Subgroups — #28407137

Order 14464.

564 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813701	C₂	1	1	yes	maximal, characteristic
3	#20299564	C₂	1	4	yes	maximal, complemented
4	#20299561	C₂	1	4	yes	maximal, complemented
5	M₁₆×#90673	C₂	1	4	yes	maximal, complemented
6	C₂×M₁₆×D₁₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14268657	C₄	1	2	yes	complemented
8	D₁₁₃⋊M₁₆	C₄	1	2	yes	complemented
9	D₁₁₃⋊M₁₆	C₄	1	2	yes	complemented
10	D₁₁₃⋊M₁₆	C₂²	1	2	yes	—
11	D₁₁₃⋊M₁₆	C₂²	1	2	yes	—
12	C₂×#14268656	C₄	1	2	yes	complemented
13	C₂×#14268655	C₂²	1	2	yes	—
14	#16813701	C₄	1	2	yes	—
15	#16813701	C₄	1	2	yes	—
16	#16813701	C₂²	1	2	yes	—
17	#16813701	C₂²	1	2	yes	—
18	C₂×C₄⋊#90673	C₄	1	2	yes	—
19	C₂×C₂²⋊#90673	C₄	1	2	yes	—
20	C₂×C₄×#90673	C₂²	1	2	yes	—
21	#16813676	C₂²	1	8	yes	complemented
22	C₈×#90673	C₂²	1	8	yes	complemented
23	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
24	C₂×#14268631	C₄	1	1	yes	characteristic, complemented
25	C₂×#14268630	C₄	1	2	yes	complemented
26	M₁₆×D₁₁₃	C₄	1	2	yes	complemented
27	M₁₆×D₁₁₃	C₄	1	2	yes	complemented
28	M₁₆×D₁₁₃	C₂²	1	2	yes	—
29	M₁₆×D₁₁₃	C₂²	1	2	yes	—
30	C₂×C₈×D₁₁₃	C₂²	1	2	yes	—
31	C₂×M₁₆×C₁₁₃	C₄	1	1	yes	characteristic, complemented
32	C₂²×#90673	C₂×C₄	1	4	yes	—
33	#14268657	C₂×C₄	1	4	yes	—
34	#14268657	C₂×C₄	1	4	yes	complemented
35	#14268656	C₂×C₄	1	4	yes	—
36	#14268656	C₂×C₄	1	4	yes	complemented
37	#14268655	C₂×C₄	1	4	yes	complemented
38	#14268655	C₂³	1	4	yes	—
39	C₂×#11744753	C₂×C₄	1	4	yes	complemented
40	C₄⋊#90673	C₂×C₄	1	4	yes	—
41	C₄⋊#90673	C₂×C₄	1	4	yes	—
42	C₂²⋊#90673	C₂×C₄	1	4	yes	—
43	C₂²⋊#90673	C₂×C₄	1	4	yes	—
44	C₄×#90673	C₂×C₄	1	4	yes	—
45	C₄×#90673	C₂³	1	4	yes	—
46	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
60	#14268631	C₂×C₄	1	2	yes	—
61	#14268631	C₂×C₄	1	2	yes	complemented
62	#14268630	C₂×C₄	1	4	yes	—
63	#14268630	C₂×C₄	1	4	yes	complemented
64	C₂×#11744751	C₂×C₄	1	2	yes	complemented
65	C₈×D₁₁₃	C₂×C₄	1	4	yes	complemented
66	C₈×D₁₁₃	C₂³	1	4	yes	—
67	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
68	M₁₆×C₁₁₃	C₂×C₄	1	2	yes	—
69	M₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
70	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	complemented
71	C₂³×C₁₁₃	C₄²	1	1	yes	characteristic
72	C₂²×D₁₁₃	—	4	2	no	—
73	C₂²×D₁₁₃	C₄²	1	1	yes	characteristic
74	C₂²×D₁₁₃	C₄²	1	1	yes	characteristic
75	C₂²×D₁₁₃	C₄²	1	1	yes	characteristic
76	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×#90673	—	4	4	no	—
80	C₂×#90673	C₂²×C₄	1	8	yes	—
81	C₂×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₁₃	C₄²	1	1	yes	characteristic
88	C₂×Dic₁₁₃	C₄²	1	1	yes	characteristic
89	C₂×Dic₁₁₃	C₄²	1	1	yes	characteristic
90	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
94	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
95	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
96	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
97	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
98	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
99	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
109	#11744753	C₂²×C₄	1	8	yes	—
110	C₈×C₁₁₃	C₂²×C₄	1	4	yes	—
111	#11744751	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₁₃	—	4	1	no	—
113	C₂²×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
114	C₂²×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×D₁₁₃	—	2	2	no	—
117	C₂×D₁₁₃	—	2	2	no	—
118	C₂×D₁₁₃	—	2	2	no	—
119	C₂×D₁₁₃	—	4	1	no	—
