Subgroups — #28407297

Order 14464.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813709	C₂	1	2	yes	maximal, complemented
2	#20299633	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299632	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299631	C₂	1	2	yes	maximal, complemented
5	#20299626	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299625	C₂	1	2	yes	maximal, complemented
7	D₄×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299601	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299571	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299143	C₂	1	1	yes	maximal, characteristic
12	#16813709	—	2	8	no	—
13	C₂×#14268657	C₂²	1	2	yes	—
14	C₂×#14268656	C₂²	1	2	yes	—
15	C₂×#14268656	C₂²	1	2	yes	—
16	C₂×#14268655	C₂²	1	2	yes	—
17	C₂²×#11744753	C₂²	1	1	yes	characteristic
18	C₂²×#11744753	C₂²	1	2	yes	—
19	C₂²×#11744753	C₂²	1	1	yes	characteristic
20	#16813689	C₂²	1	2	yes	—
21	C₄⋊#11744753	C₂²	1	1	yes	characteristic
22	C₄⋊#11744753	C₂²	1	1	yes	characteristic
23	#16813679	C₂²	1	2	yes	—
24	#16813679	C₂²	1	2	yes	—
25	C₂²⋊#11744753	C₂²	1	2	yes	—
26	C₂²⋊#11744753	C₂²	1	2	yes	—
27	C₄×#11744753	C₂²	1	1	yes	characteristic
28	C₄×#11744753	C₂²	1	1	yes	characteristic
29	C₄×#11744753	C₂²	1	1	yes	characteristic
30	C₄×#11744753	C₂²	1	1	yes	characteristic
31	C₂×#14268646	C₂²	1	2	yes	—
32	#16813621	C₄	1	1	yes	characteristic
33	#16813618	C₄	1	1	yes	characteristic
34	#16813608	C₄	1	2	yes	—
35	#16813596	C₄	1	1	yes	characteristic
36	#16813594	C₄	1	2	yes	—
37	D₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
38	#16813584	C₂²	1	1	yes	characteristic
39	#16813582	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
41	#121×C₁₁₃	C₄	1	1	yes	characteristic
42	#14268657	—	2	4	no	—
43	#14268656	—	2	4	no	—
44	#14268656	—	2	4	no	—
45	#14268655	—	2	4	no	—
46	C₂×#11744753	—	2	2	no	—
47	C₂×#11744753	—	2	4	no	—
48	C₂×#11744753	D₄	1	4	yes	complemented
49	C₂×#11744753	C₂³	1	1	yes	characteristic
50	C₂×#11744753	C₂³	1	1	yes	characteristic
51	C₂×#11744753	C₂³	1	2	yes	—
52	C₂×#11744753	C₂³	1	2	yes	—
53	C₂×#11744753	C₂³	1	1	yes	characteristic
54	C₂×#11744753	C₂³	1	1	yes	characteristic
55	#14268646	—	2	4	no	—
56	#14268646	—	2	4	no	—
57	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
58	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
62	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
64	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
66	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
67	#14268626	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
69	#14268624	C₂×C₄	1	2	yes	—
70	#14268624	C₂×C₄	1	2	yes	—
71	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
74	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
76	C₂×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
77	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
