Subgroups — #28407464

Order 14464.

402 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	#20299633	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299632	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299630	C₂	1	2	yes	maximal, complemented
4	#20299627	C₂	1	2	yes	maximal, complemented
5	#20299626	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299619	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299614	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299572	C₂	1	2	yes	maximal, complemented
10	#20299571	C₂	1	2	yes	maximal, complemented
11	#20299143	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268657	C₂²	1	2	yes	—
13	C₂×#14268656	C₂²	1	2	yes	—
14	C₂×#14268655	C₂²	1	2	yes	—
15	C₂²×#11744753	C₂²	1	2	yes	—
16	C₄⋊#11744753	C₂²	1	1	yes	characteristic
17	C₄⋊#11744753	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
23	C₂²⋊#11744753	C₂²	1	2	yes	—
24	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
25	C₂²⋊#11744753	C₂²	1	2	yes	—
26	C₂²⋊#11744753	C₂²	1	2	yes	—
27	#16813677	C₂²	1	2	yes	—
28	C₄×#11744753	C₂²	1	1	yes	characteristic
29	C₄×#11744753	C₂²	1	1	yes	characteristic
30	C₂×#14268646	C₂²	1	2	yes	—
31	#16813621	C₄	1	1	yes	characteristic
32	#16813618	C₄	1	1	yes	characteristic
33	#16813608	C₄	1	2	yes	—
34	#16813596	C₄	1	1	yes	characteristic
35	#16813594	C₄	1	2	yes	—
36	D₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
37	#16813584	C₂²	1	1	yes	characteristic
38	#16813582	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
40	#121×C₁₁₃	C₄	1	1	yes	characteristic
41	#14268657	—	4	2	no	—
42	#14268656	—	4	2	no	—
43	#14268655	—	4	2	no	—
44	C₂×#11744753	—	4	2	no	—
45	C₂×#11744753	C₂³	1	1	yes	characteristic
46	C₂×#11744753	C₂³	1	1	yes	characteristic
47	C₂×#11744753	C₂³	1	1	yes	characteristic
48	C₂×#11744753	C₂³	1	1	yes	characteristic
49	C₂×#11744753	C₂³	1	2	yes	—
50	C₂×#11744753	C₂³	1	2	yes	—
51	#14268646	—	4	2	no	—
52	#14268646	—	4	2	no	—
53	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
54	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
58	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
61	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
62	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
63	#14268626	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
65	#14268624	C₂×C₄	1	2	yes	—
66	#14268624	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
70	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
72	C₂×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
73	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
76	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
77	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
79	D₄×C₁₁₃	—	2	2	no	—
80	D₄×C₁₁₃	—	4	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	C₂²⋊D₁₁₃	—	4	2	no	—
86	C₂²⋊D₁₁₃	—	4	2	no	—
87	C₂²⋊D₁₁₃	—	4	2	no	—
88	C₂×Dic₁₁₃	—	2	1	no	—
89	C₂×Dic₁₁₃	—	2	1	no	—
90	C₂×Dic₁₁₃	—	2	2	no	—
91	C₂×Dic₁₁₃	—	2	2	no	—
92	C₂×Dic₁₁₃	—	4	1	no	—
93	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
98	D₄₅₂	—	2	2	no	—
99	C₄×D₁₁₃	—	2	2	no	—
100	C₄×D₁₁₃	—	2	2	no	—
101	Dic₂₂₆	—	4	2	no	—
102	#11744753	—	2	1	no	—
103	#11744753	—	2	2	no	—
104	#11744753	—	2	1	no	—
105	#11744753	—	2	2	no	—
106	#11744753	—	2	1	no	—
107	#11744753	—	2	1	no	—
108	C₂²×C₁₁₃	—	2	1	no	—
109	C₂²×C₁₁₃	—	2	2	no	—
110	C₂²×C₁₁₃	—	2	1	no	—
111	C₂²×C₁₁₃	—	2	2	no	—
112	C₂²×C₁₁₃	—	2	1	no	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₁₃	—	2	2	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	Dic₁₁₃	—	2	1	no	—
122	Dic₁₁₃	—	2	2	no	—
123	Dic₁₁₃	—	2	1	no	—
124	Dic₁₁₃	—	2	1	no	—
125	Dic₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
126	Dic₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₁₁₃	—	4	1	no	—
128	C₂×C₁₁₃	—	4	2	no	—
