Subgroups — #28399730

Order 14464.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299132	C₂	1	1	yes	maximal, characteristic
2	#20298725	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298722	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298675	C₂	1	1	yes	maximal, characteristic, complemented
5	#20298669	C₂	1	2	yes	maximal, complemented
6	#20298663	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₁₃	C₂	1	2	yes	maximal, complemented
8	#20298658	C₂	1	2	yes	maximal, complemented
9	#20298651	C₂	1	1	yes	maximal, characteristic, complemented
10	#20298623	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14268630	C₂²	1	2	yes	—
13	C₂×C₈×D₁₁₃	C₂²	1	2	yes	—
14	#16813616	C₄	1	2	yes	—
15	#16813614	C₄	1	1	yes	characteristic
16	#16813610	C₄	1	1	yes	characteristic
17	#16813605	C₄	1	2	yes	—
18	#16813601	C₄	1	1	yes	characteristic
19	#16813596	C₄	1	1	yes	characteristic
20	C₄×Dic₂₂₆	C₂²	1	1	yes	characteristic
21	C₄×D₄₅₂	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
23	#16813579	C₂²	1	2	yes	—
24	C₄⋊#11744751	C₂²	1	1	yes	characteristic
25	C₄⋊#11744751	C₂²	1	1	yes	characteristic
26	#16813551	C₂²	1	2	yes	—
27	#16813551	C₂²	1	2	yes	—
28	C₂²⋊#11744751	C₂²	1	1	yes	characteristic
29	C₂²⋊#11744751	C₂²	1	1	yes	characteristic
30	#16813534	C₂²	1	2	yes	—
31	#16813534	C₂²	1	2	yes	—
32	#16813532	C₂²	1	2	yes	—
33	C₄×#11744751	C₂²	1	1	yes	characteristic
34	C₄×#11744751	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₁₃	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₁₃	C₂²	1	2	yes	—
37	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
39	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
45	#14268630	—	4	2	no	—
46	C₂×#11744751	C₂³	1	1	yes	characteristic
47	C₂×#11744751	C₂³	1	1	yes	characteristic
48	C₂×#11744751	C₂³	1	1	yes	characteristic
49	C₂×#11744751	C₂³	1	1	yes	characteristic
50	C₈×D₁₁₃	—	4	2	no	—
51	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
53	#14268626	C₂×C₄	1	2	yes	—
54	#14268626	C₂×C₄	1	2	yes	—
55	#14268626	C₂×C₄	1	2	yes	—
56	#14268626	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
58	#14268624	C₂×C₄	1	2	yes	—
59	#14268624	C₂×C₄	1	2	yes	—
60	#14268624	C₂×C₄	1	2	yes	—
61	#14268624	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₁₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₁₃	—	4	2	no	—
67	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
68	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
69	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₁₃	—	2	2	no	—
76	C₂×C₄×C₁₁₃	—	2	2	no	—
77	C₂×C₄×C₁₁₃	—	2	1	no	—
78	C₂×C₄×C₁₁₃	—	2	1	no	—
79	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₁₁₃	—	2	4	no	—
86	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
89	D₄₅₂	—	2	2	no	—
90	C₄×D₁₁₃	—	2	2	no	—
91	C₄×D₁₁₃	—	2	2	no	—
92	Dic₂₂₆	—	2	2	no	—
93	C₈×C₁₁₃	—	2	2	no	—
94	C₈×C₁₁₃	—	2	2	no	—
95	#11744751	—	2	1	no	—
96	#11744751	—	2	1	no	—
97	#11744751	—	2	1	no	—
98	#11744751	—	2	1	no	—
99	C₂²×C₁₁₃	—	2	1	no	—
100	C₂²×C₁₁₃	—	2	1	no	—
101	C₂²×C₁₁₃	#131	1	2	yes	—
102	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₁₃	—	2	2	no	—
104	C₂×D₁₁₃	—	2	2	no	—
105	C₂×D₁₁₃	—	2	2	no	—
106	C₄×C₁₁₃	—	2	1	no	—
107	C₄×C₁₁₃	—	2	1	no	—
108	C₄×C₁₁₃	—	2	1	no	—
109	C₄×C₁₁₃	—	2	1	no	—
110	C₄×C₁₁₃	#131	1	2	yes	—
111	C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₁₁₃	—	2	2	no	—
114	Dic₁₁₃	—	2	2	no	—
115	Dic₁₁₃	—	2	2	no	—
116	C₂×C₁₁₃	—	2	2	no	—
117	C₂×C₁₁₃	#520	1	1	yes	characteristic
118	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
120	D₁₁₃	—	4	2	no	—
121	#2994	—	113	1	no	maximal, Sylow
122	C₁₁₃	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	—	113	1	no	—
