Subgroups — C₁₁×#34173 #14611366

Order 2816.

5126 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#519	C₂	1	2	yes	maximal, complemented
3	Q₈×C₁₁×C₂²⋊C₄	C₂	1	16	yes	maximal, complemented
4	C₁₁×#2310	C₂	1	6	yes	maximal, complemented
5	C₁₁×#2280	C₂	1	6	yes	maximal
6	C₂³×Q₈×C₁₁	—	2	16	no	—
7	C₂³×Q₈×C₁₁	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₁₁	C₂²	1	3	yes	—
9	C₂³×Q₈×C₁₁	C₂²	1	8	yes	—
10	C₂³×Q₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₁	C₂²	1	3	yes	—
12	C₁₁×#519	—	2	16	no	—
13	C₂×C₄×Q₈×C₁₁	C₂²	1	16	yes	complemented
14	C₂×C₁₁×#117	C₂²	1	12	yes	—
15	C₁₁×#391	C₂²	1	48	yes	complemented
16	C₁₁×#381	C₂²	1	12	yes	—
17	C₁₁×#380	C₂²	1	24	yes	—
18	C₁₁×#379	C₂²	1	4	yes	—
19	C₄×C₁₁×C₂²⋊C₄	C₂²	1	24	yes	—
20	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₁₁	—	2	128	no	—
22	C₂²×Q₈×C₁₁	—	2	64	no	—
23	C₂²×Q₈×C₁₁	—	2	24	no	—
24	C₂²×Q₈×C₁₁	C₂×C₄	1	64	yes	—
25	C₂²×Q₈×C₁₁	C₂×C₄	1	24	yes	—
26	C₂²×Q₈×C₁₁	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₁₁	D₄	1	16	yes	—
28	C₂²×Q₈×C₁₁	C₂³	1	12	yes	—
29	C₂²×Q₈×C₁₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₁	—	2	24	no	—
31	C₂³×C₄×C₁₁	C₂×C₄	1	24	yes	—
32	C₂³×C₄×C₁₁	C₂³	1	12	yes	—
33	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
34	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
35	C₄×Q₈×C₁₁	—	2	64	no	—
36	C₁₁×#117	—	2	48	no	—
37	C₂×C₁₁×C₄⋊C₄	C₂³	1	24	yes	—
38	C₂×C₁₁×C₂²⋊C₄	C₂³	1	8	yes	—
39	C₂×C₁₁×C₂²⋊C₄	C₂³	1	16	yes	—
40	C₂×C₄²×C₁₁	C₂³	1	24	yes	—
41	C₁₁×#95	C₂³	1	48	yes	—
42	#34173	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂⁴×C₁₁	—	2	4	no	—
44	C₂⁴×C₁₁	—	2	8	no	—
45	C₂⁴×C₁₁	C₂²×C₄	1	4	yes	—
46	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
47	C₂⁴×C₁₁	C₂⁴	1	2	yes	—
48	C₂×Q₈×C₁₁	—	2	256	no	—
49	C₂×Q₈×C₁₁	—	2	96	no	—
50	C₂×Q₈×C₁₁	—	2	48	no	—
51	C₂×Q₈×C₁₁	C₂²⋊C₄	1	64	yes	complemented
52	C₂×Q₈×C₁₁	C₂²×C₄	1	48	yes	—
53	C₂×Q₈×C₁₁	C₂²×C₄	1	16	yes	—
54	C₂×Q₈×C₁₁	C₂×D₄	1	32	yes	—
55	C₂²×C₄×C₁₁	—	2	12	no	—
56	C₂²×C₄×C₁₁	—	2	96	no	—
57	C₂²×C₄×C₁₁	—	2	48	no	—
58	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₁₁	C₂²×C₄	1	48	yes	—
60	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₁₁	C₂×D₄	1	24	yes	—
62	C₂²×C₄×C₁₁	C₂⁴	1	16	yes	—
63	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
65	C₁₁×C₄⋊C₄	—	2	48	no	—
66	C₁₁×C₂²⋊C₄	—	2	16	no	—
67	C₁₁×C₂²⋊C₄	—	2	32	no	—
68	C₄²×C₁₁	—	2	48	no	—
69	C₂⁴×Q₈	C₂×C₁₁	1	1	yes	characteristic
70	C₂×#519	C₂×C₁₁	1	2	yes	complemented
71	Q₈×C₂²⋊C₄	C₂×C₁₁	1	16	yes	complemented
72	#2310	C₂×C₁₁	1	6	yes	complemented
73	#2280	C₂×C₁₁	1	6	yes	—
74	C₂³×C₁₁	—	2	32	no	—
75	C₂³×C₁₁	—	2	8	no	—
76	C₂³×C₁₁	—	2	16	no	—