129	C₂×C₁₁₃	#518	1	1	yes	characteristic
130	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
131	C₂×C₁₁₃	#568	1	1	yes	characteristic
132	D₁₁₃	—	4	2	no	—
133	#2981	—	113	1	no	maximal, Sylow
134	C₁₁₃	#2981	1	1	yes	characteristic, complemented, Sylow
135	#532	—	113	1	no	—
136	#436	—	113	1	no	—
137	#436	—	113	1	no	—
138	#435	—	113	2	no	—
139	#435	—	113	2	no	—
140	C₈×D₄	—	113	1	no	—
141	C₈×D₄	—	113	1	no	—
142	#432	—	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	—	113	2	no	—
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	2	no	—
155	#121	—	113	1	no	—
156	#121	#90673	1	1	yes	characteristic
157	C₄×D₄	—	113	1	no	—
158	C₄×D₄	—	113	1	no	—
159	C₄×D₄	—	113	2	no	—
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	—	113	1	no	—
165	C₂²⋊C₈	—	113	1	no	—
166	C₂²⋊C₈	—	113	1	no	—
167	C₂²⋊C₈	—	113	2	no	—
168	C₂²⋊C₈	—	113	2	no	—
169	C₂²⋊C₈	—	113	2	no	—
170	C₂²⋊C₈	—	113	2	no	—
171	C₂²⋊C₈	—	113	2	no	—
172	#97	—	113	2	no	—
173	C₄×C₈	—	113	1	no	—
174	C₄×C₈	—	113	1	no	—
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	1	no	—
182	C₂×D₄	C₂×#90673	1	1	yes	characteristic
183	C₂×D₄	C₂×#90673	1	2	yes	—
184	C₂²×C₄	—	113	1	no	—
185	C₂²×C₄	—	113	1	no	—
186	C₂²×C₄	—	113	2	no	—
187	C₂²×C₄	—	113	2	no	—
188	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
189	M₁₆	—	452	2	no	—
190	M₁₆	—	452	2	no	—
191	C₂×C₈	—	113	1	no	—
192	C₂×C₈	—	113	1	no	—
193	C₂×C₈	—	113	1	no	—
194	C₂×C₈	—	113	1	no	—
195	C₂×C₈	—	113	2	no	—
196	C₂×C₈	—	113	2	no	—
197	C₂×C₈	—	452	2	no	—
198	C₂×C₈	—	452	2	no	—
199	C₄⋊C₄	—	113	2	no	—
200	C₄⋊C₄	—	113	1	no	—
201	C₄⋊C₄	C₂×#90673	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	C₂×#90673	1	2	yes	—
206	C₄²	—	113	1	no	—
207	C₄²	—	113	1	no	—
208	C₄²	—	113	1	no	—
209	C₄²	—	113	1	no	—
210	C₂³	—	113	2	no	—
211	C₂³	C₂²×#90673	1	2	yes	—
212	C₂³	C₂²×#90673	1	1	yes	characteristic
213	Q₈	—	452	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	452	2	no	—
217	D₄	—	452	2	no	—
218	D₄	—	452	2	no	—
219	D₄	—	226	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	113	1	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	—	452	1	no	—
227	C₂×C₄	—	226	2	no	—
228	C₂×C₄	—	226	2	no	—
229	C₂×C₄	—	226	2	no	—
230	C₂×C₄	—	226	2	no	—
231	C₂×C₄	—	226	1	no	—
232	C₂×C₄	—	226	1	no	—
233	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
234	C₂×C₄	C₂²×#90673	1	2	yes	—
235	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
236	C₈	—	226	1	no	—
237	C₈	—	226	1	no	—
238	C₈	—	226	2	no	—
239	C₈	—	226	2	no	—
240	C₈	—	226	1	no	—
241	C₈	—	226	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	226	2	no	—
249	C₂²	—	226	2	no	—
250	C₂²	—	226	2	no	—
251	C₂²	C₂³×#90673	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	113	1	no	—
255	C₄	—	113	1	no	—
256	C₄	—	226	1	no	—
257	C₄	—	226	2	no	—
258	C₄	—	226	1	no	—
259	C₄	—	226	1	no	—
260	C₄	#16813709	1	2	yes	—
261	C₂	—	4	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	452	2	no	—
264	C₂	#20299665	1	1	yes	characteristic
265	C₂	C₂×#16813709	1	1	yes	characteristic
266	C₂	#20299677	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#121×C₁₁₃	C₄
prime radical	#20299143	C₂
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	#121	#90673
hypocenter	C₁₁₃	#2981
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #20299633	C₂	semidirect
2. #20299632	C₂	semidirect
3. #20299630	C₂	semidirect
4. #20299627	C₂	semidirect
5. #20299626	C₂	semidirect
6. D₄×#11744753	C₂	semidirect
7. #20299619	C₂	semidirect
8. #20299614	C₂	semidirect
9. #20299572	C₂	semidirect
10. #20299571	C₂	semidirect
134. C₁₁₃	#2981	semidirect

Maximal subgroups

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

subgroup	index
#20299633	2
#20299632	2
#20299630	2
#20299627	2
#20299626	2
D₄×#11744753	2
#20299619	2
#20299614	2
#20299572	2
#20299571	2
#20299143	2
#2981	113