Subgroups — #28402943

Order 14464.

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	#20299288	C₂	1	2	yes	maximal
2	#20299000	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298940	C₂	1	2	yes	maximal, complemented
4	#20298929	C₂	1	2	yes	maximal, complemented
5	#20298906	C₂	1	2	yes	maximal, complemented
6	#20298839	C₂	1	1	yes	maximal, characteristic, complemented
7	#20298839	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#20298636	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813556	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₄₅₂	C₂²	1	1	yes	characteristic
13	#16813615	—	2	2	no	—
14	#16813614	C₂²	1	2	yes	—
15	#16813612	—	2	2	no	—
16	#16813603	—	2	2	no	—
17	#16813601	C₂²	1	2	yes	—
18	#16813601	C₂²	1	2	yes	—
19	C₄×Dic₂₂₆	C₂²	1	2	yes	—
20	#16813581	C₂²	1	2	yes	—
21	#16813578	—	2	2	no	—
22	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
23	#16813556	C₂²	1	1	yes	characteristic
24	#16813556	C₂²	1	1	yes	characteristic
25	#16813556	C₂²	1	1	yes	characteristic
26	#16813556	C₂²	1	2	yes	—
27	#16813554	C₂²	1	2	yes	—
28	#16813554	C₂²	1	2	yes	—
29	#16813547	C₂²	1	2	yes	—
30	#16813547	C₂²	1	2	yes	—
31	#16813547	C₂²	1	2	yes	—
32	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₂₆	C₂³	1	2	yes	—
40	C₂²×Dic₁₁₃	—	2	2	no	—
41	Dic₄₅₂	—	2	2	no	—
42	C₄⋊Dic₁₁₃	—	2	2	no	—
43	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
44	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
45	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
46	#14268626	—	2	2	no	—
47	#14268626	—	2	2	no	—
48	#14268626	—	2	2	no	—
49	#14268626	—	2	2	no	—
50	#14268626	—	2	2	no	—
51	C₂²⋊Dic₁₁₃	—	2	2	no	—
52	C₂²⋊Dic₁₁₃	—	2	2	no	—
53	C₂²⋊Dic₁₁₃	—	2	2	no	—
54	C₄×Dic₁₁₃	—	2	2	no	—
55	C₄×Dic₁₁₃	—	2	2	no	—
56	C₂×D₄×C₁₁₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₁₃	—	2	2	no	—
60	C₂×C₈×C₁₁₃	—	2	2	no	—
61	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
68	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
69	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
71	D₄×C₁₁₃	—	2	2	no	—
72	C₂×C₄×C₁₁₃	—	2	1	no	—
73	C₂×C₄×C₁₁₃	—	2	1	no	—
74	C₂×C₄×C₁₁₃	—	2	1	no	—
75	C₂×C₄×C₁₁₃	—	2	1	no	—
76	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₁₃	—	2	2	no	—
82	C₂×Dic₁₁₃	—	2	2	no	—
83	C₂×Dic₁₁₃	—	2	2	no	—
84	C₂×Dic₁₁₃	—	2	2	no	—
85	C₂×Dic₁₁₃	—	4	4	no	—
86	Dic₂₂₆	—	2	2	no	—
87	Dic₂₂₆	—	2	2	no	—
88	C₈×C₁₁₃	—	2	1	no	—
89	C₈×C₁₁₃	—	2	1	no	—
90	C₈×C₁₁₃	—	2	1	no	—
91	C₈×C₁₁₃	#41	1	2	yes	—
92	C₂²×C₁₁₃	—	2	1	no	—
93	C₂²×C₁₁₃	—	2	1	no	—
94	C₂²×C₁₁₃	—	2	1	no	—
95	C₂²×C₁₁₃	—	2	1	no	—
96	C₂²×C₁₁₃	—	2	1	no	—
97	C₂²×C₁₁₃	#135	1	2	yes	—
98	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₁₁₃	—	2	1	no	—
100	C₄×C₁₁₃	—	2	1	no	—
101	C₄×C₁₁₃	—	2	1	no	—
102	C₄×C₁₁₃	—	2	1	no	—
103	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
104	C₄×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
105	Dic₁₁₃	—	4	2	no	—
106	Dic₁₁₃	—	4	2	no	—
107	Dic₁₁₃	—	4	2	no	—
108	Dic₁₁₃	—	4	2	no	—
109	C₂×C₁₁₃	—	2	2	no	—
110	C₂×C₁₁₃	—	4	1	no	—
111	C₂×C₁₁₃	#547	1	1	yes	characteristic
112	C₂×C₁₁₃	C₂×#135	1	1	yes	characteristic
113	C₂×C₁₁₃	#578	1	1	yes	characteristic
114	#3336	—	113	1	no	maximal, Sylow
115	C₁₁₃	#3336	1	1	yes	characteristic, complemented, Sylow
116	#552	—	113	2	no	—
117	#499	—	113	1	no	—
118	#485	—	113	2	no	—
119	#483	—	113	2	no	—
120	#479	—	113	2	no	—
121	#467	—	113	1	no	—
