	yes	—
42	C₂²×C₄×C₁₁	C₂³	1	12	yes	—
43	C₂²×C₄×C₁₁	C₂³	1	3	yes	—
44	C₂²×C₄×C₁₁	C₂³	1	3	yes	—
45	C₁₁×C₄⋊C₄	C₂³	1	12	yes	—
46	C₁₁×C₄⋊C₄	C₂³	1	24	yes	—
47	C₁₁×C₂²⋊C₄	C₂³	1	4	yes	—
48	C₁₁×C₂²⋊C₄	C₂³	1	8	yes	—
49	C₁₁×C₂²⋊C₄	C₂³	1	24	yes	—
50	C₄²×C₁₁	C₂³	1	12	yes	—
51	#3486	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₁	—	2	2	no	—
53	C₂³×C₁₁	—	2	4	no	—
54	C₂³×C₁₁	C₂⁴	1	4	yes	—
55	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
56	C₂³×C₁₁	C₂⁴	1	2	yes	—
57	Q₈×C₁₁	—	2	24	no	—
58	Q₈×C₁₁	—	2	12	no	—
59	Q₈×C₁₁	C₂×D₄	1	16	yes	—
60	D₄×C₁₁	—	2	8	no	—
61	D₄×C₁₁	—	2	16	no	—
62	D₄×C₁₁	—	2	12	no	—
63	C₂×C₄×C₁₁	—	2	6	no	—
64	C₂×C₄×C₁₁	—	2	24	no	—
65	C₂×C₄×C₁₁	—	2	24	no	—
66	C₂×C₄×C₁₁	C₂×D₄	1	12	yes	—
67	C₂×C₄×C₁₁	C₂⁴	1	8	yes	—
68	C₂×C₄×C₁₁	C₂⁴	1	6	yes	—
69	C₂×C₄×C₁₁	C₂⁴	1	4	yes	—
70	C₂×C₄×C₁₁	C₂⁴	1	6	yes	—
71	C₂×2₋¹⁺⁴	C₂×C₁₁	1	1	yes	characteristic, complemented
72	C₂³×Q₈	C₂×C₁₁	1	1	yes	characteristic, complemented
73	D₄×Q₈	C₂×C₁₁	1	8	yes	complemented
74	#548	C₂×C₁₁	1	8	yes	complemented
75	#536	C₂×C₁₁	1	6	yes	complemented
76	#525	C₂×C₁₁	1	6	yes	complemented
77	#519	C₂×C₁₁	1	1	yes	characteristic, complemented
78	C₂²×C₁₁	—	2	8	no	—
79	C₂²×C₁₁	—	2	2	no	—
80	C₂²×C₁₁	—	2	2	no	—
81	C₂²×C₁₁	—	2	2	no	—
82	C₂²×C₁₁	—	2	4	no	—
83	C₂²×C₁₁	—	2	8	no	—
84	C₂²×C₁₁	C₂²×D₄	1	2	yes	—
85	C₂²×C₁₁	2₋¹⁺⁴	1	4	yes	complemented
86	C₂²×C₁₁	C₂⁵	1	1	yes	characteristic
87	C₄×C₁₁	—	2	4	no	—
88	C₄×C₁₁	—	2	6	no	—
89	C₄×C₁₁	—	2	8	no	—
90	C₄×C₁₁	C₂²×D₄	1	12	yes	—
91	2₋¹⁺⁴	—	2	8	no	—
92	C₂×#41	C₂²×C₁₁	1	6	yes	—
93	C₂×#41	C₂²×C₁₁	1	4	yes	complemented
94	C₂²×Q₈	—	2	8	no	—
95	C₂²×Q₈	C₂²×C₁₁	1	3	yes	—
96	C₂²×Q₈	C₂²×C₁₁	1	8	yes	—
97	C₂²×Q₈	C₂²×C₁₁	1	1	yes	characteristic
98	C₂²×Q₈	C₂²×C₁₁	1	4	yes	complemented
99	C₂³×C₄	C₂²×C₁₁	1	3	yes	—
100	C₄⋊Q₈	C₂²×C₁₁	1	12	yes	complemented
101	#124	C₂²×C₁₁	1	12	yes	complemented
102	#123	C₂²×C₁₁	1	12	yes	—
103	C₂²⋊Q₈	C₂²×C₁₁	1	12	yes	—
104	C₂²⋊Q₈	C₂²×C₁₁	1	24	yes	complemented
105	#121	C₂²×C₁₁	1	12	yes	—
106	#120	C₂²×C₁₁	1	4	yes	—
107	C₄×Q₈	C₂²×C₁₁	1	8	yes	complemented
108	C₄×D₄	C₂²×C₁₁	1	24	yes	—
109	#117	C₂²×C₁₁	1	6	yes	—
110	C₂×C₁₁	—	2	2	no	—
111	C₂×C₁₁	—	2	4	no	—
112	C₂×C₁₁	—	4	4	no	—
113	C₂×C₁₁	C₂³×D₄	1	1	yes	characteristic
114	C₂×C₁₁	C₂×2₋¹⁺⁴	1	2	yes	—
115	C₂⁴	C₂³×C₁₁	1	1	yes	characteristic
116	#41	—	2	16	no	—
117	#41	—	2	24	no	—
118	C₂×Q₈	—	2	32	no	—
119	C₂×Q₈	—	2	16	no	—
120	C₂×Q₈	—	2	12	no	—
121	C₂×Q₈	D₄×C₁₁	1	8	yes	complemented
122	C₂×Q₈	C₂³×C₁₁	1	6	yes	—
