Subgroups — #28407334

Order 14464.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299666	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299665	C₂	1	1	yes	maximal, characteristic
3	#20299637	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#90673	C₂	1	4	yes	maximal, complemented
5	#20299608	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299591	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299590	C₂	1	1	yes	maximal, characteristic
8	#20299561	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₁₁₃	C₂	1	1	yes	maximal, characteristic
10	D₁₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#90673	—	2	2	no	—
12	Q₈×#90673	C₂²	1	4	yes	complemented
13	D₄×#90673	—	2	2	no	—
14	D₄×#90673	C₂²	1	4	yes	—
15	#16813701	—	2	2	no	—
16	#16813701	C₂²	1	1	yes	characteristic
17	#16813701	C₂²	1	1	yes	characteristic
18	#16813701	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#90673	—	2	2	no	—
20	#16813691	C₂²	1	2	yes	complemented
21	#16813690	C₂²	1	2	yes	complemented
22	#16813685	C₂²	1	4	yes	complemented
23	#16813684	C₂²	1	4	yes	—
24	#16813676	C₂²	1	2	yes	complemented
25	C₈×#90673	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₁₁₃	C₂²	1	1	yes	characteristic
28	C₂×#14268636	C₄	1	1	yes	characteristic, complemented
29	C₂×#14268635	C₄	1	1	yes	characteristic, complemented
30	C₂×#14268634	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₁₁₃	C₄	1	2	yes	complemented
32	SD₁₆×D₁₁₃	C₄	1	2	yes	complemented
33	SD₁₆×D₁₁₃	C₂²	1	2	yes	—
34	SD₁₆×D₁₁₃	C₂²	1	2	yes	—
35	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₁₃	C₄	1	1	yes	characteristic, complemented
37	C₂²×#90673	—	2	4	no	—
38	#14268657	—	2	1	no	—
39	#14268656	—	2	1	no	—
40	#14268655	C₂³	1	2	yes	—
41	C₄⋊#90673	—	2	4	no	—
42	C₄⋊#90673	D₄	1	2	yes	—
43	C₄⋊#90673	C₂³	1	2	yes	—
44	C₄⋊#90673	C₂³	1	2	yes	—
45	C₂²⋊#90673	—	2	1	no	—
46	C₂²⋊#90673	—	2	1	no	—
47	C₂²⋊#90673	—	2	4	no	—
48	C₂²⋊#90673	D₄	1	2	yes	—
49	C₄×#90673	—	2	4	no	—
50	C₄×#90673	—	2	1	no	—
51	C₄×#90673	—	2	1	no	—
52	C₄×#90673	C₂³	1	2	yes	—
53	C₂³×D₁₁₃	—	2	1	no	—
54	C₂×Dic₂₂₆	—	2	1	no	—
55	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₁₁₃	—	2	1	no	—
57	Q₈×D₁₁₃	—	2	1	no	—
58	Q₈×D₁₁₃	C₂×C₄	1	2	yes	complemented
59	Q₈×D₁₁₃	C₂³	1	2	yes	—
60	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
62	D₄×D₁₁₃	—	2	1	no	—
63	D₄×D₁₁₃	—	2	1	no	—
64	D₄×D₁₁₃	C₂×C₄	1	2	yes	—
65	D₄×D₁₁₃	C₂³	1	2	yes	—
66	C₂×C₄×D₁₁₃	—	2	1	no	—
67	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
68	#14268636	C₂×C₄	1	2	yes	complemented
69	#14268636	C₂×C₄	1	2	yes	complemented
70	#14268635	C₂×C₄	1	2	yes	complemented