122	#467	—	113	1	no	—
123	C₄×Q₁₆	—	113	1	no	—
124	C₈×D₄	D₁₁₃	1	1	yes	characteristic
125	#427	—	113	1	no	—
126	C₂×#107	—	113	1	no	—
127	C₂×Q₁₆	—	113	1	no	—
128	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
129	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
130	#126	—	226	2	no	—
131	#125	—	113	2	no	—
132	#125	—	226	2	no	—
133	#123	—	226	2	no	—
134	C₂²⋊Q₈	—	113	2	no	—
135	C₂²⋊Q₈	—	113	2	no	—
136	C₄×Q₈	—	113	2	no	—
137	C₄×D₄	C₂×D₁₁₃	1	1	yes	characteristic
138	#117	—	113	2	no	—
139	#117	—	226	2	no	—
140	C₂×C₄⋊C₄	—	113	2	no	—
141	#107	—	113	1	no	—
142	#107	—	113	1	no	—
143	#107	—	113	2	no	—
144	#107	—	113	1	no	—
145	#106	—	113	2	no	—
146	#106	—	113	2	no	—
147	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
148	#103	—	113	2	no	—
149	#103	—	113	2	no	—
150	#103	—	113	2	no	—
151	C₂²⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
152	C₂²⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
153	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
154	C₂×Q₈	—	113	2	no	—
155	C₂×D₄	D₄₅₂	1	1	yes	characteristic
156	C₂²×C₄	—	226	2	no	—
157	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
159	Q₁₆	—	226	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
166	C₄⋊C₄	—	113	2	no	—
167	C₄⋊C₄	—	113	2	no	—
168	C₄⋊C₄	—	113	2	no	—
169	C₄⋊C₄	—	226	2	no	—
170	C₄⋊C₄	—	226	2	no	—
171	C₄⋊C₄	—	226	2	no	—
172	C₄⋊C₄	—	226	2	no	—
173	C₄⋊C₄	—	226	2	no	—
174	C₄⋊C₄	—	226	2	no	—
175	C₄⋊C₄	D₄₅₂	1	1	yes	characteristic
176	C₂²⋊C₄	—	226	2	no	—
177	C₂²⋊C₄	—	226	2	no	—
178	C₂²⋊C₄	—	226	2	no	—
179	C₂²⋊C₄	D₄₅₂	1	1	yes	characteristic
180	C₂²⋊C₄	D₄₅₂	1	1	yes	characteristic
181	C₄²	—	226	2	no	—
182	C₄²	—	226	2	no	—
183	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
184	C₂³	C₂×D₄₅₂	1	1	yes	characteristic
185	C₂³	C₂×D₄₅₂	1	1	yes	characteristic
186	Q₈	—	226	2	no	—
187	Q₈	—	226	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	—	452	4	no	—
194	C₂×C₄	—	226	2	no	—
195	C₂×C₄	—	226	2	no	—
196	C₂×C₄	—	226	2	no	—
197	C₂×C₄	—	226	2	no	—
198	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
201	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#14268646	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	#16813644	1	2	yes	—
213	C₂²	C₂²×D₄₅₂	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	—	452	2	no	—
219	C₄	—	452	2	no	—
220	C₄	—	452	2	no	—
221	C₄	—	452	2	no	—
222	C₄	C₂×#14268646	1	1	yes	characteristic
223	C₄	#16813669	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	#20299252	1	1	yes	characteristic
227	C₂	C₂×#16813644	1	1	yes	characteristic
228	C₂	#20299415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₅₂
commutator, cosocle	C₂×C₄×C₁₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	C₈×D₄×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
prime radical	C₂×D₄×C₁₁₃	D₄
hypercenter	C₈×D₄	D₁₁₃
hypocenter	C₁₁₃	#3336
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #20299000	C₂	semidirect
3. #20298940	C₂	semidirect
4. #20298929	C₂	semidirect
5. #20298906	C₂	semidirect
6. #20298839	C₂	semidirect
7. #20298839	C₂	semidirect
8. C₄×Dic₄₅₂	C₂	semidirect
9. #20298636	C₂	semidirect
10. C₂×#16813556	C₂	semidirect
115. C₁₁₃	#3336	semidirect

Maximal subgroups

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

subgroup	index
#20299288	2
#20299000	2
#20298940	2
#20298929	2
#20298906	2
#20298839	2
C₄×Dic₄₅₂	2
#20298636	2
C₂×#16813556	2
C₈×D₄×C₁₁₃	2
#3336	113