Subgroups — #28393475

Order 14464.

776 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813584	C₂	1	1	yes	maximal, characteristic
3	C₂×#16813584	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₄×D₁₁₃	C₂	1	2	yes	maximal, complemented
5	#20298497	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298472	C₂	1	2	yes	maximal, complemented
7	#20298469	C₂	1	2	yes	maximal, complemented
8	#20298467	C₂	1	1	yes	maximal, characteristic, complemented
9	#20298457	C₂	1	2	yes	maximal, complemented
10	C₄×#14268626	C₂	1	1	yes	maximal, characteristic, complemented
11	#389×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
16	C₄×D₄₅₂	—	2	4	no	—
17	#16813584	—	2	4	no	—
18	#16813584	C₄	1	8	yes	complemented
19	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×#14268626	C₂²	1	2	yes	—
22	C₂×#14268626	C₂²	1	1	yes	characteristic
23	C₂×#14268626	C₂²	1	2	yes	—
24	C₂×#14268626	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₁₃	—	2	8	no	—
26	C₂×#14268624	C₂²	1	2	yes	—
27	C₂×#14268624	C₂²	1	2	yes	—
28	C₂×#14268624	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
31	#16813529	C₂²	1	2	yes	—
32	#16813529	C₂²	1	1	yes	characteristic
33	#16813529	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
38	#95×C₁₁₃	C₂²	1	2	yes	—
39	C₂³×D₁₁₃	C₂³	1	2	yes	—
40	C₂×D₄₅₂	C₂×C₄	1	4	yes	—
41	C₂×D₄₅₂	D₄	1	4	yes	—
42	C₂×D₄₅₂	Q₈	1	4	yes	—
43	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₁₃	—	2	4	no	—
49	C₂×C₄×D₁₁₃	—	2	4	no	—
50	C₂×C₄×D₁₁₃	—	2	4	no	—
51	C₂×C₄×D₁₁₃	—	2	4	no	—
52	C₂×C₄×D₁₁₃	—	2	4	no	—
53	C₂×C₄×D₁₁₃	—	2	4	no	—
54	C₂×C₄×D₁₁₃	—	4	2	no	—
55	C₂×C₄×D₁₁₃	—	4	2	no	—
56	C₂×C₄×D₁₁₃	C₂×C₄	1	8	yes	complemented
57	C₄⋊Dic₁₁₃	—	2	2	no	—
58	C₄⋊Dic₁₁₃	—	2	4	no	—
59	#14268626	—	2	4	no	—
60	#14268626	—	2	2	no	—
61	#14268626	—	4	2	no	—
62	#14268626	D₄	1	4	yes	complemented
63	#14268624	—	2	4	no	—
64	#14268624	—	2	4	no	—
65	#14268624	C₂×C₄	1	8	yes	complemented
66	C₄×Dic₁₁₃	—	2	2	no	—
67	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	complemented
68	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₁₃	—	2	4	no	—
74	C₄⋊C₄×C₁₁₃	—	2	2	no	—
75	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
76	C₄²×C₁₁₃	—	2	2	no	—
77	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²×D₁₁₃	—	2	2	no	—
79	C₂²×D₁₁₃	—	2	2	no	—
80	C₂²×D₁₁₃	—	2	2	no	—
81	C₂²×D₁₁₃	—	2	2	no	—
82	C₂²×D₁₁₃	C₂²×C₄	1	4	yes	—
83	C₂²×D₁₁₃	C₂×D₄	1	4	yes	—
84	C₂²×D₁₁₃	C₂×Q₈	1	4	yes	—
85	C₂×C₄×C₁₁₃	—	2	1	no	—
86	C₂×C₄×C₁₁₃	—	2	2	no	—
87	C₂×C₄×C₁₁₃	—	2	2	no	—
88	C₂×C₄×C₁₁₃	—	2	1	no	—
89	C₂×C₄×C₁₁₃	—	2	1	no	—
90	C₂×C₄×C₁₁₃	—	2	2	no	—
91	C₂×C₄×C₁₁₃	—	2	2	no	—
92	C₂×C₄×C₁₁₃	—	2	1	no	—
93	C₂×C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₄×C₁₁₃	—	2	1	no	—
95	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
100	C₂×Dic₁₁₃	—	2	1	no	—
101	C₂×Dic₁₁₃	—	2	1	no	—
102	C₂×Dic₁₁₃	—	2	1	no	—
103	C₂×Dic₁₁₃	—	2	1	no	—
104	C₂×Dic₁₁₃	—	2	1	no	—
105	C₂×Dic₁₁₃	—	2	2	no	—
106	C₂×Dic₁₁₃	—	2	2	no	—
107	C₂×Dic₁₁₃	—	2	1	no	—
108	C₂×Dic₁₁₃	—	2	2	no	—
109	C₂×Dic₁₁₃	—	2	1	no	—
110	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