71	#14268635	C₂×C₄	1	2	yes	complemented
72	#14268634	C₂×C₄	1	2	yes	complemented
73	#14268634	C₂×C₄	1	2	yes	complemented
74	C₂×#11744751	C₂×C₄	1	1	yes	characteristic, complemented
75	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
76	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
77	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
78	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
82	SD₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
83	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂³×C₁₁₃	—	2	1	no	—
85	C₂²×D₁₁₃	—	2	1	no	—
86	C₂²×D₁₁₃	—	2	2	no	—
87	C₂²×D₁₁₃	—	2	2	no	—
88	C₂²×D₁₁₃	—	8	2	no	—
89	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×#90673	—	2	2	no	—
91	C₂×#90673	—	2	2	no	—
92	C₂×#90673	—	8	4	no	—
93	C₂×#90673	C₂×D₄	1	4	yes	—
94	Q₈×C₁₁₃	—	2	1	no	—
95	Q₈×C₁₁₃	C₂²×C₄	1	2	yes	—
96	D₄×C₁₁₃	—	2	1	no	—
97	D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₁₃	—	2	1	no	—
99	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₁₁₃	—	2	2	no	—
101	C₂²⋊D₁₁₃	—	2	2	no	—
102	C₂×Dic₁₁₃	—	2	1	no	—
103	C₂×Dic₁₁₃	#41	1	1	yes	characteristic
104	D₄₅₂	—	2	1	no	—
105	D₄₅₂	C₂²×C₄	1	2	yes	—
106	C₄×D₁₁₃	—	2	2	no	—
107	C₄×D₁₁₃	—	2	2	no	—
108	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄×D₁₁₃	#41	1	1	yes	characteristic
111	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
112	Dic₂₂₆	—	2	1	no	—
113	Dic₂₂₆	—	2	2	no	—
114	Dic₂₂₆	—	2	2	no	—
115	Dic₂₂₆	C₂²×C₄	1	2	yes	—
116	#11744753	—	2	2	no	—
117	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
119	#11744751	C₂²×C₄	1	1	yes	characteristic
120	#11744751	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₁₁₃	—	2	2	no	—
122	C₂²×C₁₁₃	—	8	1	no	—
123	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×D₁₁₃	—	2	2	no	—
125	C₂×D₁₁₃	—	2	2	no	—
126	C₂×D₁₁₃	—	4	4	no	—
127	C₂×D₁₁₃	—	8	2	no	—
128	C₂×D₁₁₃	—	8	1	no	—
129	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×D₁₁₃	C₂²×D₄	1	1	yes	characteristic
131	#90673	—	2	4	no	—
132	#90673	—	4	2	no	—
133	#90673	—	4	2	no	—
134	C₄×C₁₁₃	—	2	2	no	—
135	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
137	Dic₁₁₃	—	2	2	no	—
138	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
139	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₁₁₃	—	2	1	no	—
141	C₂×C₁₁₃	—	4	2	no	—
142	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
143	D₁₁₃	—	2	1	no	—
144	D₁₁₃	—	4	2	no	—
145	D₁₁₃	#577	1	2	yes	—
146	#2949	—	113	1	no	maximal, Sylow
147	C₁₁₃	#2949	1	1	yes	characteristic, complemented, Sylow
148	C₂²×SD₁₆	—	113	1	no	—
149	#519	—	113	1	no	—
150	#518	—	113	1	no	—
151	#440	—	113	4	no	—
152	C₄×SD₁₆	—	113	4	no	—
