Subgroups — C₁₁×#2368 #13065206

Order 1408.

548 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#391	C₂	1	2	yes	maximal, complemented
3	C₁₁×#387	C₂	1	2	yes	maximal, complemented
4	C₁₁×#387	C₂	1	2	yes	maximal, complemented
5	C₁₁×#385	C₂	1	2	yes	maximal
6	C₄×C₁₁×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₁×C₂²⋊C₄	C₂	1	2	yes	maximal
8	C₄×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#95	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₁₁×C₂²⋊Q₈	C₄	1	8	yes	complemented
14	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₁×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₁×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₁×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
24	C₁₁×#95	C₂²	1	1	yes	characteristic
25	C₁₁×#95	C₂²	1	1	yes	characteristic
26	C₁₁×#95	C₂²	1	2	yes	—
27	C₁₁×#95	C₂²	1	2	yes	—
28	C₁₁×#95	C₂²	1	2	yes	—
29	C₁₁×#95	C₂²	1	2	yes	—
30	C₁₁×#95	C₂²	1	2	yes	—
31	C₁₁×#95	C₂²	1	2	yes	—
32	C₂⁴×C₁₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₁	—	4	2	no	—
34	C₂×Q₈×C₁₁	C₂×C₄	1	4	yes	complemented
35	C₂²×C₄×C₁₁	—	2	4	no	—
36	C₂²×C₄×C₁₁	—	2	4	no	—
37	C₂²×C₄×C₁₁	—	4	2	no	—
38	C₂²×C₄×C₁₁	—	4	2	no	—
39	C₂²×C₄×C₁₁	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
48	C₁₁×C₄⋊C₄	—	2	4	no	—
49	C₁₁×C₄⋊C₄	—	4	2	no	—
50	C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
51	C₁₁×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
52	C₁₁×C₂²⋊C₄	—	2	2	no	—
53	C₁₁×C₂²⋊C₄	C₂×C₄	1	8	yes	—
54	C₁₁×C₂²⋊C₄	Q₈	1	4	yes	complemented
55	C₄²×C₁₁	—	2	4	no	—
56	C₄²×C₁₁	—	4	2	no	—
57	C₄²×C₁₁	—	4	2	no	—
58	#2368	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₁	—	2	2	no	—
60	C₂³×C₁₁	—	2	2	no	—
61	C₂³×C₁₁	C₂²×C₄	1	2	yes	—
62	C₂³×C₁₁	C₂×Q₈	1	2	yes	—
63	C₂³×C₁₁	#41	1	2	yes	—
64	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
65	Q₈×C₁₁	—	4	4	no	—
66	C₂×C₄×C₁₁	—	2	2	no	—
67	C₂×C₄×C₁₁	—	2	2	no	—
68	C₂×C₄×C₁₁	—	2	2	no	—
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	2	2	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	—	2	2	no	—
73	C₂×C₄×C₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	2	no	—
75	C₂×C₄×C₁₁	—	2	2	no	—
76	C₂×C₄×C₁₁	—	2	2	no	—
77	C₂×C₄×C₁₁	—	2	2	no	—
78	C₂×C₄×C₁₁	—	2	2	no	—
79	C₂×C₄×C₁₁	—	2	2	no	—
80	C₂×C₄×C₁₁	—	2	2	no	—
81	C₂×C₄×C₁₁	—	2	2	no	—
82	C₂×C₄×C₁₁	—	4	4	no	—
83	C₂×C₄×C₁₁	—	4	8	no	—
84	C₂×C₄×C₁₁	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₁₁	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	—
89	C₂×C₄×C₁₁	#41	1	4	yes	—
90	C₂×C₂²⋊Q₈	C₂×C₁₁	1	1	yes	characteristic
91	#391	C₂×C₁₁	1	2	yes	complemented
92	#387	C₂×C₁₁	1	2	yes	complemented
93	#387	C₂×C₁₁	1	2	yes	complemented
94	#385	C₂×C₁₁	1	2	yes	—
95	C₄×C₄⋊C₄	C₂×C₁₁	1	2	yes	complemented
96	C₄×C₂²⋊C₄	C₂×C₁₁	1	2	yes	—
97	C₄×C₂²⋊C₄	C₂×C₁₁	1	1	yes	characteristic
98	C₂×#95	C₂×C₁₁	1	1	yes	characteristic
99	C₂²×C₁₁	—	2	2	no	—
100	C₂²×C₁₁	—	2	2	no	—
101	C₂²×C₁₁	—	2	4	no	—
102	C₂²×C₁₁	—	2	4	no	—
103	C₂²×C₁₁	C₄×Q₈	1	4	yes	—
104	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
107	C₂²×C₁₁	C₂×#41	1	2	yes	—
108	C₂²×C₁₁	H(5,𝔽₂)	1	2	yes	—
109	C₄×C₁₁	—	2	4	no	—
110	C₄×C₁₁	—	2	2	no	—
111	C₄×C₁₁	—	2	2	no	—
112	C₄×C₁₁	—	2	4	no	—
113	C₄×C₁₁	—	2	4	no	—
114	C₄×C₁₁	—	2	4	no	—
115	C₄×C₁₁	—	4	2	no	—
116	C₄×C₁₁	—	4	2	no	—
117	C₂²×Q₈	C₂²×C₁₁	1	1	yes	characteristic
118	C₂³×C₄	C₂²×C₁₁	1	2	yes	—
119	C₂³×C₄	C₂²×C₁₁	1	1	yes	characteristic
