Subgroups — #28407388

Order 14464.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299649	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299633	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299630	C₂	1	2	yes	maximal, complemented
5	#20299625	C₂	1	2	yes	maximal, complemented
6	D₄×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299622	C₂	1	2	yes	maximal, complemented
8	#20299620	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299562	C₂	1	2	yes	maximal, complemented
10	C₄×#14268655	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299127	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268656	C₂²	1	2	yes	—
13	C₂×#14268655	C₂²	1	1	yes	characteristic
14	C₂×#14268655	C₂²	1	1	yes	characteristic
15	C₂²×#11744753	C₂²	1	2	yes	—
16	#16813689	C₂²	1	2	yes	—
17	C₄⋊#11744753	C₂²	1	1	yes	characteristic
18	C₄⋊#11744753	C₂²	1	1	yes	characteristic
19	C₄⋊#11744753	C₂²	1	2	yes	—
20	#16813679	C₂²	1	2	yes	—
21	C₂²⋊#11744753	C₂²	1	2	yes	—
22	C₂²⋊#11744753	C₂²	1	2	yes	—
23	#16813677	C₂²	1	2	yes	—
24	#16813677	C₂²	1	2	yes	—
25	C₄×#11744753	C₂²	1	1	yes	characteristic
26	C₄×#11744753	C₂²	1	1	yes	characteristic
27	C₄×#11744753	C₂²	1	1	yes	characteristic
28	C₄×#11744753	C₂²	1	1	yes	characteristic
29	C₄×#11744753	C₂²	1	2	yes	—
30	#16813615	C₄	1	2	yes	—
31	#16813614	C₄	1	1	yes	characteristic
32	#16813604	C₄	1	2	yes	—
33	#16813598	C₄	1	1	yes	characteristic
34	#16813593	C₄	1	1	yes	characteristic
35	Q₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
37	#16813582	C₂²	1	2	yes	—
38	#16813578	C₂²	1	2	yes	—
39	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
40	#124×C₁₁₃	C₄	1	1	yes	characteristic
41	#14268656	—	2	4	no	—
42	#14268655	—	2	2	no	—
43	#14268655	—	2	2	no	—
44	C₂×#11744753	—	2	4	no	—
45	C₂×#11744753	C₂³	1	1	yes	characteristic
46	C₂×#11744753	C₂³	1	1	yes	characteristic
47	C₂×#11744753	C₂³	1	2	yes	—
48	C₂×#11744753	C₂³	1	2	yes	—
49	C₂×#11744753	C₂³	1	1	yes	characteristic
50	C₂×#11744753	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
52	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
53	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
56	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
59	#14268626	C₂×C₄	1	2	yes	—
60	#14268626	C₂×C₄	1	2	yes	—
61	#14268626	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
63	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
64	#14268624	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
68	C₄×Dic₁₁₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
72	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
73	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
74	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₁₃	—	2	2	no	—
77	D₄×C₁₁₃	—	2	2	no	—
78	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₁₁₃	—	2	4	no	—
83	C₂×Dic₁₁₃	—	2	2	no	—
84	C₂×Dic₁₁₃	—	2	2	no	—
85	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
90	C₄×D₁₁₃	—	2	2	no	—
91	C₄×D₁₁₃	—	2	2	no	—
92	C₄×D₁₁₃	—	2	1	no	—
93	C₄×D₁₁₃	—	2	1	no	—
94	#11744753	—	2	2	no	—
95	#11744753	—	2	1	no	—
96	#11744753	—	2	1	no	—
97	#11744753	—	2	2	no	—
98	#11744753	#41	1	2	yes	—
99	#11744753	#41	1	2	yes	—
100	C₂²×C₁₁₃	—	2	2	no	—
101	C₂²×C₁₁₃	—	2	2	no	—
102	C₂²×C₁₁₃	—	2	2	no	—
103	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₁₁₃	—	2	1	no	—
105	C₂×D₁₁₃	—	2	1	no	—
106	C₂×D₁₁₃	#131	1	2	yes	—
107	C₄×C₁₁₃	—	2	1	no	—
108	C₄×C₁₁₃	—	2	1	no	—
109	C₄×C₁₁₃	—	2	2	no	—
110	C₄×C₁₁₃	#131	1	2	yes	—
111	Dic₁₁₃	—	2	2	no	—
112	Dic₁₁₃	—	2	1	no	—
113	Dic₁₁₃	—	2	1	no	—
114	Dic₁₁₃	—	2	2	no	—
115	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
116	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₁₃	—	4	2	no	—
118	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
119	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
120	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
