Subgroups — #28407269

Order 14464.

392 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299623	C₂	1	2	yes	maximal, complemented
2	#20299622	C₂	1	2	yes	maximal, complemented
3	#20299620	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299619	C₂	1	1	yes	maximal, characteristic, complemented
5	#20299617	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299614	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299601	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊#11744753	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299571	C₂	1	2	yes	maximal
10	#20299562	C₂	1	2	yes	maximal, complemented
11	C₄×#14268646	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268656	C₂²	1	2	yes	—
13	C₂²×#11744753	C₂²	1	2	yes	—
14	#16813689	C₂²	1	2	yes	—
15	C₄⋊#11744753	C₂²	1	1	yes	characteristic
16	C₄⋊#11744753	C₂²	1	1	yes	characteristic
17	C₄⋊#11744753	C₂²	1	2	yes	—
18	C₄⋊#11744753	C₂²	1	1	yes	characteristic
19	C₄⋊#11744753	C₂²	1	1	yes	characteristic
20	#16813679	C₂²	1	2	yes	—
21	#16813679	C₂²	1	2	yes	—
22	C₂²⋊#11744753	C₂²	1	2	yes	—
23	C₂²⋊#11744753	C₂²	1	2	yes	—
24	#16813677	C₂²	1	2	yes	—
25	#16813677	C₂²	1	2	yes	—
26	C₄×#11744753	C₂²	1	2	yes	—
27	C₄×#11744753	C₂²	1	2	yes	—
28	C₂×#14268646	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₂₆	C₄	1	1	yes	characteristic
30	#16813587	C₄	1	1	yes	characteristic
31	D₄×Dic₁₁₃	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
34	#16813582	C₄	1	1	yes	characteristic
35	#16813582	C₄	1	1	yes	characteristic
36	#16813580	C₂²	1	1	yes	characteristic
37	#16813578	C₂²	1	1	yes	characteristic
38	#16813578	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
41	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₁₁₃	C₄	1	1	yes	characteristic
43	#14268656	—	2	4	no	—
44	C₂×#11744753	—	2	4	no	—
45	C₂×#11744753	C₂³	1	2	yes	—
46	C₂×#11744753	C₂³	1	2	yes	—
47	C₂×#11744753	C₂³	1	2	yes	—
48	C₂×#11744753	C₂³	1	2	yes	—
49	#14268646	—	2	2	no	—
50	#14268646	—	2	2	no	—
51	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
62	#14268626	C₂×C₄	1	1	yes	characteristic
63	#14268626	C₂×C₄	1	1	yes	characteristic
64	#14268626	C₂×C₄	1	1	yes	characteristic
65	#14268626	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
68	#14268624	C₂×C₄	1	1	yes	characteristic
69	#14268624	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
85	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
86	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
89	D₄×C₁₁₃	—	2	2	no	—
90	C₂×C₄×C₁₁₃	—	2	2	no	—
91	C₂×C₄×C₁₁₃	—	2	2	no	—
92	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊D₁₁₃	—	2	2	no	—
98	C₂²⋊D₁₁₃	—	2	2	no	—
99	C₂×Dic₁₁₃	—	2	1	no	—
100	C₂×Dic₁₁₃	—	2	1	no	—
101	C₂×Dic₁₁₃	—	4	1	no	—
102	C₂×Dic₁₁₃	—	4	1	no	—
103	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₁₁₃	#41	1	2	yes	—
110	C₂×Dic₁₁₃	#41	1	2	yes	—