77	C₂³×C₁₁	—	2	4	no	—
78	C₂³×C₁₁	—	2	4	no	—
79	C₂³×C₁₁	—	2	4	no	—
80	C₂³×C₁₁	C₂³×C₄	1	2	yes	—
81	C₂³×C₁₁	C₂³×C₄	1	4	yes	—
82	C₂³×C₁₁	C₂²×D₄	1	4	yes	—
83	C₂³×C₁₁	2₋¹⁺⁴	1	8	yes	—
84	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
85	Q₈×C₁₁	—	2	96	no	—
86	Q₈×C₁₁	C₂×C₂²⋊C₄	1	64	yes	—
87	C₂×C₄×C₁₁	—	2	24	no	—
88	C₂×C₄×C₁₁	—	2	12	no	—
89	C₂×C₄×C₁₁	—	2	96	no	—
90	C₂×C₄×C₁₁	—	2	16	no	—
91	C₂×C₄×C₁₁	—	2	32	no	—
92	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	48	yes	—
93	C₂×C₄×C₁₁	C₂³×C₄	1	12	yes	—
94	C₂×C₄×C₁₁	C₂³×C₄	1	12	yes	—
95	C₂×C₄×C₁₁	C₂²×D₄	1	24	yes	—
96	C₂³×Q₈	—	2	16	no	—
97	C₂³×Q₈	C₄×C₁₁	1	16	yes	complemented
98	C₂³×Q₈	C₂²×C₁₁	1	3	yes	—
99	C₂³×Q₈	C₂²×C₁₁	1	8	yes	—
100	C₂³×Q₈	C₂²×C₁₁	1	1	yes	characteristic
101	C₂⁴×C₄	C₂²×C₁₁	1	3	yes	—
102	#519	—	2	16	no	—
103	C₂×C₄×Q₈	C₂²×C₁₁	1	16	yes	complemented
104	C₂×#117	C₂²×C₁₁	1	12	yes	—
105	#391	C₂²×C₁₁	1	48	yes	complemented
106	#381	C₂²×C₁₁	1	12	yes	—
107	#380	C₂²×C₁₁	1	24	yes	—
108	#379	C₂²×C₁₁	1	4	yes	—
109	C₄×C₂²⋊C₄	C₂²×C₁₁	1	24	yes	—
110	C₂²×C₁₁	—	2	32	no	—
111	C₂²×C₁₁	—	2	4	no	—
112	C₂²×C₁₁	—	2	4	no	—
113	C₂²×C₁₁	—	2	4	no	—
114	C₂²×C₁₁	—	2	8	no	—
115	C₂²×C₁₁	—	2	16	no	—
116	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	4	yes	—
117	C₂²×C₁₁	#519	1	8	yes	complemented
118	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
119	C₂²×C₁₁	C₂³×D₄	1	2	yes	—
120	C₂²×C₁₁	C₂×2₋¹⁺⁴	1	4	yes	—
121	C₄×C₁₁	—	2	12	no	—
122	C₄×C₁₁	—	4	16	no	—
123	C₄×C₁₁	C₂²×C₂²⋊C₄	1	24	yes	—
124	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
125	C₂²×Q₈	—	2	128	no	—
126	C₂²×Q₈	—	2	64	no	—
127	C₂²×Q₈	—	2	24	no	—
128	C₂²×Q₈	C₂×C₄×C₁₁	1	64	yes	—
129	C₂²×Q₈	C₂×C₄×C₁₁	1	24	yes	—
130	C₂²×Q₈	C₂×C₄×C₁₁	1	8	yes	—
131	C₂²×Q₈	D₄×C₁₁	1	16	yes	—
132	C₂²×Q₈	C₂³×C₁₁	1	12	yes	—
133	C₂²×Q₈	C₂³×C₁₁	1	4	yes	—
134	C₂³×C₄	—	2	24	no	—
135	C₂³×C₄	C₂×C₄×C₁₁	1	24	yes	—
136	C₂³×C₄	C₂³×C₁₁	1	12	yes	—
137	C₂³×C₄	C₂³×C₁₁	1	3	yes	—
138	C₂³×C₄	C₂³×C₁₁	1	3	yes	—
139	C₄×Q₈	—	2	64	no	—
140	#117	—	2	48	no	—
141	C₂×C₄⋊C₄	C₂³×C₁₁	1	24	yes	—
142	C₂×C₂²⋊C₄	C₂³×C₁₁	1	8	yes	—
143	C₂×C₂²⋊C₄	C₂³×C₁₁	1	16	yes	—
144	C₂×C₄²	C₂³×C₁₁	1	24	yes	—
145	#95	C₂³×C₁₁	1	48	yes	—
146	C₂×C₁₁	—	2	4	no	—
147	C₂×C₁₁	—	2	8	no	—
148	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₁₁	C₂×#519	1	2	yes	—
150	C₂×C₁₁	#3486	1	4	yes	—
151	C₂⁴	—	2	4	no	—
152	C₂⁴	—	2	8	no	—
153	C₂⁴	C₂²×C₄×C₁₁	1	4	yes	—
154	C₂⁴	C₂⁴×C₁₁	1	1	yes	characteristic
155	C₂⁴	C₂⁴×C₁₁	1	2	yes	—
156	C₂×Q₈	—	2	256	no	—
157	C₂×Q₈	—	2	96	no	—
158	C₂×Q₈	—	2	48	no	—
159	C₂×Q₈	C₁₁×C₂²⋊C₄	1	64	yes	complemented
160	C₂×Q₈	C₂²×C₄×C₁₁	1	48	yes	—
161	C₂×Q₈	C₂²×C₄×C₁₁	1	16	yes	—