121	D₁₁₃	—	2	2	no	—
122	#2966	—	113	1	no	maximal, Sylow
123	C₁₁₃	#2966	1	1	yes	characteristic, complemented, Sylow
124	#529	—	113	1	no	—
125	#446	—	113	1	no	—
126	C₈×Q₈	—	113	1	no	—
127	#436	—	113	1	no	—
128	#435	—	113	2	no	—
129	C₈×D₄	—	113	1	no	—
130	C₈×D₄	—	113	2	no	—
131	#433	—	113	2	no	—
132	#432	—	113	1	no	—
133	#405	—	113	2	no	—
134	C₂×C₄×C₈	—	113	1	no	—
135	C₂×M₁₆	—	113	2	no	—
136	C₂²×C₈	—	113	2	no	—
137	C₂²×C₈	—	113	1	no	—
138	C₂²×C₈	—	113	1	no	—
139	#126	—	113	2	no	—
140	#125	—	113	1	no	—
141	#124	#90673	1	1	yes	characteristic
142	#123	—	113	2	no	—
143	C₂²⋊Q₈	—	113	1	no	—
144	#121	—	113	1	no	—
145	C₄×Q₈	—	113	1	no	—
146	C₄×D₄	—	113	1	no	—
147	C₄×D₄	—	113	2	no	—
148	#117	—	113	2	no	—
149	C₂×C₄²	—	113	1	no	—
150	C₄⋊C₈	—	113	1	no	—
151	C₄⋊C₈	—	113	1	no	—
152	C₄⋊C₈	—	113	2	no	—
153	C₄⋊C₈	—	113	2	no	—
154	C₂²⋊C₈	—	113	2	no	—
155	C₂²⋊C₈	—	113	2	no	—
156	C₂²⋊C₈	—	113	2	no	—
157	#97	—	113	2	no	—
158	#97	—	113	2	no	—
159	C₄×C₈	—	113	1	no	—
160	C₄×C₈	—	113	1	no	—
161	C₄×C₈	—	113	1	no	—
162	C₄×C₈	—	113	1	no	—
163	C₄×C₈	—	113	2	no	—
164	C₂×Q₈	C₂×#90673	1	1	yes	characteristic
165	C₂×D₄	—	113	2	no	—
166	C₂×D₄	C₂×#90673	1	1	yes	characteristic
167	C₂²×C₄	—	113	2	no	—
168	C₂²×C₄	—	113	1	no	—
169	C₂²×C₄	—	113	1	no	—
170	C₂²×C₄	—	113	1	no	—
171	M₁₆	—	226	4	no	—
172	C₂×C₈	—	113	1	no	—
173	C₂×C₈	—	113	1	no	—
174	C₂×C₈	—	113	1	no	—
175	C₂×C₈	—	113	1	no	—
176	C₂×C₈	—	113	2	no	—
177	C₂×C₈	—	113	2	no	—
178	C₂×C₈	—	226	4	no	—
179	C₂×C₈	—	226	2	no	—
180	C₂×C₈	—	226	2	no	—
181	C₄⋊C₄	—	113	2	no	—
182	C₄⋊C₄	—	113	2	no	—
183	C₄⋊C₄	—	113	1	no	—
184	C₄⋊C₄	—	113	2	no	—
185	C₄⋊C₄	—	113	1	no	—
186	C₄⋊C₄	—	113	2	no	—
187	C₂²⋊C₄	—	113	2	no	—
188	C₂²⋊C₄	—	113	2	no	—
189	C₂²⋊C₄	—	113	2	no	—
190	C₂²⋊C₄	C₂×#90673	1	2	yes	—
191	C₂²⋊C₄	C₂×#90673	1	2	yes	—
192	C₄²	—	113	1	no	—
193	C₄²	—	113	1	no	—
194	C₄²	—	113	2	no	—
195	C₄²	—	113	1	no	—
196	C₄²	C₂×#90673	1	1	yes	characteristic
197	C₂³	—	113	1	no	—
198	C₂³	C₂²×#90673	1	2	yes	—
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	226	4	no	—
202	C₂×C₄	—	113	1	no	—
203	C₂×C₄	—	113	1	no	—
204	C₂×C₄	—	113	1	no	—
205	C₂×C₄	—	113	2	no	—
206	C₂×C₄	—	113	2	no	—
207	C₂×C₄	—	226	1	no	—
208	C₂×C₄	—	226	1	no	—
209	C₂×C₄	—	226	2	no	—
210	C₂×C₄	—	226	2	no	—
211	C₂×C₄	—	226	2	no	—
212	C₂×C₄	—	226	2	no	—
213	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
214	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
215	C₂×C₄	C₂²×#90673	1	2	yes	—
216	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
217	C₈	—	113	2	no	—
218	C₈	—	113	2	no	—
219	C₈	—	226	1	no	—
220	C₈	—	226	1	no	—
221	C₈	—	226	2	no	—
222	C₈	—	226	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	113	2	no	—
227	C₂²	—	226	1	no	—
228	C₂²	—	226	1	no	—
229	C₂²	C₂³×#90673	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	113	1	no	—
234	C₄	—	113	1	no	—
235	C₄	—	226	2	no	—
236	C₄	—	226	1	no	—
237	C₄	—	226	1	no	—
238	C₄	—	226	2	no	—
239	C₄	#16813709	1	2	yes	—
240	C₂	—	4	2	no	—
241	C₂	—	226	2	no	—
242	C₂	#41×#90673	1	1	yes	characteristic
243	C₂	C₂×#16813709	1	1	yes	characteristic
244	C₂	D₁₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#124×C₁₁₃	C₄
prime radical	#16813593	C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	#124	#90673
hypocenter	C₁₁₃	#2966
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20299649	C₂	semidirect
2. Q₈×#11744753	C₂	semidirect
3. #20299633	C₂	semidirect
4. #20299630	C₂	semidirect
5. #20299625	C₂	semidirect
6. D₄×#11744753	C₂	semidirect
7. #20299622	C₂	semidirect
8. #20299620	C₂	semidirect
9. #20299562	C₂	semidirect
10. C₄×#14268655	C₂	semidirect
123. C₁₁₃	#2966	semidirect

Maximal subgroups

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

subgroup	index
#20299649	2
Q₈×#11744753	2
#20299633	2
#20299630	2
#20299625	2
D₄×#11744753	2
#20299622	2
#20299620	2
#20299562	2
C₄×#14268655	2
#20299127	2
#2966	113