Subgroups — #13080985

Order 1408.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10623072	C₂	1	2	yes	maximal
2	C₈×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic
3	#10622729	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622729	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	#10622442	C₂	1	2	yes	maximal, complemented
8	#10622336	C₂	1	2	yes	maximal, complemented
9	#10622334	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₄	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	#66988	—	2	2	no	—
16	#66979	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
18	#66977	C₂²	1	2	yes	—
19	#66977	C₂²	1	2	yes	—
20	C₂×Dic₄₄	C₂²	1	1	yes	characteristic
21	#66942	—	2	2	no	—
22	#66932	—	2	2	no	—
23	#66930	—	2	2	no	—
24	#66923	C₂²	1	2	yes	—
25	C₄×Dic₂₂	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	#66882	C₂²	1	1	yes	characteristic
31	#66882	C₂²	1	1	yes	characteristic
32	#66882	C₂²	1	1	yes	characteristic
33	#66882	C₂²	1	2	yes	—
34	#66881	C₂²	1	2	yes	—
35	#66881	C₂²	1	2	yes	—
36	#66880	C₂²	1	2	yes	—
37	#66880	C₂²	1	2	yes	—
38	#66880	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	C₂²×Dic₁₁	—	2	2	no	—
43	C₂×Dic₂₂	C₂³	1	2	yes	—
44	C₂×C₈×C₁₁	—	2	2	no	—
45	C₂×C₈×C₁₁	—	2	2	no	—
46	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₁₁×C₂²⋊C₄	D₄	1	1	yes	characteristic
52	C₁₁×C₂²⋊C₄	D₄	1	1	yes	characteristic
53	C₄²×C₁₁	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₁₁	—	2	2	no	—
55	C₂²⋊Dic₁₁	—	2	2	no	—
56	C₂²⋊Dic₁₁	—	2	2	no	—
57	C₄⋊Dic₁₁	—	2	2	no	—
58	C₄⋊Dic₁₁	C₂³	1	2	yes	—
59	C₄⋊Dic₁₁	C₂³	1	2	yes	—
60	C₄⋊Dic₁₁	C₂³	1	2	yes	—
61	Dic₁₁⋊C₄	—	2	2	no	—
62	Dic₁₁⋊C₄	—	2	2	no	—
63	Dic₁₁⋊C₄	—	2	2	no	—
64	Dic₁₁⋊C₄	—	2	2	no	—
65	Dic₁₁⋊C₄	—	2	2	no	—
66	C₄×Dic₁₁	—	2	2	no	—
67	C₄×Dic₁₁	—	2	2	no	—
68	Dic₄₄	—	2	2	no	—
69	#3336	—	11	1	no	maximal, Sylow
70	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
72	D₄×C₁₁	—	2	2	no	—
73	C₂×C₄×C₁₁	—	2	1	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	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₁	—	2	2	no	—
83	C₂×Dic₁₁	—	2	2	no	—
84	C₂×Dic₁₁	—	2	2	no	—
85	C₂×Dic₁₁	—	2	2	no	—
86	C₂×Dic₁₁	—	4	4	no	—
87	Dic₂₂	—	2	2	no	—
88	Dic₂₂	—	2	2	no	—
89	C₈×C₁₁	—	2	1	no	—
90	C₈×C₁₁	—	2	1	no	—
91	C₈×C₁₁	—	2	1	no	—
92	C₈×C₁₁	#41	1	2	yes	—
93	#552	—	11	2	no	—
94	#499	—	11	1	no	—
95	#485	—	11	2	no	—
96	#483	—	11	2	no	—
97	#479	—	11	2	no	—
98	#467	—	11	1	no	—
99	#467	—	11	1	no	—
100	C₄×Q₁₆	—	11	1	no	—
101	C₈×D₄	D₁₁	1	1	yes	characteristic
102	#427	—	11	1	no	—
103	C₂×#107	—	11	1	no	—
104	C₂²×C₁₁	—	2	1	no	—
105	C₂²×C₁₁	—	2	1	no	—
106	C₂²×C₁₁	—	2	1	no	—
107	C₂²×C₁₁	—	2	1	no	—
108	C₂²×C₁₁	—	2	1	no	—
109	C₂²×C₁₁	#135	1	2	yes	—
110	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₁	—	2	1	no	—
112	C₄×C₁₁	—	2	1	no	—
113	C₄×C₁₁	—	2	1	no	—
114	C₄×C₁₁	—	2	1	no	—
115	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
117	Dic₁₁	—	4	2	no	—
118	Dic₁₁	—	4	2	no	—
119	Dic₁₁	—	4	2	no	—
120	Dic₁₁	—	4	2	no	—
121	C₂×Q₁₆	—	11	1	no	—
122	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
124	#126	—	22	2	no	—
125	#125	—	11	2	no	—
126	#125	—	22	2	no	—
127	#123	—	22	2	no	—