111	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
112	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
113	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
114	C₂×Dic₁₁₃	#41	1	2	yes	—
115	D₄₅₂	C₄⋊C₄	1	16	yes	complemented
116	C₄×D₁₁₃	—	2	16	no	—
117	C₄×D₁₁₃	—	4	8	no	—
118	C₄×D₁₁₃	—	4	8	no	—
119	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂×D₁₁₃	—	2	4	no	—
127	C₂×D₁₁₃	—	2	4	no	—
128	C₂×D₁₁₃	—	2	4	no	—
129	C₂×D₁₁₃	—	2	4	no	—
130	C₂×D₁₁₃	—	2	4	no	—
131	C₂×D₁₁₃	—	2	4	no	—
132	C₂×D₁₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
133	C₄×C₁₁₃	—	2	4	no	—
134	C₄×C₁₁₃	—	2	2	no	—
135	C₄×C₁₁₃	—	4	2	no	—
136	C₄×C₁₁₃	—	4	1	no	—
137	C₄×C₁₁₃	C₂×C₄⋊C₄	1	4	yes	complemented
138	Dic₁₁₃	—	2	2	no	—
139	Dic₁₁₃	—	2	2	no	—
140	Dic₁₁₃	—	4	2	no	—
141	Dic₁₁₃	—	4	1	no	—
142	Dic₁₁₃	C₄×D₄	1	4	yes	complemented
143	C₂×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₁₃	#520	1	1	yes	characteristic
146	C₂×C₁₁₃	D₄²	1	1	yes	characteristic
147	C₂×C₁₁₃	#547	1	1	yes	characteristic
148	C₂×C₁₁₃	D₄×Q₈	1	1	yes	characteristic
149	C₂×C₁₁₃	#554	1	1	yes	characteristic
150	D₁₁₃	—	2	8	no	—
151	D₄×C₄⋊C₄	—	113	1	no	maximal, Sylow
152	C₁₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄×D₄	—	113	1	no	—
154	C₂×C₄×D₄	—	113	1	no	—
155	C₂×C₄×D₄	—	113	1	no	—
156	C₂²×C₄⋊C₄	—	113	2	no	—
157	#389	—	113	1	no	—
158	#389	D₁₁₃	1	1	yes	characteristic, complemented
159	#385	—	113	2	no	—
160	#385	—	113	2	no	—
161	#384	—	113	1	no	—
162	#380	—	113	2	no	—
163	C₄×C₄⋊C₄	—	113	1	no	—
164	C₂²×D₄	—	113	1	no	—
165	C₂³×C₄	—	113	2	no	—
166	C₂³×C₄	—	113	2	no	—
167	C₂³×C₄	—	113	2	no	—
168	C₄×D₄	—	113	8	no	—
169	C₄×D₄	—	226	4	no	—
170	C₄×D₄	—	226	4	no	—
171	C₂×C₄⋊C₄	—	113	1	no	—
172	C₂×C₄⋊C₄	—	113	2	no	—
173	C₂×C₄⋊C₄	—	113	2	no	—
174	C₂×C₄⋊C₄	—	113	2	no	—
175	C₂×C₄⋊C₄	—	113	1	no	—
176	C₂×C₄⋊C₄	—	113	1	no	—
177	C₂×C₄⋊C₄	—	226	8	no	—
178	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	—	113	2	no	—
182	C₂×C₂²⋊C₄	—	113	2	no	—
183	C₂×C₂²⋊C₄	—	113	2	no	—
184	C₂×C₄²	—	113	1	no	—
185	C₂×C₄²	—	113	1	no	—
186	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
187	#95	—	113	1	no	—
188	#95	—	113	1	no	—
189	#95	—	113	2	no	—
190	#95	C₂×D₁₁₃	1	2	yes	—
191	C₂⁴	—	113	2	no	—
192	C₂×D₄	—	113	4	no	—
193	C₂×D₄	—	113	4	no	—
194	C₂×D₄	—	113	4	no	—
195	C₂²×C₄	—	113	1	no	—
196	C₂²×C₄	—	113	1	no	—
197	C₂²×C₄	—	113	1	no	—
198	C₂²×C₄	—	113	8	no	—
199	C₂²×C₄	—	113	1	no	—
200	C₂²×C₄	—	113	2	no	—
201	C₂²×C₄	—	452	2	no	—
202	C₂²×C₄	—	452	2	no	—
203	C₂²×C₄	—	226	4	no	—
204	C₂²×C₄	—	226	4	no	—
205	C₂²×C₄	—	226	4	no	—
206	C₂²×C₄	—	226	4	no	—
207	C₂²×C₄	—	226	4	no	—
208	C₂²×C₄	—	226	4	no	—
209	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
211	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
212	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	113	4	no	—
217	C₄⋊C₄	—	452	2	no	—
218	C₄⋊C₄	—	226	4	no	—
219	C₄⋊C₄	—	226	4	no	—
220	C₄⋊C₄	—	226	2	no	—
221	C₄⋊C₄	—	226	2	no	—
222	C₄⋊C₄	C₄×D₁₁₃	1	4	yes	complemented
223	C₂²⋊C₄	—	113	8	no	—
224	C₂²⋊C₄	—	226	4	no	—