111	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
112	C₄×D₁₁₃	—	2	2	no	—
113	C₄×D₁₁₃	—	2	1	no	—
114	C₄×D₁₁₃	—	2	1	no	—
115	C₄×D₁₁₃	—	4	1	no	—
116	Dic₂₂₆	—	2	2	no	—
117	#11744753	—	2	2	no	—
118	#11744753	—	2	2	no	—
119	#11744753	—	2	2	no	—
120	#11744753	—	2	2	no	—
121	C₂²×C₁₁₃	—	2	1	no	—
122	C₂²×C₁₁₃	—	2	1	no	—
123	C₂²×C₁₁₃	—	2	1	no	—
124	C₂²×C₁₁₃	—	2	1	no	—
125	C₂²×C₁₁₃	—	2	1	no	—
126	C₂²×C₁₁₃	#117	1	2	yes	—
127	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×D₁₁₃	—	2	1	no	—
129	C₂×D₁₁₃	—	2	1	no	—
130	C₂×D₁₁₃	—	2	1	no	—
131	C₄×C₁₁₃	—	2	1	no	—
132	C₄×C₁₁₃	—	2	1	no	—
133	C₄×C₁₁₃	—	2	1	no	—
134	C₄×C₁₁₃	—	2	1	no	—
135	C₄×C₁₁₃	#131	1	2	yes	—
136	Dic₁₁₃	—	2	1	no	—
137	Dic₁₁₃	—	2	1	no	—
138	Dic₁₁₃	—	2	1	no	—
139	Dic₁₁₃	—	2	1	no	—
140	Dic₁₁₃	—	2	1	no	—
141	Dic₁₁₃	#117	1	2	yes	—
142	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
143	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₁₁₃	—	2	2	no	—
145	C₂×C₁₁₃	—	4	1	no	—
146	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
147	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
148	C₂×C₁₁₃	#568	1	1	yes	characteristic
149	D₁₁₃	—	4	1	no	—
150	#2925	—	113	1	no	maximal, Sylow
151	C₁₁₃	#2925	1	1	yes	characteristic, complemented, Sylow
152	C₄×#41	—	113	1	no	—
153	#433	—	113	2	no	—
154	#433	—	113	2	no	—
155	#432	—	113	1	no	—
156	#432	—	113	1	no	—
157	#431	—	113	1	no	—
158	#431	—	113	1	no	—
159	#423	—	113	1	no	—
160	C₂×C₄⋊C₈	—	113	1	no	—
161	#408	—	113	2	no	—
162	#405	—	113	2	no	—
163	C₂×#41	—	113	1	no	—
164	C₂×M₁₆	—	113	2	no	—
165	C₂²×C₈	—	113	2	no	—
166	C₄×Q₈	—	113	1	no	—
167	C₄×Q₈	—	113	1	no	—
168	C₄×D₄	—	113	1	no	—
169	C₄×D₄	—	113	1	no	—
170	C₄×D₄	—	113	1	no	—
171	C₄×D₄	—	113	1	no	—
172	C₄×D₄	—	113	1	no	—
173	C₄×D₄	#90673	1	1	yes	characteristic
174	#117	—	113	1	no	—
175	#117	—	113	1	no	—
176	#117	—	113	1	no	—
177	C₂×C₄²	—	113	1	no	—
178	C₂×C₄²	—	113	1	no	—
179	C₂×C₄²	—	113	1	no	—
180	C₄⋊C₈	—	113	1	no	—
181	C₄⋊C₈	—	113	1	no	—
182	C₄⋊C₈	—	113	2	no	—
183	C₄⋊C₈	—	113	1	no	—
184	C₄⋊C₈	—	113	1	no	—
185	C₄⋊C₈	—	113	2	no	—
186	C₂²⋊C₈	—	113	2	no	—
187	C₂²⋊C₈	—	113	2	no	—
188	C₂²⋊C₈	—	113	2	no	—
189	C₂²⋊C₈	—	113	2	no	—
190	#97	—	113	2	no	—
191	#97	—	113	2	no	—
192	C₄×C₈	—	113	2	no	—
193	C₄×C₈	—	113	2	no	—
194	#41	—	226	2	no	—
195	#41	—	226	2	no	—
196	C₂×Q₈	—	113	1	no	—
197	C₂×D₄	—	113	1	no	—
198	C₂×D₄	—	113	1	no	—
199	C₂×D₄	C₂×#90673	1	1	yes	characteristic
200	C₂²×C₄	—	113	1	no	—
201	C₂²×C₄	—	113	1	no	—
202	C₂²×C₄	—	113	1	no	—
203	C₂²×C₄	—	113	1	no	—
204	C₂²×C₄	—	113	1	no	—
205	C₂²×C₄	—	113	1	no	—
206	C₂²×C₄	—	113	1	no	—
207	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
208	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
209	M₁₆	—	226	4	no	—
210	C₂×C₈	—	113	2	no	—
211	C₂×C₈	—	113	2	no	—
212	C₂×C₈	—	113	2	no	—
213	C₂×C₈	—	113	2	no	—
214	C₂×C₈	—	226	4	no	—
215	C₄⋊C₄	—	113	1	no	—
216	C₄⋊C₄	—	113	1	no	—
