Subgroups — #28399444

Order 14464.

484 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	C₂×#16813641	C₂	1	1	yes	maximal, characteristic
2	#20299077	C₂	1	2	yes	maximal, complemented
3	#20298698	C₂	1	2	yes	maximal, complemented
4	#20298695	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298692	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#20298596	C₂	1	2	yes	maximal, complemented
10	#20298552	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14268646	C₂²	1	2	yes	—
13	#16813641	C₄	1	8	yes	complemented
14	C₂×Dic₄₅₂	C₂²	1	1	yes	characteristic
15	C₂×#14268635	C₂²	1	2	yes	—
16	C₂×#14268630	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₂₆	C₂²	1	2	yes	—
19	#16813587	C₂²	1	2	yes	—
20	D₄×Dic₁₁₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₁₁₃	—	2	2	no	—
22	#16813582	—	2	2	no	—
23	#16813579	C₂²	1	1	yes	characteristic
24	#16813578	—	2	2	no	—
25	C₂×#14268626	—	2	2	no	—
26	C₄⋊C₄×D₁₁₃	C₂²	1	2	yes	—
27	#16813556	C₂²	1	1	yes	characteristic
28	#16813555	C₂²	1	2	yes	—
29	#16813547	C₂²	1	2	yes	—
30	#16813545	C₂²	1	2	yes	—
31	#16813543	C₂²	1	2	yes	—
32	#16813533	C₂²	1	1	yes	characteristic
33	#16813532	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
37	#107×C₁₁₃	C₂²	1	1	yes	characteristic
38	#102×C₁₁₃	C₂²	1	2	yes	—
39	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
40	#14268646	—	2	4	no	—
41	#14268646	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₂₂₆	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₁₃	—	2	2	no	—
44	C₂²×Dic₁₁₃	—	2	2	no	—
45	C₂²×Dic₁₁₃	—	2	2	no	—
46	C₂×C₄×D₁₁₃	—	2	2	no	—
47	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
48	Dic₄₅₂	C₂×C₄	1	4	yes	complemented
49	#14268635	C₂×C₄	1	8	yes	complemented
50	#14268630	—	2	2	no	—
51	C₂×#11744751	C₂³	1	1	yes	characteristic
52	C₂×#11744751	C₂³	1	1	yes	characteristic
53	C₈×D₁₁₃	C₂×C₄	1	4	yes	complemented
54	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
55	#14268626	—	2	2	no	—
56	#14268626	—	2	2	no	—
57	#14268626	—	2	2	no	—
58	#14268626	D₄	1	1	yes	characteristic
59	#14268626	D₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₁₃	—	2	2	no	—
61	#14268624	D₄	1	1	yes	characteristic
62	#14268624	D₄	1	1	yes	characteristic
63	C₄×Dic₁₁₃	—	2	2	no	—
64	C₄×Dic₁₁₃	—	2	2	no	—
65	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₁₃	—	2	2	no	—
68	D₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
73	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
74	C₂³×C₁₁₃	—	2	2	no	—
75	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
76	D₄×C₁₁₃	—	2	2	no	—
77	D₄×C₁₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₁₃	—	2	2	no	—
79	C₂×C₄×C₁₁₃	—	8	2	no	—
80	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₁₃	—	2	8	no	—
84	C₂×Dic₁₁₃	—	2	2	no	—
85	C₂×Dic₁₁₃	—	2	8	no	—
86	C₂×Dic₁₁₃	—	2	2	no	—
87	C₂×Dic₁₁₃	—	8	2	no	—
88	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄×D₁₁₃	—	8	2	no	—
92	C₄×D₁₁₃	C₂²×C₄	1	2	yes	—
93	C₄×D₁₁₃	#41	1	2	yes	—
94	Dic₂₂₆	—	2	2	no	—
95	Dic₂₂₆	C₂²×C₄	1	4	yes	—
96	C₈×C₁₁₃	—	2	1	no	—
97	C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
98	#11744751	—	2	1	no	—
99	#11744751	C₂²×C₄	1	2	yes	—
100	C₂²×C₁₁₃	—	2	4	no	—
101	C₂²×C₁₁₃	—	4	2	no	—
102	C₂²×C₁₁₃	—	4	2	no	—
103	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₁₃	—	2	1	no	—
105	C₂×D₁₁₃	—	2	1	no	—
106	C₂×D₁₁₃	C₄×D₄	1	2	yes	—
107	C₄×C₁₁₃	—	2	1	no	—
108	C₄×C₁₁₃	—	2	1	no	—
109	C₄×C₁₁₃	—	4	2	no	—
110	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
112	Dic₁₁₃	—	2	4	no	—
113	Dic₁₁₃	—	2	1	no	—
114	Dic₁₁₃	—	2	1	no	—
115	Dic₁₁₃	—	4	2	no	—
116	Dic₁₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₁₁₃	—	4	4	no	—
118	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₁₃	#576	1	1	yes	characteristic
120	C₂×C₁₁₃	#578	1	1	yes	characteristic
121	D₁₁₃	—	2	2	no	—
122	#2951	—	113	1	no	maximal, Sylow
123	C₁₁₃	#2951	1	1	yes	characteristic, complemented, Sylow
124	C₂×#135	—	113	1	no	—