80	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
83	D₄×C₁₁₃	—	2	4	no	—
84	D₄×C₁₁₃	—	2	2	no	—
85	C₂×C₄×C₁₁₃	—	2	2	no	—
86	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊D₁₁₃	—	2	4	no	—
90	C₂²⋊D₁₁₃	—	2	4	no	—
91	C₂²⋊D₁₁₃	—	2	4	no	—
92	C₂×Dic₁₁₃	—	2	2	no	—
93	C₂×Dic₁₁₃	—	2	2	no	—
94	C₂×Dic₁₁₃	—	2	2	no	—
95	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
100	C₂×Dic₁₁₃	#41	1	2	yes	—
101	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
102	D₄₅₂	—	2	2	no	—
103	C₄×D₁₁₃	—	2	2	no	—
104	C₄×D₁₁₃	—	2	2	no	—
105	Dic₂₂₆	—	2	4	no	—
106	#11744753	—	2	2	no	—
107	#11744753	—	2	2	no	—
108	#11744753	—	2	1	no	—
109	#11744753	—	2	1	no	—
110	#11744753	C₂×D₄	1	2	yes	—
111	#11744753	C₂×D₄	1	2	yes	—
112	C₂²×C₁₁₃	—	2	1	no	—
113	C₂²×C₁₁₃	—	2	1	no	—
114	C₂²×C₁₁₃	—	2	2	no	—
115	C₂²×C₁₁₃	—	2	2	no	—
116	C₂²×C₁₁₃	—	2	2	no	—
117	C₂²×C₁₁₃	C₄×D₄	1	2	yes	—
118	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₁₁₃	—	2	2	no	—
120	C₂×D₁₁₃	—	2	2	no	—
121	C₂×D₁₁₃	—	2	2	no	—
122	C₄×C₁₁₃	—	2	2	no	—
123	C₄×C₁₁₃	—	2	1	no	—
124	C₄×C₁₁₃	#131	1	2	yes	complemented
125	Dic₁₁₃	—	2	2	no	—
126	Dic₁₁₃	—	2	1	no	—
127	Dic₁₁₃	C₄×D₄	1	2	yes	—
128	Dic₁₁₃	#131	1	2	yes	complemented
129	Dic₁₁₃	C₂²×D₄	1	1	yes	characteristic
130	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₁₁₃	—	2	2	no	—
132	C₂×C₁₁₃	—	4	2	no	—
133	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
135	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
136	D₁₁₃	—	4	2	no	—
137	#2931	—	113	1	no	maximal, Sylow
138	C₁₁₃	#2931	1	1	yes	characteristic, complemented, Sylow
139	C₂×#131	—	113	2	no	—
140	#532	—	113	1	no	—
141	#436	—	113	2	no	—
142	#436	—	113	1	no	—
143	#436	—	113	1	no	—
144	C₈×D₄	—	113	1	no	—
145	C₈×D₄	—	113	2	no	—
146	C₈×D₄	—	113	1	no	—
147	#423	—	113	1	no	—
148	#408	—	113	2	no	—
149	C₂×C₄×C₈	—	113	1	no	—
150	C₂×#41	—	113	2	no	—
151	#131	—	226	8	no	—
152	C₂×M₁₆	—	113	2	no	—
153	C₂×M₁₆	—	113	2	no	—
154	C₂×M₁₆	—	113	2	no	—
155	C₂²×C₈	—	113	1	no	—
156	C₂²×C₈	—	113	2	no	—
157	C₂²×C₈	—	113	2	no	—
158	C₂²×C₈	—	113	1	no	—
159	C₄⋊Q₈	—	113	1	no	—
160	#127	—	113	1	no	—
161	#124	—	113	2	no	—
162	#121	—	113	2	no	—
163	#121	—	113	1	no	—
164	#121	#90673	1	1	yes	characteristic
165	C₄×D₄	—	113	1	no	—
166	C₄×D₄	—	113	1	no	—
167	C₄×D₄	—	113	2	no	—
168	C₂×C₄²	—	113	1	no	—
169	C₄⋊C₈	—	113	1	no	—
170	C₄⋊C₈	—	113	1	no	—
171	C₄⋊C₈	—	113	2	no	—
172	C₂²⋊C₈	—	113	2	no	—
173	C₂²⋊C₈	—	113	2	no	—
174	C₂²⋊C₈	—	113	2	no	—
175	C₂²⋊C₈	—	113	2	no	—
176	C₄×C₈	—	113	1	no	—
177	C₄×C₈	—	113	1	no	—
178	C₄×C₈	—	113	1	no	—
179	C₄×C₈	—	113	1	no	—
180	#41	—	226	4	no	—
181	#41	—	226	4	no	—
182	C₂×Q₈	—	113	2	no	—
183	C₂×D₄	—	113	2	no	—
184	C₂×D₄	—	113	2	no	—
185	C₂×D₄	—	113	2	no	—
186	C₂×D₄	—	113	1	no	—
187	C₂×D₄	C₂×#90673	1	1	yes	characteristic
188	C₂×D₄	C₂×#90673	1	2	yes	—
189	C₂²×C₄	—	113	1	no	—
