Subgroups — #28399639

Order 14464.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299145	C₂	1	1	yes	maximal, characteristic
2	#20298675	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298674	C₂	1	2	yes	maximal, complemented
4	C₈×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298661	C₂	1	2	yes	maximal, complemented
7	#20298650	C₂	1	1	yes	maximal, characteristic, complemented
8	#20298649	C₂	1	1	yes	maximal, characteristic, complemented
9	#20298566	C₂	1	2	yes	maximal, complemented
10	#20298563	C₂	1	2	yes	maximal, complemented
11	#436×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14268647	C₂²	1	2	yes	—
13	C₂×#14268630	C₂²	1	2	yes	—
14	C₂²×#11744751	C₂²	1	2	yes	—
15	C₂×C₈×D₁₁₃	C₂²	1	2	yes	—
16	#16813621	C₄	1	1	yes	characteristic
17	#16813618	C₄	1	1	yes	characteristic
18	#16813608	C₄	1	2	yes	—
19	#16813597	C₄	1	1	yes	characteristic
20	#16813595	C₄	1	2	yes	—
21	#16813593	C₄	1	1	yes	characteristic
22	C₄×D₄₅₂	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
24	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
25	C₄⋊#11744751	C₂²	1	1	yes	characteristic
26	#16813551	C₂²	1	1	yes	characteristic
27	#16813551	C₂²	1	1	yes	characteristic
28	C₂²⋊#11744751	C₂²	1	2	yes	—
29	#16813534	C₂²	1	1	yes	characteristic
30	#16813534	C₂²	1	2	yes	—
31	#16813534	C₂²	1	1	yes	characteristic
32	#16813534	C₂²	1	2	yes	—
33	#16813534	C₂²	1	2	yes	—
34	C₄×#11744751	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₁₃	C₂²	1	2	yes	—
36	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
37	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
41	#14268647	—	4	2	no	—
42	#14268647	—	4	2	no	—
43	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
44	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄₅₂	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
53	#14268630	—	4	2	no	—
54	C₂×#11744751	—	4	2	no	—
55	C₂×#11744751	C₂³	1	2	yes	—
56	C₂×#11744751	C₂³	1	1	yes	characteristic
57	C₂×#11744751	C₂³	1	1	yes	characteristic
58	C₈×D₁₁₃	—	4	2	no	—
59	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
60	#14268626	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
62	#14268624	C₂×C₄	1	2	yes	—
63	#14268624	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
69	M₁₆×C₁₁₃	—	4	2	no	—
70	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
72	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
75	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
76	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₁₁₃	—	2	2	no	—
80	C₂×C₄×C₁₁₃	—	2	2	no	—
81	C₂×C₄×C₁₁₃	—	2	2	no	—
82	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₁₁₃	—	2	4	no	—
87	C₂²⋊D₁₁₃	—	4	2	no	—
88	C₂²⋊D₁₁₃	—	4	2	no	—
89	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
93	D₄₅₂	—	2	2	no	—
94	D₄₅₂	—	4	2	no	—
95	C₄×D₁₁₃	—	2	2	no	—
96	C₄×D₁₁₃	—	2	2	no	—
97	C₄×D₁₁₃	—	2	2	no	—
98	C₄×D₁₁₃	—	2	1	no	—
99	C₄×D₁₁₃	—	2	1	no	—
100	C₄×D₁₁₃	—	2	2	no	—
101	Dic₂₂₆	—	4	2	no	—
102	C₈×C₁₁₃	—	2	1	no	—
103	C₈×C₁₁₃	—	2	2	no	—
104	C₈×C₁₁₃	—	2	1	no	—
105	#11744751	—	2	2	no	—
106	#11744751	—	2	1	no	—
107	#11744751	—	2	1	no	—
108	C₂²×C₁₁₃	—	2	2	no	—
109	C₂²×C₁₁₃	—	2	2	no	—
110	C₂²×C₁₁₃	—	2	2	no	—
111	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₁₃	—	2	1	no	—
113	C₂×D₁₁₃	—	2	2	no	—
114	C₂×D₁₁₃	—	2	1	no	—
115	C₂×D₁₁₃	—	2	1	no	—
116	C₂×D₁₁₃	—	2	2	no	—
117	C₂×D₁₁₃	—	2	2	no	—
118	C₄×C₁₁₃	—	2	2	no	—
119	C₄×C₁₁₃	—	2	1	no	—
120	C₄×C₁₁₃	#131	1	2	yes	—
121	C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₁₁₃	—	2	1	no	—
124	Dic₁₁₃	—	2	2	no	—
125	Dic₁₁₃	—	2	1	no	—
126	Dic₁₁₃	#131	1	2	yes	—
127	C₂×C₁₁₃	—	4	2	no	—
128	C₂×C₁₁₃	#518	1	1	yes	characteristic
129	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
131	D₁₁₃	—	4	1	no	—
132	D₁₁₃	—	4	2	no	—
133	#2979	—	113	1	no	maximal, Sylow
134	C₁₁₃	#2979	1	1	yes	characteristic, complemented, Sylow
135	#532	—	113	1	no	—
136	#436	—	113	2	no	—
137	#436	—	113	1	no	—
