Subgroups — #13077533

Order 1408.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#135	C₂	1	1	yes	maximal, characteristic
2	#10623350	C₂	1	2	yes	maximal, complemented
3	#10622849	C₂	1	2	yes	maximal, complemented
4	#10622810	C₂	1	4	yes	maximal, complemented
5	#10622809	C₂	1	4	yes	maximal, complemented
6	C₂×#66883	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622714	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#41	C₂²	1	2	yes	—
9	C₁₁×#135	C₄	1	2	yes	complemented
10	C₁₁×#135	C₄	1	2	yes	complemented
11	C₁₁×#135	C₂²	1	2	yes	—
12	C₁₁×#135	C₂²	1	2	yes	—
13	C₂×C₁₁×Q₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂×C₁₁×SD₁₆	C₄	1	2	yes	complemented
15	C₂×C₁₁×D₈	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
17	#67001	—	2	4	no	—
18	#67001	C₂²	1	8	yes	complemented
19	#66979	C₂²	1	1	yes	characteristic
20	C₂×#4318	—	2	4	no	—
21	C₂×#4318	C₂²	1	2	yes	—
22	C₂×#4318	C₂²	1	1	yes	characteristic
23	#66901	C₂²	1	8	yes	complemented
24	#66883	C₂²	1	2	yes	complemented
25	#66883	C₂²	1	2	yes	complemented
26	#66879	C₂²	1	2	yes	complemented
27	C₈×Dic₁₁	C₂²	1	2	yes	complemented
28	C₁₁×#41	—	2	2	no	—
29	C₁₁×#41	—	2	2	no	—
30	C₁₁×#41	C₂×C₄	1	4	yes	—
31	C₁₁×#41	C₂³	1	4	yes	—
32	C₂×Q₈×C₁₁	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₁₁	—	2	2	no	—
34	C₂×D₄×C₁₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₁	—	2	2	no	—
36	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
37	C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
38	C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
39	C₁₁×SD₁₆	C₂×C₄	1	4	yes	—
40	C₁₁×SD₁₆	C₂×C₄	1	4	yes	—
41	C₁₁×D₈	C₂×C₄	1	2	yes	—
42	C₁₁×D₈	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₁₁	—	2	1	no	—
50	C₄⋊Dic₁₁	—	2	1	no	—
51	C₄×Dic₁₁	C₂³	1	2	yes	—
52	#4318	—	2	8	no	—
53	#4318	—	2	2	no	—
54	#4318	D₄	1	2	yes	—
55	#4318	D₄	1	2	yes	—
56	#4318	C₂³	1	4	yes	—
57	C₂×C₁₁⋊C₈	—	2	8	no	—
58	C₂×C₁₁⋊C₈	—	2	2	no	—
59	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
60	#2957	—	11	1	no	maximal, Sylow
61	C₂³×C₁₁	—	2	2	no	—
62	C₂³×C₁₁	#41	1	1	yes	characteristic
63	Q₈×C₁₁	—	2	2	no	—
64	Q₈×C₁₁	C₂²×C₄	1	4	yes	—
65	D₄×C₁₁	—	2	4	no	—
66	D₄×C₁₁	—	2	4	no	—
67	D₄×C₁₁	—	2	2	no	—
68	D₄×C₁₁	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	2	4	no	—
71	C₂×C₄×C₁₁	—	2	4	no	—
72	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₁	—	2	2	no	—
79	C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₁₁⋊C₈	—	2	4	no	—
84	C₁₁⋊C₈	C₂×D₄	1	4	yes	—
85	C₂×#135	D₁₁	1	1	yes	characteristic
86	#568	—	11	2	no	—
87	#444	—	11	4	no	—
88	#443	—	11	4	no	—
89	C₂×#108	—	11	1	no	—
90	#421	—	11	2	no	—
91	#405	—	11	1	no	—
92	C₂²×C₁₁	—	2	4	no	—
93	C₂²×C₁₁	—	4	1	no	—
94	C₂²×C₁₁	—	8	2	no	—
95	C₂²×C₁₁	C₄×D₄	1	1	yes	characteristic
96	C₂²×C₁₁	C₄×D₄	1	1	yes	characteristic
97	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
98	C₄×C₁₁	—	2	4	no	—
99	C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
100	C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
101	C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
103	Dic₁₁	—	4	2	no	—
104	C₂×#41	C₂×D₁₁	1	2	yes	—
105	#135	Dic₁₁	1	2	yes	complemented
106	#135	Dic₁₁	1	2	yes	complemented
107	#135	C₂×D₁₁	1	2	yes	—
108	#135	C₂×D₁₁	1	2	yes	—
109	C₂×Q₁₆	Dic₁₁	1	1	yes	characteristic, complemented
110	C₂×SD₁₆	Dic₁₁	1	2	yes	complemented
111	C₂×D₈	Dic₁₁	1	1	yes	characteristic, complemented
112	#131	—	11	8	no	—
113	#131	—	22	4	no	—
114	C₂×M₁₆	—	11	1	no	—