217	C₄⋊C₄	—	113	1	no	—
218	C₄⋊C₄	—	113	1	no	—
219	C₄⋊C₄	—	113	1	no	—
220	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
221	C₂²⋊C₄	—	113	1	no	—
222	C₂²⋊C₄	—	113	1	no	—
223	C₂²⋊C₄	—	113	1	no	—
224	C₂²⋊C₄	—	113	1	no	—
225	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
227	C₄²	—	113	1	no	—
228	C₄²	—	113	1	no	—
229	C₄²	—	113	1	no	—
230	C₄²	—	113	1	no	—
231	C₄²	—	113	1	no	—
232	C₄²	—	113	1	no	—
233	C₄²	—	113	1	no	—
234	C₄²	—	113	1	no	—
235	C₄²	—	113	1	no	—
236	C₄²	C₂×#90673	1	1	yes	characteristic
237	C₂³	—	113	1	no	—
238	C₂³	C₂²×#90673	1	1	yes	characteristic
239	C₂³	C₂²×#90673	1	1	yes	characteristic
240	Q₈	—	226	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	226	2	no	—
243	D₄	—	226	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	113	2	no	—
247	C₂×C₄	—	113	2	no	—
248	C₂×C₄	—	113	1	no	—
249	C₂×C₄	—	113	1	no	—
250	C₂×C₄	—	113	1	no	—
251	C₂×C₄	—	113	1	no	—
252	C₂×C₄	—	113	1	no	—
253	C₂×C₄	—	113	1	no	—
254	C₂×C₄	—	113	1	no	—
255	C₂×C₄	—	452	1	no	—
256	C₂×C₄	—	452	1	no	—
257	C₂×C₄	—	452	1	no	—
258	C₂×C₄	—	226	1	no	—
259	C₂×C₄	—	226	1	no	—
260	C₂×C₄	—	226	1	no	—
261	C₂×C₄	—	226	1	no	—
262	C₂×C₄	—	226	2	no	—
263	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
264	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
265	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
266	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
267	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
268	C₈	—	226	2	no	—
269	C₈	—	226	2	no	—
270	C₈	—	226	2	no	—
271	C₈	—	226	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	226	1	no	—
278	C₂²	—	226	1	no	—
279	C₂²	—	226	1	no	—
280	C₂²	#16813701	1	2	yes	—
281	C₂²	C₂³×#90673	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	113	1	no	—
287	C₄	—	113	1	no	—
288	C₄	—	113	2	no	—
289	C₄	—	226	1	no	—
290	C₄	—	226	1	no	—
291	C₄	—	226	1	no	—
292	C₄	—	226	1	no	—
293	C₄	—	226	1	no	—
294	C₄	#16813709	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	452	1	no	—
298	C₂	C₂×#16813701	1	1	yes	characteristic
299	C₂	C₂×#16813709	1	1	yes	characteristic
300	C₂	#20299677	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₄×D₄×C₁₁₃	C₄
prime radical	C₂×#14268646	C₂²
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₄×D₄	#90673
hypocenter	C₁₁₃	#2925
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20299623	C₂	semidirect
2. #20299622	C₂	semidirect
3. #20299620	C₂	semidirect
4. #20299619	C₂	semidirect
5. #20299617	C₂	semidirect
6. #20299614	C₂	semidirect
7. #20299601	C₂	semidirect
8. C₂×C₄⋊#11744753	C₂	semidirect
10. #20299562	C₂	semidirect
151. C₁₁₃	#2925	semidirect

Maximal subgroups

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

subgroup	index
#20299623	2
#20299622	2
#20299620	2
#20299619	2
#20299617	2
#20299614	2
#20299601	2
C₂×C₄⋊#11744753	2
#20299571	2
#20299562	2
C₄×#14268646	2
#2925	113