225	C₂²⋊C₄	—	226	4	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	113	4	no	—
228	C₄²	—	226	2	no	—
229	C₂³	—	113	4	no	—
230	C₂³	—	113	4	no	—
231	C₂³	—	113	4	no	—
232	C₂³	—	226	2	no	—
233	C₂³	—	226	2	no	—
234	C₂³	—	226	2	no	—
235	C₂³	—	226	2	no	—
236	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
237	D₄	—	113	16	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	113	2	no	—
249	C₂×C₄	—	113	2	no	—
250	C₂×C₄	—	113	2	no	—
251	C₂×C₄	—	113	2	no	—
252	C₂×C₄	—	113	2	no	—
253	C₂×C₄	—	452	8	no	—
254	C₂×C₄	—	452	8	no	—
255	C₂×C₄	—	226	16	no	—
256	C₂×C₄	—	226	1	no	—
257	C₂×C₄	—	226	2	no	—
258	C₂×C₄	—	226	2	no	—
259	C₂×C₄	—	226	1	no	—
260	C₂×C₄	—	226	2	no	—
261	C₂×C₄	—	226	1	no	—
262	C₂×C₄	—	226	1	no	—
263	C₂×C₄	—	226	1	no	—
264	C₂×C₄	—	226	1	no	—
265	C₂×C₄	—	226	1	no	—
266	C₂×C₄	C₂×C₄×D₁₁₃	1	4	yes	—
267	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
268	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
269	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
270	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
271	C₂²	—	113	8	no	—
272	C₂²	—	226	4	no	—
273	C₂²	—	226	4	no	—
274	C₂²	—	226	4	no	—
275	C₂²	—	226	4	no	—
276	C₂²	—	226	4	no	—
277	C₂²	—	226	4	no	—
278	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
279	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
280	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
281	C₂²	C₂×Q₈×D₁₁₃	1	1	yes	characteristic
282	C₂²	#41×D₁₁₃	1	1	yes	characteristic
283	C₂²	#16813668	1	1	yes	characteristic
284	C₂²	#16813670	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	113	4	no	—
290	C₄	—	452	2	no	—
291	C₄	—	452	1	no	—
292	C₄	—	226	2	no	—
293	C₄	—	226	2	no	—
294	C₄	C₄⋊C₄×D₁₁₃	1	4	yes	complemented
295	C₂	—	226	8	no	—
296	C₂	C₂×C₄⋊C₄×D₁₁₃	1	1	yes	characteristic
297	C₂	C₄×D₄×D₁₁₃	1	1	yes	characteristic
298	C₂	#20299080	1	1	yes	characteristic
299	C₂	#20299232	1	1	yes	characteristic
300	C₂	#20299250	1	1	yes	characteristic
301	C₂	#20299263	1	1	yes	characteristic
302	C₂	#20299304	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Fitting	#389×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×D₄₅₂	C₂²
hypercenter	#389	D₁₁₃
hypocenter	C₁₁₃	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
4. C₂×C₄⋊C₄×D₁₁₃	C₂	semidirect
5. #20298497	C₂	semidirect
6. #20298472	C₂	semidirect
7. #20298469	C₂	semidirect
8. #20298467	C₂	semidirect
9. #20298457	C₂	semidirect
10. C₄×#14268626	C₂	semidirect
11. #389×C₁₁₃	C₂	semidirect
18. #16813584	C₄	semidirect
56. C₂×C₄×D₁₁₃	C₂×C₄	semidirect
62. #14268626	D₄	semidirect
65. #14268624	C₂×C₄	semidirect
67. C₄×Dic₁₁₃	C₂×C₄	semidirect
75. C₄⋊C₄×C₁₁₃	C₂×C₄	semidirect
115. D₄₅₂	C₄⋊C₄	semidirect
132. C₂×D₁₁₃	C₂×C₄⋊C₄	semidirect
137. C₄×C₁₁₃	C₂×C₄⋊C₄	semidirect
142. Dic₁₁₃	C₄×D₄	semidirect
152. C₁₁₃	D₄×C₄⋊C₄	semidirect
158. #389	D₁₁₃	semidirect
222. C₄⋊C₄	C₄×D₁₁₃	semidirect
294. C₄	C₄⋊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×C₄×D₄₅₂	2
C₂×#16813584	2
C₂×C₄⋊C₄×D₁₁₃	2
#20298497	2
#20298472	2
#20298469	2
#20298467	2
#20298457	2
C₄×#14268626	2
#389×C₁₁₃	2
D₄×C₄⋊C₄	113