162	C₂×Q₈	C₂×D₄×C₁₁	1	32	yes	—
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	2	96	no	—
165	C₂²×C₄	—	2	48	no	—
166	C₂²×C₄	C₂²×C₄×C₁₁	1	12	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₁	1	48	yes	—
168	C₂²×C₄	C₂²×C₄×C₁₁	1	12	yes	—
169	C₂²×C₄	C₂×D₄×C₁₁	1	24	yes	—
170	C₂²×C₄	C₂⁴×C₁₁	1	16	yes	—
171	C₂²×C₄	C₂⁴×C₁₁	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₁₁	1	6	yes	—
173	C₄⋊C₄	—	2	48	no	—
174	C₂²⋊C₄	—	2	16	no	—
175	C₂²⋊C₄	—	2	32	no	—
176	C₄²	—	2	48	no	—
177	C₁₁	#34173	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³	—	2	32	no	—
179	C₂³	—	2	8	no	—
180	C₂³	—	2	16	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	4	no	—
184	C₂³	C₂³×C₄×C₁₁	1	2	yes	—
185	C₂³	C₂³×C₄×C₁₁	1	4	yes	—
186	C₂³	C₂²×D₄×C₁₁	1	4	yes	—
187	C₂³	C₁₁×2₋¹⁺⁴	1	8	yes	—
188	C₂³	C₂⁵×C₁₁	1	1	yes	characteristic
189	Q₈	—	2	96	no	—
190	Q₈	C₂×C₁₁×C₂²⋊C₄	1	64	yes	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	96	no	—
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	32	no	—
196	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	48	yes	—
197	C₂×C₄	C₂³×C₄×C₁₁	1	12	yes	—
198	C₂×C₄	C₂³×C₄×C₁₁	1	12	yes	—
199	C₂×C₄	C₂²×D₄×C₁₁	1	24	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	8	no	—
205	C₂²	—	2	16	no	—
206	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	4	yes	—
207	C₂²	C₁₁×#519	1	8	yes	complemented
208	C₂²	C₂⁴×C₄×C₁₁	1	1	yes	characteristic
209	C₂²	C₂³×D₄×C₁₁	1	2	yes	—
210	C₂²	C₂×C₁₁×2₋¹⁺⁴	1	4	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	4	16	no	—
213	C₄	C₂²×C₁₁×C₂²⋊C₄	1	24	yes	—
214	C₂	—	2	4	no	—
215	C₂	—	2	8	no	—
216	C₂	C₂³×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
217	C₂	C₂×C₁₁×#519	1	2	yes	—
218	C₂	C₁₁×#3486	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₁
Frattini, cosocle	C₂³	C₂⁵×C₁₁
prime radical	C₂⁵×C₁₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₁×#519	C₂	semidirect
3. Q₈×C₁₁×C₂²⋊C₄	C₂	semidirect
4. C₁₁×#2310	C₂	semidirect
7. C₂³×Q₈×C₁₁	C₄	semidirect
13. C₂×C₄×Q₈×C₁₁	C₂²	semidirect
15. C₁₁×#391	C₂²	semidirect
42. #34173	C₁₁	direct (N × H)
51. C₂×Q₈×C₁₁	C₂²⋊C₄	semidirect
70. C₂×#519	C₂×C₁₁	semidirect
71. Q₈×C₂²⋊C₄	C₂×C₁₁	semidirect
72. #2310	C₂×C₁₁	semidirect
97. C₂³×Q₈	C₄×C₁₁	semidirect
103. C₂×C₄×Q₈	C₂²×C₁₁	semidirect
105. #391	C₂²×C₁₁	semidirect
117. C₂²×C₁₁	#519	semidirect
159. C₂×Q₈	C₁₁×C₂²⋊C₄	semidirect
177. C₁₁	#34173	direct (N × H)
207. C₂²	C₁₁×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₁	2
C₂×C₁₁×#519	2
Q₈×C₁₁×C₂²⋊C₄	2
C₁₁×#2310	2
C₁₁×#2280	2
#34173	11

Subgroups — C11×#34173 #14611366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#34173 #14611366