123	C₂×Q₈	C₂³×C₁₁	1	12	yes	—
124	C₂×Q₈	C₂³×C₁₁	1	4	yes	—
125	C₂×D₄	C₂³×C₁₁	1	8	yes	—
126	C₂×D₄	C₂³×C₁₁	1	4	yes	—
127	C₂×D₄	C₂³×C₁₁	1	6	yes	—
128	C₂²×C₄	—	2	12	no	—
129	C₂²×C₄	C₂³×C₁₁	1	12	yes	—
130	C₂²×C₄	C₂³×C₁₁	1	12	yes	—
131	C₂²×C₄	C₂³×C₁₁	1	3	yes	—
132	C₂²×C₄	C₂³×C₁₁	1	3	yes	—
133	C₄⋊C₄	C₂³×C₁₁	1	12	yes	—
134	C₄⋊C₄	C₂³×C₁₁	1	24	yes	—
135	C₂²⋊C₄	C₂³×C₁₁	1	4	yes	—
136	C₂²⋊C₄	C₂³×C₁₁	1	8	yes	—
137	C₂²⋊C₄	C₂³×C₁₁	1	24	yes	—
138	C₄²	C₂³×C₁₁	1	12	yes	—
139	C₁₁	#3486	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	C₂⁴×C₁₁	1	4	yes	—
143	C₂³	C₂⁴×C₁₁	1	1	yes	characteristic
144	C₂³	C₂⁴×C₁₁	1	2	yes	—
145	Q₈	—	2	24	no	—
146	Q₈	—	2	12	no	—
147	Q₈	C₂×D₄×C₁₁	1	16	yes	—
148	D₄	—	2	8	no	—
149	D₄	—	2	16	no	—
150	D₄	—	2	12	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	C₂×D₄×C₁₁	1	12	yes	—
155	C₂×C₄	C₂⁴×C₁₁	1	8	yes	—
156	C₂×C₄	C₂⁴×C₁₁	1	6	yes	—
157	C₂×C₄	C₂⁴×C₁₁	1	4	yes	—
158	C₂×C₄	C₂⁴×C₁₁	1	6	yes	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₂²×D₄×C₁₁	1	2	yes	—
166	C₂²	C₁₁×2₋¹⁺⁴	1	4	yes	complemented
167	C₂²	C₂⁵×C₁₁	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	—	2	8	no	—
171	C₄	C₂²×D₄×C₁₁	1	12	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂³×D₄×C₁₁	1	1	yes	characteristic
176	C₂	C₂×C₁₁×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×2₋¹⁺⁴	C₂	semidirect
2. C₂³×Q₈×C₁₁	C₂	semidirect
3. D₄×Q₈×C₁₁	C₂	semidirect
4. C₁₁×#548	C₂	semidirect
5. C₁₁×#536	C₂	semidirect
6. C₁₁×#525	C₂	semidirect
7. C₁₁×#519	C₂	semidirect
10. C₂×C₁₁×#41	C₂²	semidirect
15. C₂²×Q₈×C₁₁	C₂²	semidirect
17. C₁₁×C₄⋊Q₈	C₂²	semidirect
18. C₁₁×#124	C₂²	semidirect
21. C₁₁×C₂²⋊Q₈	C₂²	semidirect
24. C₄×Q₈×C₁₁	C₂²	semidirect
33. C₂×Q₈×C₁₁	D₄	semidirect
51. #3486	C₁₁	direct (N × H)
71. C₂×2₋¹⁺⁴	C₂×C₁₁	semidirect
72. C₂³×Q₈	C₂×C₁₁	semidirect
73. D₄×Q₈	C₂×C₁₁	semidirect
74. #548	C₂×C₁₁	semidirect
75. #536	C₂×C₁₁	semidirect
76. #525	C₂×C₁₁	semidirect
77. #519	C₂×C₁₁	semidirect
85. C₂²×C₁₁	2₋¹⁺⁴	semidirect
93. C₂×#41	C₂²×C₁₁	semidirect
98. C₂²×Q₈	C₂²×C₁₁	semidirect
100. C₄⋊Q₈	C₂²×C₁₁	semidirect
101. #124	C₂²×C₁₁	semidirect
104. C₂²⋊Q₈	C₂²×C₁₁	semidirect
107. C₄×Q₈	C₂²×C₁₁	semidirect
121. C₂×Q₈	D₄×C₁₁	semidirect
139. C₁₁	#3486	direct (N × H)
166. C₂²	C₁₁×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×2₋¹⁺⁴	2
C₂³×Q₈×C₁₁	2
D₄×Q₈×C₁₁	2
C₁₁×#548	2
C₁₁×#536	2
C₁₁×#525	2
C₁₁×#519	2
#3486	11

Subgroups — C11×#3486 #13066324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#3486 #13066324