153	#427	—	113	1	no	—
154	#419	—	113	1	no	—
155	#418	—	113	1	no	—
156	#405	—	113	1	no	—
157	C₂²×Q₈	—	113	1	no	—
158	C₂²×D₄	—	113	1	no	—
159	C₂×SD₁₆	—	113	1	no	—
160	C₂×SD₁₆	—	113	1	no	—
161	C₂×SD₁₆	—	113	2	no	—
162	C₂×SD₁₆	—	113	2	no	—
163	C₂×SD₁₆	—	113	2	no	—
164	C₂×SD₁₆	—	113	2	no	—
165	C₂×SD₁₆	—	113	1	no	—
166	C₂×SD₁₆	#90673	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	113	1	no	—
168	C₂²×C₈	—	113	1	no	—
169	C₄×Q₈	—	113	4	no	—
170	C₄×Q₈	—	226	2	no	—
171	C₄×D₄	—	113	4	no	—
172	C₄×D₄	—	226	2	no	—
173	#117	—	113	1	no	—
174	#117	—	113	1	no	—
175	#117	—	113	1	no	—
176	#117	—	226	2	no	—
177	C₂×C₂²⋊C₄	—	226	2	no	—
178	#107	—	113	2	no	—
179	#106	—	113	2	no	—
180	#103	—	113	4	no	—
181	#102	—	113	4	no	—
182	#97	—	113	2	no	—
183	C₄×C₈	—	113	2	no	—
184	C₂⁴	—	226	1	no	—
185	C₂×Q₈	—	113	2	no	—
186	C₂×Q₈	—	113	2	no	—
187	C₂×Q₈	—	113	1	no	—
188	C₂×Q₈	—	226	1	no	—
189	C₂×Q₈	—	226	1	no	—
190	C₂×Q₈	—	226	1	no	—
191	C₂×Q₈	C₂×#90673	1	1	yes	characteristic, complemented
192	C₂×D₄	—	113	2	no	—
193	C₂×D₄	—	113	2	no	—
194	C₂×D₄	—	113	1	no	—
195	C₂×D₄	—	226	1	no	—
196	C₂×D₄	—	226	1	no	—
197	C₂×D₄	—	226	1	no	—
198	C₂×D₄	C₂×#90673	1	1	yes	characteristic
199	C₂²×C₄	—	113	1	no	—
200	C₂²×C₄	—	226	1	no	—
201	C₂²×C₄	—	226	4	no	—
202	SD₁₆	—	113	2	no	—
203	SD₁₆	—	113	2	no	—
204	SD₁₆	—	113	2	no	—
205	SD₁₆	—	113	2	no	—
206	SD₁₆	—	113	2	no	—
207	SD₁₆	—	113	2	no	—
208	SD₁₆	C₂×#90673	1	2	yes	complemented
209	SD₁₆	C₂×#90673	1	2	yes	complemented
210	M₁₆	—	226	1	no	—
211	M₁₆	—	226	1	no	—
212	C₂×C₈	—	113	1	no	—
213	C₂×C₈	—	113	1	no	—
214	C₂×C₈	—	113	1	no	—
215	C₂×C₈	—	113	1	no	—
216	C₂×C₈	—	113	1	no	—
217	C₂×C₈	—	113	2	no	—
218	C₂×C₈	C₂×#90673	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	113	2	no	—
220	C₄⋊C₄	—	113	2	no	—
221	C₄⋊C₄	—	113	2	no	—
222	C₄⋊C₄	—	226	4	no	—
223	C₂²⋊C₄	—	113	2	no	—
224	C₂²⋊C₄	—	226	4	no	—
225	C₂²⋊C₄	—	226	1	no	—
226	C₂²⋊C₄	—	226	1	no	—
227	C₄²	—	113	2	no	—
228	C₄²	—	226	1	no	—
229	C₄²	—	226	1	no	—
230	C₄²	—	226	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	113	1	no	—
233	C₂³	—	904	2	no	—
234	C₂³	—	226	2	no	—
235	C₂³	—	226	2	no	—
236	C₂³	—	226	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	113	2	no	—
239	Q₈	—	226	2	no	—
240	Q₈	—	226	2	no	—
241	Q₈	—	226	1	no	—
242	Q₈	C₂²×#90673	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	113	2	no	—
245	D₄	—	226	2	no	—
246	D₄	—	226	2	no	—
247	D₄	—	226	1	no	—
248	D₄	C₂²×#90673	1	2	yes	—
249	C₂×C₄	—	2	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	—	113	4	no	—
256	C₂×C₄	—	904	4	no	—
257	C₂×C₄	—	226	2	no	—
258	C₂×C₄	—	226	2	no	—
259	C₂×C₄	—	226	2	no	—
