Subgroups — #13081019

Order 1408.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10623096	C₂	1	2	yes	maximal, complemented
2	C₈×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622728	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622728	C₂	1	1	yes	maximal, characteristic, complemented
5	#10622719	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#66882	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622460	C₂	1	2	yes	maximal, complemented
8	#10622350	C₂	1	2	yes	maximal, complemented
9	#10622348	C₂	1	2	yes	maximal, complemented
10	C₄×D₈₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#10622296	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₁	C₂²	1	1	yes	characteristic
15	#66992	—	2	2	no	—
16	#66983	C₂²	1	2	yes	—
17	#66983	C₂²	1	2	yes	—
18	#66979	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
20	C₂×D₈₈	C₂²	1	1	yes	characteristic
21	#66951	—	2	2	no	—
22	#66936	—	2	2	no	—
23	(C₂²⋊D₁₁)⋊C₄	—	2	2	no	—
24	C₄×D₄₄	C₂²	1	2	yes	—
25	#66923	C₂²	1	2	yes	—
26	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
30	#66885	C₂²	1	2	yes	—
31	#66885	C₂²	1	2	yes	—
32	#66885	C₂²	1	2	yes	—
33	#66882	C₂²	1	1	yes	characteristic
34	#66882	C₂²	1	1	yes	characteristic
35	#66882	C₂²	1	1	yes	characteristic
36	#66882	C₂²	1	2	yes	—
37	#66881	C₂²	1	2	yes	—
38	#66881	C₂²	1	2	yes	—
39	C₂×D₄×C₁₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₁	—	2	2	no	—
43	C₂×C₂²⋊D₁₁	—	2	2	no	—
44	C₂²×Dic₁₁	—	2	2	no	—
45	C₂×D₄₄	C₂³	1	2	yes	—
46	C₂×C₄×D₁₁	—	2	2	no	—
47	C₂×C₈×C₁₁	—	2	2	no	—
48	C₂×C₈×C₁₁	—	2	2	no	—
49	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
53	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
54	C₁₁×C₂²⋊C₄	D₄	1	1	yes	characteristic
55	C₁₁×C₂²⋊C₄	D₄	1	1	yes	characteristic
56	C₄²×C₁₁	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₁	—	2	2	no	—
58	(C₂×D₁₁)⋊C₄	—	2	2	no	—
59	(C₂×D₁₁)⋊C₄	—	2	2	no	—
60	(C₂×D₁₁)⋊C₄	—	2	2	no	—
61	C₄⋊Dic₁₁	—	2	2	no	—
62	C₄⋊Dic₁₁	C₂³	1	2	yes	—
63	C₄⋊Dic₁₁	C₂³	1	2	yes	—
64	C₄⋊Dic₁₁	C₂³	1	2	yes	—
65	Dic₁₁⋊C₄	—	2	2	no	—
66	Dic₁₁⋊C₄	—	2	2	no	—
67	C₄×Dic₁₁	—	2	2	no	—
68	D₈₈	—	2	2	no	—
69	#3339	—	11	1	no	maximal, Sylow
70	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₁	—	2	2	no	—
73	D₄×C₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	1	no	—
75	C₂×C₄×C₁₁	—	2	1	no	—
76	C₂×C₄×C₁₁	—	2	1	no	—
77	C₂×C₄×C₁₁	—	2	1	no	—
78	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₁	—	4	4	no	—
84	C₂²⋊D₁₁	—	8	2	no	—
85	C₂×Dic₁₁	—	2	2	no	—
86	C₂×Dic₁₁	—	2	2	no	—
87	C₂×Dic₁₁	—	2	2	no	—
88	C₂×Dic₁₁	—	8	2	no	—
89	D₄₄	—	2	2	no	—
90	D₄₄	—	2	2	no	—
91	C₄×D₁₁	—	8	2	no	—
92	C₈×C₁₁	—	2	1	no	—
93	C₈×C₁₁	—	2	1	no	—
94	C₈×C₁₁	—	2	1	no	—
95	C₈×C₁₁	#41	1	2	yes	—
96	#553	—	11	2	no	—
97	#499	—	11	1	no	—
98	#482	—	11	2	no	—
99	#481	—	11	2	no	—
100	#478	—	11	2	no	—
101	#466	—	11	1	no	—
102	#466	—	11	1	no	—
103	C₄×D₈	—	11	1	no	—
104	C₈×D₄	D₁₁	1	1	yes	characteristic, complemented
105	#427	—	11	1	no	—
106	C₂×#107	—	11	1	no	—
107	C₂²×C₁₁	—	2	1	no	—
108	C₂²×C₁₁	—	2	1	no	—
109	C₂²×C₁₁	—	2	1	no	—
110	C₂²×C₁₁	—	2	1	no	—
111	C₂²×C₁₁	—	2	1	no	—
112	C₂²×C₁₁	#135	1	2	yes	—
113	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₁	—	4	2	no	—
115	C₂×D₁₁	—	4	2	no	—
116	C₂×D₁₁	—	4	2	no	—
117	C₄×C₁₁	—	2	1	no	—
118	C₄×C₁₁	—	2	1	no	—
119	C₄×C₁₁	—	2	1	no	—
120	C₄×C₁₁	—	2	1	no	—
121	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
123	Dic₁₁	—	4	2	no	—