128	C₂²⋊Q₈	—	11	2	no	—
129	C₂²⋊Q₈	—	11	2	no	—
130	C₄×Q₈	—	11	2	no	—
131	C₄×D₄	C₂×D₁₁	1	1	yes	characteristic
132	#117	—	11	2	no	—
133	#117	—	22	2	no	—
134	C₂×C₄⋊C₄	—	11	2	no	—
135	#107	—	11	1	no	—
136	#107	—	11	1	no	—
137	#107	—	11	2	no	—
138	#107	—	11	1	no	—
139	#106	—	11	2	no	—
140	#106	—	11	2	no	—
141	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
142	#103	—	11	2	no	—
143	#103	—	11	2	no	—
144	#103	—	11	2	no	—
145	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
146	C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
147	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
148	C₂×C₁₁	—	2	2	no	—
149	C₂×C₁₁	—	4	1	no	—
150	C₂×C₁₁	#547	1	1	yes	characteristic
151	C₂×C₁₁	C₂×#135	1	1	yes	characteristic
152	C₂×C₁₁	#578	1	1	yes	characteristic
153	C₂×Q₈	—	11	2	no	—
154	C₂×D₄	D₄₄	1	1	yes	characteristic
155	C₂²×C₄	—	22	2	no	—
156	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
158	Q₁₆	—	22	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
162	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
164	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
165	C₄⋊C₄	—	11	2	no	—
166	C₄⋊C₄	—	11	2	no	—
167	C₄⋊C₄	—	11	2	no	—
168	C₄⋊C₄	—	22	2	no	—
169	C₄⋊C₄	—	22	2	no	—
170	C₄⋊C₄	—	22	2	no	—
171	C₄⋊C₄	—	22	2	no	—
172	C₄⋊C₄	—	22	2	no	—
173	C₄⋊C₄	—	22	2	no	—
174	C₄⋊C₄	D₄₄	1	1	yes	characteristic
175	C₂²⋊C₄	—	22	2	no	—
176	C₂²⋊C₄	—	22	2	no	—
177	C₂²⋊C₄	—	22	2	no	—
178	C₂²⋊C₄	D₄₄	1	1	yes	characteristic
179	C₂²⋊C₄	D₄₄	1	1	yes	characteristic
180	C₄²	—	22	2	no	—
181	C₄²	—	22	2	no	—
182	C₄²	C₂²×D₁₁	1	1	yes	characteristic
183	C₁₁	#3336	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂×D₄₄	1	1	yes	characteristic
185	C₂³	C₂×D₄₄	1	1	yes	characteristic
186	Q₈	—	22	2	no	—
187	Q₈	—	22	2	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	22	2	no	—
194	C₂×C₄	—	22	2	no	—
195	C₂×C₄	—	22	2	no	—
196	C₂×C₄	—	22	2	no	—
197	C₂×C₄	—	44	4	no	—
198	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
201	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#4341	1	2	yes	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#66957	1	2	yes	—
213	C₂²	C₂²×D₄₄	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	44	2	no	—
219	C₄	—	44	2	no	—
220	C₄	—	44	2	no	—
221	C₄	—	44	2	no	—
222	C₄	C₂×#4341	1	1	yes	characteristic
223	C₄	#67043	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	#10623086	1	1	yes	characteristic
227	C₂	C₂×#66957	1	1	yes	characteristic
228	C₂	#10623284	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
prime radical	C₂×D₄×C₁₁	D₄
hypercenter	C₈×D₄	D₁₁
hypocenter	C₁₁	#3336
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
3. #10622729	C₂	semidirect
4. #10622729	C₂	semidirect
5. #10622719	C₂	semidirect
6. C₂×#66882	C₂	semidirect
7. #10622442	C₂	semidirect
8. #10622336	C₂	semidirect
9. #10622334	C₂	semidirect
10. C₄×Dic₄₄	C₂	semidirect
11. #10622296	C₂	semidirect
183. C₁₁	#3336	semidirect

Maximal subgroups

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

subgroup	index
#10623072	2
C₈×D₄×C₁₁	2
#10622729	2
#10622719	2
C₂×#66882	2
#10622442	2
#10622336	2
#10622334	2
C₄×Dic₄₄	2
#10622296	2
#3336	11