124	#446	—	113	1	no	—
125	C₈×Q₈	—	113	1	no	—
126	#436	—	113	1	no	—
127	#435	—	113	2	no	—
128	C₈×D₄	—	113	2	no	—
129	C₈×D₄	—	113	1	no	—
130	#433	—	113	2	no	—
131	#431	—	113	1	no	—
132	#424	—	113	2	no	—
133	C₂×C₄⋊C₈	D₁₁₃	1	1	yes	characteristic, complemented
134	C₂×M₁₆	—	113	2	no	—
135	C₂²×C₈	—	113	2	no	—
136	C₂²×C₈	C₂×D₁₁₃	1	2	yes	—
137	#126	—	113	2	no	—
138	#125	—	113	1	no	—
139	#124	—	113	1	no	—
140	#123	—	113	2	no	—
141	C₂²⋊Q₈	—	113	1	no	—
142	#121	—	113	1	no	—
143	C₄×Q₈	—	113	1	no	—
144	C₄×D₄	—	113	2	no	—
145	C₄×D₄	—	113	1	no	—
146	#117	—	113	2	no	—
147	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
148	C₄⋊C₈	—	113	2	no	—
149	C₄⋊C₈	—	113	2	no	—
150	C₄⋊C₈	—	113	1	no	—
151	C₄⋊C₈	—	113	1	no	—
152	C₄⋊C₈	C₂×D₁₁₃	1	2	yes	—
153	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
155	C₂²⋊C₈	—	113	1	no	—
156	C₂²⋊C₈	—	113	1	no	—
157	C₂²⋊C₈	—	113	2	no	—
158	C₂²⋊C₈	—	113	2	no	—
159	#97	—	113	2	no	—
160	C₄×C₈	—	113	2	no	—
161	C₄×C₈	—	113	1	no	—
162	C₄×C₈	—	113	1	no	—
163	C₂×Q₈	—	113	1	no	—
164	C₂×D₄	—	113	2	no	—
165	C₂×D₄	—	113	1	no	—
166	C₂²×C₄	—	113	2	no	—
167	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
170	M₁₆	—	452	2	no	—
171	C₂×C₈	—	4	2	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	—	452	2	no	—
177	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
178	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
179	C₄⋊C₄	—	113	2	no	—
180	C₄⋊C₄	—	113	2	no	—
181	C₄⋊C₄	—	113	1	no	—
182	C₄⋊C₄	—	113	2	no	—
183	C₄⋊C₄	—	113	1	no	—
184	C₄⋊C₄	—	113	2	no	—
185	C₂²⋊C₄	—	113	2	no	—
186	C₂²⋊C₄	—	113	2	no	—
187	C₂²⋊C₄	—	113	2	no	—
188	C₂²⋊C₄	—	113	2	no	—
189	C₂²⋊C₄	—	113	2	no	—
190	C₄²	—	113	2	no	—
191	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
192	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
193	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
194	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
195	C₂³	—	113	2	no	—
196	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
197	Q₈	—	226	2	no	—
198	D₄	—	226	4	no	—
199	D₄	—	226	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	113	2	no	—
205	C₂×C₄	—	113	2	no	—
206	C₂×C₄	—	113	2	no	—
207	C₂×C₄	—	226	2	no	—
208	C₂×C₄	—	226	2	no	—
209	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	226	1	no	—
218	C₈	—	226	1	no	—
219	C₈	—	226	1	no	—
220	C₈	—	226	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	226	2	no	—
224	C₂²	—	226	2	no	—
225	C₂²	—	226	2	no	—
226	C₂²	#16813630	1	2	yes	—
227	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	226	2	no	—
233	C₄	—	226	2	no	—
234	C₄	—	226	2	no	—
235	C₄	#16813628	1	2	yes	—
236	C₄	#16813667	1	1	yes	characteristic
237	C₄	#16813670	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	452	2	no	—
240	C₂	#20299079	1	1	yes	characteristic
241	C₂	C₂×#16813628	1	1	yes	characteristic
242	C₂	C₂×#16813630	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₁₁₃	C₂
prime radical	#16813596	C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₁₃
hypocenter	C₁₁₃	#2994
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20298725	C₂	semidirect
3. #20298722	C₂	semidirect
4. #20298675	C₂	semidirect
5. #20298669	C₂	semidirect
6. #20298663	C₂	semidirect
7. C₈×C₂²⋊D₁₁₃	C₂	semidirect
8. #20298658	C₂	semidirect
9. #20298651	C₂	semidirect
10. #20298623	C₂	semidirect
11. C₂×C₄⋊C₈×C₁₁₃	C₂	semidirect
122. C₁₁₃	#2994	semidirect
133. C₂×C₄⋊C₈	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299132	2
#20298725	2
#20298722	2
#20298675	2
#20298669	2
#20298663	2
C₈×C₂²⋊D₁₁₃	2
#20298658	2
#20298651	2
#20298623	2
C₂×C₄⋊C₈×C₁₁₃	2
#2994	113