125	#520	—	113	2	no	—
126	#440	—	113	2	no	—
127	#440	—	113	2	no	—
128	C₄×Q₁₆	—	113	1	no	—
129	C₄×Q₁₆	—	113	1	no	—
130	C₄×D₈	—	113	1	no	—
131	C₄×D₈	D₁₁₃	1	1	yes	characteristic, complemented
132	C₂×#107	—	113	1	no	—
133	#417	—	113	2	no	—
134	#405	—	113	1	no	—
135	C₂×#41	—	113	2	no	—
136	#135	—	113	8	no	—
137	C₂×Q₁₆	—	113	1	no	—
138	C₂×SD₁₆	—	113	2	no	—
139	C₂×D₈	C₂×D₁₁₃	1	1	yes	characteristic
140	C₂×M₁₆	—	113	1	no	—
141	C₂²×C₈	—	113	1	no	—
142	C₄×Q₈	—	113	2	no	—
143	C₄×Q₈	—	113	2	no	—
144	C₄×D₄	—	113	2	no	—
145	C₄×D₄	—	226	2	no	—
146	C₄×D₄	—	226	2	no	—
147	C₄×D₄	C₂×D₁₁₃	1	2	yes	—
148	#117	—	113	1	no	—
149	#117	—	226	2	no	—
150	C₂×C₄⋊C₄	—	113	2	no	—
151	C₂×C₄⋊C₄	—	226	2	no	—
152	#107	—	113	1	no	—
153	#107	—	113	2	no	—
154	#107	C₂×D₁₁₃	1	1	yes	characteristic
155	#103	—	113	2	no	—
156	#103	—	113	2	no	—
157	#102	—	113	2	no	—
158	#102	C₂×D₁₁₃	1	2	yes	—
159	#97	—	113	1	no	—
160	#97	—	113	1	no	—
161	C₄×C₈	—	113	1	no	—
162	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
163	#41	—	113	8	no	—
164	#41	—	226	4	no	—
165	C₂×Q₈	—	113	2	no	—
166	C₂×D₄	—	226	2	no	—
167	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	113	1	no	—
170	C₂²×C₄	—	226	2	no	—
171	C₂²×C₄	—	226	2	no	—
172	C₂²×C₄	—	226	2	no	—
173	Q₁₆	—	113	4	no	—
174	SD₁₆	—	113	8	no	—
175	D₈	C₄×D₁₁₃	1	4	yes	complemented
176	M₁₆	—	226	2	no	—
177	C₂×C₈	—	113	1	no	—
178	C₂×C₈	—	113	4	no	—
179	C₂×C₈	—	113	1	no	—
180	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	113	1	no	—
183	C₄⋊C₄	—	113	1	no	—
184	C₄⋊C₄	—	113	2	no	—
185	C₄⋊C₄	—	226	2	no	—
186	C₄⋊C₄	—	226	2	no	—
187	C₄⋊C₄	—	226	2	no	—
188	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	113	1	no	—
191	C₂²⋊C₄	—	113	1	no	—
192	C₂²⋊C₄	—	226	2	no	—
193	C₄²	—	113	1	no	—
194	C₄²	—	226	2	no	—
195	C₄²	—	226	2	no	—
196	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	113	1	no	—
199	Q₈	—	113	4	no	—
200	Q₈	—	226	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	226	8	no	—
203	D₄	C₂×C₄×D₁₁₃	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	113	1	no	—
207	C₂×C₄	—	113	2	no	—
208	C₂×C₄	—	113	2	no	—
209	C₂×C₄	—	113	1	no	—
210	C₂×C₄	—	113	1	no	—
211	C₂×C₄	—	904	2	no	—
212	C₂×C₄	—	904	2	no	—
213	C₂×C₄	—	226	8	no	—
214	C₂×C₄	—	226	2	no	—
215	C₂×C₄	—	226	2	no	—
216	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
217	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	113	2	no	—
221	C₈	—	226	1	no	—
222	C₈	C₂×C₄×D₁₁₃	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	113	2	no	—
227	C₂²	—	226	1	no	—
228	C₂²	—	226	1	no	—
229	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	113	2	no	—
234	C₄	—	452	2	no	—
235	C₄	—	226	4	no	—
236	C₄	—	226	1	no	—
237	C₄	—	226	1	no	—
238	C₄	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
239	C₄	#41×D₁₁₃	1	1	yes	characteristic
240	C₂	—	4	4	no	—
241	C₂	—	226	2	no	—
242	C₂	C₄×D₄×D₁₁₃	1	1	yes	characteristic
243	C₂	#20299399	1	1	yes	characteristic
244	C₂	#20299412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	C₄×D₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
prime radical	C₂×#16813641	C₂
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₄×D₈	D₁₁₃
hypocenter	C₁₁₃	#2951
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #20299077	C₂	semidirect
3. #20298698	C₂	semidirect
4. #20298695	C₂	semidirect
5. C₄×Dic₄₅₂	C₂	semidirect
6. #20298692	C₂	semidirect
7. D₈×Dic₁₁₃	C₂	semidirect
8. #107×D₁₁₃	C₂	semidirect
9. #20298596	C₂	semidirect
10. #20298552	C₂	semidirect
11. C₄×D₈×C₁₁₃	C₂	semidirect
13. #16813641	C₄	semidirect
41. #14268646	C₂×C₄	semidirect
48. Dic₄₅₂	C₂×C₄	semidirect
49. #14268635	C₂×C₄	semidirect
53. C₈×D₁₁₃	C₂×C₄	semidirect
68. D₈×C₁₁₃	C₂×C₄	semidirect
123. C₁₁₃	#2951	semidirect
131. C₄×D₈	D₁₁₃	semidirect
175. D₈	C₄×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813641	2
#20299077	2
#20298698	2
#20298695	2
C₄×Dic₄₅₂	2
#20298692	2
D₈×Dic₁₁₃	2
#107×D₁₁₃	2
#20298596	2
#20298552	2
C₄×D₈×C₁₁₃	2
#2951	113