260	C₂×C₄	—	226	2	no	—
261	C₂×C₄	—	226	1	no	—
262	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
263	C₈	—	113	1	no	—
264	C₈	—	113	1	no	—
265	C₈	—	226	2	no	—
266	C₈	C₂²×#90673	1	1	yes	characteristic
267	C₈	C₂²×#90673	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	113	1	no	—
271	C₂²	—	113	1	no	—
272	C₂²	—	904	1	no	—
273	C₂²	—	904	2	no	—
274	C₂²	—	452	4	no	—
275	C₂²	—	226	2	no	—
276	C₂²	—	226	2	no	—
277	C₂²	D₄×#90673	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	113	1	no	—
280	C₄	—	113	1	no	—
281	C₄	—	452	2	no	—
282	C₄	—	452	2	no	—
283	C₄	—	226	4	no	—
284	C₄	—	226	2	no	—
285	C₄	D₄×#90673	1	1	yes	characteristic
286	C₄	C₂³×#90673	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	113	2	no	—
290	C₂	—	452	2	no	—
291	C₂	—	226	1	no	—
292	C₂	C₂×D₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#90673
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₄	C₂³×#90673
Fitting	C₂×SD₁₆×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₂×C₄×D₄
prime radical	C₂×D₄×D₁₁₃	C₂²
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	C₂×SD₁₆	#90673
hypocenter	C₁₁₃	#2949
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20299666	C₂	semidirect
3. #20299637	C₂	semidirect
4. SD₁₆×#90673	C₂	semidirect
5. #20299608	C₂	semidirect
6. #20299591	C₂	semidirect
8. #20299561	C₂	semidirect
12. Q₈×#90673	C₂²	semidirect
20. #16813691	C₂²	semidirect
21. #16813690	C₂²	semidirect
22. #16813685	C₂²	semidirect
24. #16813676	C₂²	semidirect
25. C₈×#90673	C₂²	semidirect
28. C₂×#14268636	C₄	semidirect
29. C₂×#14268635	C₄	semidirect
30. C₂×#14268634	C₄	semidirect
31. SD₁₆×D₁₁₃	C₄	semidirect
32. SD₁₆×D₁₁₃	C₄	semidirect
36. C₂×SD₁₆×C₁₁₃	C₄	semidirect
55. C₂×Dic₂₂₆	C₂×C₄	semidirect
58. Q₈×D₁₁₃	C₂×C₄	semidirect
68. #14268636	C₂×C₄	semidirect
69. #14268636	C₂×C₄	semidirect
70. #14268635	C₂×C₄	semidirect
71. #14268635	C₂×C₄	semidirect
72. #14268634	C₂×C₄	semidirect
73. #14268634	C₂×C₄	semidirect
74. C₂×#11744751	C₂×C₄	semidirect
75. C₈×D₁₁₃	C₂×C₄	semidirect
76. C₈×D₁₁₃	C₂×C₄	semidirect
79. C₂×Q₈×C₁₁₃	C₂×C₄	semidirect
81. SD₁₆×C₁₁₃	C₂×C₄	semidirect
82. SD₁₆×C₁₁₃	C₂×C₄	semidirect
83. C₂×C₈×C₁₁₃	C₂×C₄	semidirect
147. C₁₁₃	#2949	semidirect
166. C₂×SD₁₆	#90673	semidirect
191. C₂×Q₈	C₂×#90673	semidirect
208. SD₁₆	C₂×#90673	semidirect
209. SD₁₆	C₂×#90673	semidirect
218. C₂×C₈	C₂×#90673	semidirect

Maximal subgroups

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

subgroup	index
#20299666	2
#20299665	2
#20299637	2
SD₁₆×#90673	2
#20299608	2
#20299591	2
#20299590	2
#20299561	2
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