138	#436	D₁₁₃	1	1	yes	characteristic, complemented
139	C₈×D₄	—	113	1	no	—
140	C₈×D₄	—	113	2	no	—
141	C₈×D₄	—	113	1	no	—
142	#431	—	113	1	no	—
143	#431	—	113	1	no	—
144	#408	—	113	2	no	—
145	#408	—	113	2	no	—
146	C₂×#41	—	113	2	no	—
147	C₂×M₁₆	—	113	2	no	—
148	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	—
149	C₂²×C₈	—	113	2	no	—
150	C₂²×C₈	—	113	2	no	—
151	C₄⋊Q₈	—	113	1	no	—
152	#127	—	113	1	no	—
153	#124	—	113	2	no	—
154	#121	—	113	1	no	—
155	#121	—	113	1	no	—
156	#121	—	113	2	no	—
157	C₄×D₄	—	113	2	no	—
158	C₄×D₄	—	113	1	no	—
159	C₄×D₄	C₂×D₁₁₃	1	1	yes	characteristic
160	C₂×C₄²	—	113	1	no	—
161	C₄⋊C₈	—	113	1	no	—
162	C₄⋊C₈	—	113	1	no	—
163	C₄⋊C₈	—	113	1	no	—
164	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
165	C₂²⋊C₈	—	113	2	no	—
166	C₂²⋊C₈	—	113	2	no	—
167	C₂²⋊C₈	—	113	1	no	—
168	C₂²⋊C₈	—	113	1	no	—
169	C₂²⋊C₈	—	113	2	no	—
170	C₂²⋊C₈	—	113	2	no	—
171	C₂²⋊C₈	C₂×D₁₁₃	1	2	yes	—
172	C₄×C₈	—	113	2	no	—
173	C₄×C₈	—	113	1	no	—
174	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
175	#41	—	452	2	no	—
176	#41	—	452	2	no	—
177	C₂×Q₈	—	113	2	no	—
178	C₂×D₄	—	113	2	no	—
179	C₂×D₄	—	113	2	no	—
180	C₂×D₄	—	113	2	no	—
181	C₂×D₄	—	113	2	no	—
182	C₂×D₄	—	113	1	no	—
183	C₂×D₄	C₄×D₁₁₃	1	1	yes	characteristic
184	C₂²×C₄	—	113	1	no	—
185	C₂²×C₄	—	113	1	no	—
186	C₂²×C₄	—	113	1	no	—
187	C₂²×C₄	—	113	2	no	—
188	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	452	2	no	—
191	C₂×C₈	—	113	2	no	—
192	C₂×C₈	—	113	1	no	—
193	C₂×C₈	—	113	1	no	—
194	C₂×C₈	—	452	2	no	—
195	C₂×C₈	—	452	2	no	—
196	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
197	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
199	C₄⋊C₄	—	113	2	no	—
200	C₄⋊C₄	—	113	1	no	—
201	C₄⋊C₄	C₄×D₁₁₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	113	2	no	—
203	C₂²⋊C₄	—	113	2	no	—
204	C₂²⋊C₄	—	113	2	no	—
205	C₂²⋊C₄	C₄×D₁₁₃	1	2	yes	—
206	C₄²	—	113	1	no	—
207	C₄²	—	113	1	no	—
208	C₄²	—	113	1	no	—
209	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
210	C₂³	—	113	2	no	—
211	C₂³	—	113	1	no	—
212	C₂³	C₂×C₄×D₁₁₃	1	2	yes	—
213	Q₈	—	452	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	452	2	no	—
216	D₄	—	452	2	no	—
217	D₄	—	452	2	no	—
218	D₄	—	226	4	no	—
219	D₄	—	226	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	113	1	no	—
223	C₂×C₄	—	113	1	no	—
224	C₂×C₄	—	113	1	no	—
225	C₂×C₄	—	113	2	no	—
226	C₂×C₄	—	226	2	no	—
227	C₂×C₄	—	226	2	no	—
228	C₂×C₄	—	226	2	no	—
229	C₂×C₄	—	226	1	no	—
230	C₂×C₄	—	226	2	no	—
231	C₂×C₄	—	226	1	no	—
232	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	226	1	no	—
240	C₈	—	226	2	no	—
241	C₈	—	226	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	226	1	no	—
246	C₂²	—	226	2	no	—
247	C₂²	—	226	1	no	—
248	C₂²	—	226	2	no	—
249	C₂²	—	226	1	no	—
250	C₂²	—	226	2	no	—
251	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	113	2	no	—
255	C₄	—	226	1	no	—
256	C₄	—	226	2	no	—
257	C₄	—	226	1	no	—
258	C₄	#16813630	1	2	yes	—
259	C₄	#16813667	1	1	yes	characteristic
260	C₄	#16813668	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	452	1	no	—
263	C₂	—	452	2	no	—
264	C₂	#20299070	1	1	yes	characteristic
265	C₂	#131×D₁₁₃	1	1	yes	characteristic
266	C₂	C₂×#16813630	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#436×C₁₁₃	C₂
prime radical	#20299145	C₂
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#436	D₁₁₃
hypocenter	C₁₁₃	#2979
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #20298675	C₂	semidirect
3. #20298674	C₂	semidirect
4. C₈×D₄₅₂	C₂	semidirect
5. D₄×#11744751	C₂	semidirect
6. #20298661	C₂	semidirect
7. #20298650	C₂	semidirect
8. #20298649	C₂	semidirect
9. #20298566	C₂	semidirect
10. #20298563	C₂	semidirect
11. #436×C₁₁₃	C₂	semidirect
134. C₁₁₃	#2979	semidirect
138. #436	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299145	2
#20298675	2
#20298674	2
C₈×D₄₅₂	2
D₄×#11744751	2
#20298661	2
#20298650	2
#20298649	2
#20298566	2
#20298563	2
#436×C₁₁₃	2
#2979	113