124	Dic₁₁	—	4	2	no	—
125	Dic₁₁	—	4	2	no	—
126	C₂×D₈	—	11	1	no	—
127	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
129	#126	—	22	2	no	—
130	#125	—	11	2	no	—
131	#123	—	22	2	no	—
132	#121	—	11	2	no	—
133	#121	—	11	2	no	—
134	#121	—	22	2	no	—
135	C₄×D₄	—	11	2	no	—
136	C₄×D₄	—	22	2	no	—
137	C₄×D₄	C₂×D₁₁	1	1	yes	characteristic
138	#117	—	11	2	no	—
139	C₂×C₄⋊C₄	—	11	2	no	—
140	#107	—	11	1	no	—
141	#107	—	11	1	no	—
142	#107	—	11	2	no	—
143	#107	—	11	1	no	—
144	#106	—	11	2	no	—
145	#106	—	11	2	no	—
146	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
147	#102	—	11	2	no	—
148	#102	—	11	2	no	—
149	#102	—	11	2	no	—
150	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
151	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
152	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
153	C₂×C₁₁	—	2	2	no	—
154	C₂×C₁₁	—	4	1	no	—
155	C₂×C₁₁	#547	1	1	yes	characteristic
156	C₂×C₁₁	C₂×#135	1	1	yes	characteristic
157	C₂×C₁₁	#576	1	1	yes	characteristic
158	D₁₁	—	8	2	no	—
159	C₂×D₄	—	11	2	no	—
160	C₂×D₄	—	22	2	no	—
161	C₂×D₄	—	22	2	no	—
162	C₂×D₄	D₄₄	1	1	yes	characteristic
163	C₂²×C₄	—	22	2	no	—
164	C₂²×C₄	—	22	2	no	—
165	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
167	D₈	—	22	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
174	C₄⋊C₄	—	11	2	no	—
175	C₄⋊C₄	—	11	2	no	—
176	C₄⋊C₄	—	11	2	no	—
177	C₄⋊C₄	—	22	2	no	—
178	C₄⋊C₄	—	22	2	no	—
179	C₄⋊C₄	—	22	2	no	—
180	C₄⋊C₄	D₄₄	1	1	yes	characteristic
181	C₂²⋊C₄	—	22	2	no	—
182	C₂²⋊C₄	—	22	2	no	—
183	C₂²⋊C₄	—	22	2	no	—
184	C₂²⋊C₄	—	22	2	no	—
185	C₂²⋊C₄	D₄₄	1	1	yes	characteristic
186	C₂²⋊C₄	D₄₄	1	1	yes	characteristic
187	C₄²	—	22	2	no	—
188	C₄²	C₂²×D₁₁	1	1	yes	characteristic
189	C₁₁	#3339	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	22	2	no	—
191	C₂³	C₂×D₄₄	1	1	yes	characteristic
192	C₂³	C₂×D₄₄	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	22	2	no	—
195	D₄	—	22	2	no	—
196	D₄	—	44	4	no	—
197	D₄	—	88	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	22	2	no	—
203	C₂×C₄	—	22	2	no	—
204	C₂×C₄	—	22	2	no	—
205	C₂×C₄	—	88	2	no	—
206	C₂×C₄	—	88	2	no	—
207	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#4341	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	44	2	no	—
222	C₂²	—	44	2	no	—
223	C₂²	—	44	2	no	—
224	C₂²	#66957	1	2	yes	—
225	C₂²	C₂²×D₄₄	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	44	2	no	—
231	C₄	—	44	2	no	—
232	C₄	—	44	2	no	—
233	C₄	C₂×#4341	1	1	yes	characteristic
234	C₄	#67043	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	88	2	no	—
238	C₂	#10623086	1	1	yes	characteristic
239	C₂	C₂×#66957	1	1	yes	characteristic
240	C₂	#10623283	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₈×D₄×C₁₁	C₂
socle	C₂²×C₁₁	C₂²×D₄
hypercenter	C₈×D₄	D₁₁
hypocenter	C₁₁	#3339
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #10623096	C₂	semidirect
2. C₈×D₄×C₁₁	C₂	semidirect
3. #10622728	C₂	semidirect
4. #10622728	C₂	semidirect
5. #10622719	C₂	semidirect
6. C₂×#66882	C₂	semidirect
7. #10622460	C₂	semidirect
8. #10622350	C₂	semidirect
9. #10622348	C₂	semidirect
10. C₄×D₈₈	C₂	semidirect
11. #10622296	C₂	semidirect
104. C₈×D₄	D₁₁	semidirect
189. C₁₁	#3339	semidirect

Maximal subgroups

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

subgroup	index
#10623096	2
C₈×D₄×C₁₁	2
#10622728	2
#10622719	2
C₂×#66882	2
#10622460	2
#10622350	2
#10622348	2
C₄×D₈₈	2
#10622296	2
#3339	11