190	C₂²×C₄	—	113	2	no	—
191	C₂²×C₄	—	113	1	no	—
192	C₂²×C₄	—	113	2	no	—
193	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
194	M₁₆	—	226	4	no	—
195	M₁₆	—	226	4	no	—
196	M₁₆	—	226	4	no	—
197	C₂×C₈	—	113	1	no	—
198	C₂×C₈	—	113	1	no	—
199	C₂×C₈	—	113	4	no	—
200	C₂×C₈	—	113	1	no	—
201	C₂×C₈	—	113	1	no	—
202	C₂×C₈	—	113	2	no	—
203	C₂×C₈	—	113	2	no	—
204	C₂×C₈	—	226	4	no	—
205	C₂×C₈	—	226	4	no	—
206	C₂×C₈	—	226	2	no	—
207	C₄⋊C₄	—	113	2	no	—
208	C₄⋊C₄	—	113	1	no	—
209	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
210	C₂²⋊C₄	—	113	2	no	—
211	C₂²⋊C₄	—	113	2	no	—
212	C₂²⋊C₄	—	113	2	no	—
213	C₂²⋊C₄	C₂×#90673	1	2	yes	—
214	C₄²	—	113	1	no	—
215	C₄²	—	113	1	no	—
216	C₄²	—	113	1	no	—
217	C₄²	—	113	1	no	—
218	C₂³	—	113	2	no	—
219	C₂³	C₂²×#90673	1	2	yes	—
220	C₂³	C₂²×#90673	1	1	yes	characteristic
221	Q₈	—	226	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	226	4	no	—
225	D₄	—	226	4	no	—
226	D₄	—	226	4	no	—
227	D₄	—	226	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	113	2	no	—
230	C₂×C₄	—	113	2	no	—
231	C₂×C₄	—	113	1	no	—
232	C₂×C₄	—	113	1	no	—
233	C₂×C₄	—	113	1	no	—
234	C₂×C₄	—	113	1	no	—
235	C₂×C₄	—	113	2	no	—
236	C₂×C₄	—	226	2	no	—
237	C₂×C₄	—	226	2	no	—
238	C₂×C₄	—	226	2	no	—
239	C₂×C₄	—	226	2	no	—
240	C₂×C₄	—	226	2	no	—
241	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
242	C₂×C₄	C₂²×#90673	1	2	yes	—
243	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
244	C₈	—	113	2	no	—
245	C₈	—	113	2	no	—
246	C₈	—	226	2	no	—
247	C₈	—	226	2	no	—
248	C₈	—	226	1	no	—
249	C₈	—	226	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	226	2	no	—
256	C₂²	—	226	2	no	—
257	C₂²	—	226	2	no	—
258	C₂²	D₄×#90673	1	2	yes	—
259	C₂²	C₂³×#90673	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	113	1	no	—
263	C₄	—	113	1	no	—
264	C₄	—	113	2	no	—
265	C₄	—	113	2	no	—
266	C₄	—	226	2	no	—
267	C₄	—	226	1	no	—
268	C₄	#16813709	1	2	yes	complemented
269	C₂	—	2	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	452	2	no	—
272	C₂	C₂×D₄×#90673	1	1	yes	characteristic
273	C₂	C₂×#16813709	1	1	yes	characteristic
274	C₂	D₁₁₃⋊#131	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	#2931
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. C₂×#16813709	C₂	semidirect
2. #20299633	C₂	semidirect
3. #20299632	C₂	semidirect
4. #20299631	C₂	semidirect
5. #20299626	C₂	semidirect
6. #20299625	C₂	semidirect
7. D₄×#11744753	C₂	semidirect
8. #20299601	C₂	semidirect
9. #20299571	C₂	semidirect
10. C₂×C₄×#11744753	C₂	semidirect
48. C₂×#11744753	D₄	semidirect
124. C₄×C₁₁₃	#131	semidirect
128. Dic₁₁₃	#131	semidirect
138. C₁₁₃	#2931	semidirect
268. C₄	#16813709	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813709	2
#20299633	2
#20299632	2
#20299631	2
#20299626	2
#20299625	2
D₄×#11744753	2
#20299601	2
#20299571	2
C₂×C₄×#11744753	2
#20299143	2
#2931	113