120	C₂²⋊Q₈	C₄×C₁₁	1	8	yes	complemented
121	C₂×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	C₂²×C₁₁	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₁₁	1	2	yes	—
129	C₂×C₄²	C₂²×C₁₁	1	2	yes	—
130	C₂×C₄²	C₂²×C₁₁	1	2	yes	—
131	#95	C₂²×C₁₁	1	1	yes	characteristic
132	#95	C₂²×C₁₁	1	1	yes	characteristic
133	#95	C₂²×C₁₁	1	2	yes	—
134	#95	C₂²×C₁₁	1	2	yes	—
135	#95	C₂²×C₁₁	1	2	yes	—
136	#95	C₂²×C₁₁	1	2	yes	—
137	#95	C₂²×C₁₁	1	2	yes	—
138	#95	C₂²×C₁₁	1	2	yes	—
139	C₂×C₁₁	—	2	4	no	—
140	C₂×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂×C₁₁	C₄×#41	1	1	yes	characteristic
142	C₂×C₁₁	#518	1	1	yes	characteristic
143	C₂×C₁₁	#551	1	2	yes	—
144	C₂×C₁₁	#554	1	2	yes	—
145	C₂⁴	C₂³×C₁₁	1	1	yes	characteristic
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	C₂×C₄×C₁₁	1	4	yes	complemented
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×C₄×C₁₁	1	4	yes	—
153	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
154	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
155	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
158	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
160	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	C₂×C₄×C₁₁	1	4	yes	complemented
164	C₄⋊C₄	C₂×C₄×C₁₁	1	8	yes	complemented
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	C₂×C₄×C₁₁	1	8	yes	—
167	C₂²⋊C₄	Q₈×C₁₁	1	4	yes	complemented
168	C₄²	—	2	4	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₁₁	#2368	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	C₂²×C₄×C₁₁	1	2	yes	—
175	C₂³	C₂×Q₈×C₁₁	1	2	yes	—
176	C₂³	C₁₁×#41	1	2	yes	—
177	C₂³	C₂⁴×C₁₁	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	C₂²×C₄×C₁₁	1	4	yes	—
198	C₂×C₄	C₂²×C₄×C₁₁	1	4	yes	—
199	C₂×C₄	C₂²×C₄×C₁₁	1	2	yes	—
200	C₂×C₄	C₂²×C₄×C₁₁	1	2	yes	—
201	C₂×C₄	C₂×Q₈×C₁₁	1	4	yes	—
202	C₂×C₄	C₁₁×#41	1	4	yes	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	C₄×Q₈×C₁₁	1	4	yes	—
208	C₂²	C₂³×C₄×C₁₁	1	1	yes	characteristic
209	C₂²	C₂²×Q₈×C₁₁	1	1	yes	characteristic
210	C₂²	C₂×C₁₁×#41	1	1	yes	characteristic
211	C₂²	C₂×C₁₁×#41	1	2	yes	—
212	C₂²	C₁₁×H(5,𝔽₂)	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	4	no	—
218	C₄	—	2	4	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₂	—	2	4	no	—
222	C₂	C₂×C₄×Q₈×C₁₁	1	1	yes	characteristic
223	C₂	C₄×C₁₁×#41	1	1	yes	characteristic
224	C₂	C₁₁×#518	1	1	yes	characteristic
225	C₂	C₁₁×#551	1	2	yes	—
226	C₂	C₁₁×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₁
Frattini, cosocle	C₂³	C₂⁴×C₁₁
prime radical	C₂⁴×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. C₁₁×#391	C₂	semidirect
3. C₁₁×#387	C₂	semidirect
4. C₁₁×#387	C₂	semidirect
6. C₄×C₁₁×C₄⋊C₄	C₂	semidirect
13. C₁₁×C₂²⋊Q₈	C₄	semidirect
34. C₂×Q₈×C₁₁	C₂×C₄	semidirect
50. C₁₁×C₄⋊C₄	C₂×C₄	semidirect
51. C₁₁×C₄⋊C₄	C₂×C₄	semidirect
54. C₁₁×C₂²⋊C₄	Q₈	semidirect
58. #2368	C₁₁	direct (N × H)
91. #391	C₂×C₁₁	semidirect
92. #387	C₂×C₁₁	semidirect
93. #387	C₂×C₁₁	semidirect
95. C₄×C₄⋊C₄	C₂×C₁₁	semidirect
120. C₂²⋊Q₈	C₄×C₁₁	semidirect
147. C₂×Q₈	C₂×C₄×C₁₁	semidirect
163. C₄⋊C₄	C₂×C₄×C₁₁	semidirect
164. C₄⋊C₄	C₂×C₄×C₁₁	semidirect
167. C₂²⋊C₄	Q₈×C₁₁	semidirect
171. C₁₁	#2368	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×C₂²⋊Q₈	2
C₁₁×#391	2
C₁₁×#387	2
C₁₁×#385	2
C₄×C₁₁×C₄⋊C₄	2
C₄×C₁₁×C₂²⋊C₄	2
C₂×C₁₁×#95	2
#2368	11

Subgroups — C11×#2368 #13065206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#2368 #13065206