115	C₂×M₁₆	—	11	2	no	—
116	C₂×M₁₆	—	22	4	no	—
117	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
118	#117	—	11	1	no	—
119	#108	—	11	2	no	—
120	#108	—	11	2	no	—
121	#104	—	11	8	no	—
122	#97	—	11	2	no	—
123	C₄×C₈	—	11	2	no	—
124	C₂×C₁₁	—	2	1	no	—
125	C₂×C₁₁	—	2	1	no	—
126	C₂×C₁₁	—	2	1	no	—
127	C₂×C₁₁	—	4	4	no	—
128	C₂×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
129	#41	—	2	2	no	—
130	#41	—	2	2	no	—
131	#41	C₂×Dic₁₁	1	4	yes	—
132	#41	C₂²×D₁₁	1	4	yes	—
133	C₂×Q₈	C₂×Dic₁₁	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂×Dic₁₁	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
138	Q₁₆	C₂×Dic₁₁	1	2	yes	—
139	Q₁₆	C₂×Dic₁₁	1	2	yes	—
140	SD₁₆	C₂×Dic₁₁	1	4	yes	—
141	SD₁₆	C₂×Dic₁₁	1	4	yes	—
142	D₈	C₂×Dic₁₁	1	2	yes	—
143	D₈	C₂×Dic₁₁	1	2	yes	—
144	M₁₆	—	11	2	no	—
145	M₁₆	—	11	2	no	—
146	M₁₆	—	11	4	no	—
147	M₁₆	—	22	8	no	—
148	M₁₆	—	22	2	no	—
149	C₂×C₈	—	11	2	no	—
150	C₂×C₈	—	22	8	no	—
151	C₂×C₈	—	22	2	no	—
152	C₂×C₈	C₂×Dic₁₁	1	1	yes	characteristic
153	C₂×C₈	C₂×Dic₁₁	1	1	yes	characteristic
154	C₂×C₈	C₂×Dic₁₁	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₁₁	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
158	C₄⋊C₄	—	22	1	no	—
159	C₂²⋊C₄	—	22	1	no	—
160	C₄²	—	11	2	no	—
161	C₁₁	#2957	1	1	yes	characteristic, complemented, Sylow
162	C₂³	—	2	2	no	—
163	C₂³	#4341	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	Q₈	C₂²×Dic₁₁	1	4	yes	—
166	D₄	—	2	4	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	D₄	C₂²×Dic₁₁	1	4	yes	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	22	2	no	—
174	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
175	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
176	C₂×C₄	#4341	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
180	C₈	—	11	4	no	—
181	C₈	—	22	4	no	—
182	C₈	C₂²×Dic₁₁	1	1	yes	characteristic
183	C₈	C₂²×Dic₁₁	1	1	yes	characteristic
184	C₈	C₂²×Dic₁₁	1	1	yes	characteristic
185	C₈	C₂²×Dic₁₁	1	1	yes	characteristic
186	C₂²	—	2	4	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	8	2	no	—
189	C₂²	D₄×Dic₁₁	1	1	yes	characteristic
190	C₂²	D₄×Dic₁₁	1	1	yes	characteristic
191	C₂²	C₂×#4341	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	44	2	no	—
194	C₄	D₄×Dic₁₁	1	1	yes	characteristic
195	C₄	D₄×Dic₁₁	1	1	yes	characteristic
196	C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
197	C₄	C₂³×Dic₁₁	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×D₄×Dic₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁
Fitting, prime radical	C₂×C₁₁×#135	C₂
socle	C₂×C₁₁	C₂×C₄×D₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	C₂×#135	D₁₁
hypocenter	C₁₁	#2957
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. #10623350	C₂	semidirect
3. #10622849	C₂	semidirect
4. #10622810	C₂	semidirect
5. #10622809	C₂	semidirect
6. C₂×#66883	C₂	semidirect
7. #10622714	C₂	semidirect
9. C₁₁×#135	C₄	semidirect
10. C₁₁×#135	C₄	semidirect
13. C₂×C₁₁×Q₁₆	C₄	semidirect
14. C₂×C₁₁×SD₁₆	C₄	semidirect
15. C₂×C₁₁×D₈	C₄	semidirect
18. #67001	C₂²	semidirect
23. #66901	C₂²	semidirect
24. #66883	C₂²	semidirect
25. #66883	C₂²	semidirect
26. #66879	C₂²	semidirect
27. C₈×Dic₁₁	C₂²	semidirect
105. #135	Dic₁₁	semidirect
106. #135	Dic₁₁	semidirect
109. C₂×Q₁₆	Dic₁₁	semidirect
110. C₂×SD₁₆	Dic₁₁	semidirect
111. C₂×D₈	Dic₁₁	semidirect
161. C₁₁	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#135	2
#10623350	2
#10622849	2
#10622810	2
#10622809	2
C₂×#66883	2
#10622714	2
#2957	11