120	C₂×D₁₁₃	—	4	1	no	—
121	C₂×D₁₁₃	—	4	1	no	—
122	C₂×D₁₁₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×D₁₁₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×D₁₁₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×D₁₁₃	C₂³×C₄	1	1	yes	characteristic
126	#90673	—	2	8	no	—
127	C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
128	C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
129	C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
130	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
131	Dic₁₁₃	C₂×C₄²	1	1	yes	characteristic
132	Dic₁₁₃	C₂×C₄²	1	1	yes	characteristic
133	Dic₁₁₃	C₂×C₄²	1	1	yes	characteristic
134	Dic₁₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₁₁₃	—	2	1	no	—
136	C₂×C₁₁₃	—	2	1	no	—
137	C₂×C₁₁₃	—	2	1	no	—
138	C₂×C₁₁₃	C₂²×C₄²	1	1	yes	characteristic
139	D₁₁₃	—	2	1	no	—
140	D₁₁₃	—	2	1	no	—
141	D₁₁₃	—	2	1	no	—
142	D₁₁₃	#568	1	2	yes	—
143	#2873	—	113	1	no	maximal, Sylow
144	C₁₁₃	#2873	1	1	yes	characteristic, complemented, Sylow
145	C₂²×M₁₆	—	113	1	no	—
146	C₂²×M₁₆	—	113	1	no	—
147	C₂×#117	—	113	1	no	—
148	#405	—	113	4	no	—
149	#405	—	113	4	no	—
150	C₄×M₁₆	—	113	4	no	—
151	C₂³×C₄	—	113	1	no	—
152	C₂×M₁₆	—	113	2	no	—
153	C₂×M₁₆	—	113	2	no	—
154	C₂×M₁₆	—	113	2	no	—
155	C₂×M₁₆	—	113	2	no	—
156	C₂×M₁₆	—	113	2	no	—
157	C₂×M₁₆	—	113	2	no	—
158	C₂×M₁₆	—	113	1	no	—
159	C₂×M₁₆	—	113	2	no	—
160	C₂×M₁₆	—	113	2	no	—
161	C₂×M₁₆	—	113	2	no	—
162	C₂×M₁₆	—	113	2	no	—
163	C₂×M₁₆	—	113	2	no	—
164	C₂×M₁₆	#90673	1	1	yes	characteristic, complemented
165	C₂²×C₈	—	113	2	no	—
166	C₂²×C₈	—	113	2	no	—
167	#117	—	113	2	no	—
168	#117	—	113	2	no	—
169	#117	—	113	2	no	—
170	#117	—	113	2	no	—
171	C₂×C₄⋊C₄	—	113	2	no	—
172	C₂×C₂²⋊C₄	—	113	2	no	—
173	C₂×C₄²	—	113	2	no	—
174	#97	—	113	8	no	—
175	C₄×C₈	—	113	8	no	—
176	C₂⁴	—	113	1	no	—
177	C₂²×C₄	—	113	1	no	—
178	C₂²×C₄	—	113	1	no	—
179	C₂²×C₄	—	113	1	no	—
180	C₂²×C₄	—	113	1	no	—
181	C₂²×C₄	—	113	1	no	—
182	C₂²×C₄	—	113	1	no	—
183	C₂²×C₄	—	113	1	no	—
184	C₂²×C₄	—	113	1	no	—
185	C₂²×C₄	—	113	1	no	—
186	C₂²×C₄	—	113	1	no	—
187	C₂²×C₄	—	113	1	no	—
188	C₂²×C₄	—	113	1	no	—
189	C₂²×C₄	—	113	1	no	—
190	C₂²×C₄	—	113	4	no	—
191	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
192	M₁₆	—	113	4	no	—
193	M₁₆	—	113	4	no	—
194	M₁₆	—	113	4	no	—
195	M₁₆	—	113	4	no	—
196	M₁₆	—	113	2	no	—
197	M₁₆	—	113	2	no	—
198	M₁₆	—	113	4	no	—
199	M₁₆	—	113	4	no	—
200	M₁₆	C₂×#90673	1	2	yes	—
201	M₁₆	C₂×#90673	1	2	yes	complemented
202	C₂×C₈	—	113	4	no	—
203	C₂×C₈	—	113	4	no	—
204	C₂×C₈	—	113	4	no	—
205	C₂×C₈	—	113	2	no	—
206	C₂×C₈	—	113	4	no	—
207	C₂×C₈	—	113	4	no	—
208	C₂×C₈	C₂×#90673	1	2	yes	complemented
209	C₄⋊C₄	—	113	4	no	—
210	C₄⋊C₄	—	113	4	no	—
211	C₂²⋊C₄	—	113	4	no	—
212	C₂²⋊C₄	—	113	4	no	—
213	C₄²	—	113	4	no	—
214	C₄²	—	113	4	no	—
215	C₂³	—	113	1	no	—
216	C₂³	—	113	1	no	—
217	C₂³	—	113	1	no	—
218	C₂³	—	113	1	no	—
219	C₂³	—	113	1	no	—
220	C₂³	—	113	1	no	—
221	C₂³	—	452	2	no	—
222	C₂³	C₄×#90673	1	1	yes	characteristic
223	C₂×C₄	—	113	1	no	—
224	C₂×C₄	—	113	1	no	—
225	C₂×C₄	—	113	1	no	—
226	C₂×C₄	—	113	1	no	—
227	C₂×C₄	—	113	1	no	—
228	C₂×C₄	—	113	1	no	—
229	C₂×C₄	—	113	1	no	—
230	C₂×C₄	—	113	1	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	1	no	—
236	C₂×C₄	—	113	1	no	—
237	C₂×C₄	—	113	1	no	—
238	C₂×C₄	—	113	1	no	—
239	C₂×C₄	—	113	1	no	—
240	C₂×C₄	—	113	1	no	—
241	C₂×C₄	—	113	1	no	—
242	C₂×C₄	—	113	1	no	—
243	C₂×C₄	—	113	1	no	—
244	C₂×C₄	—	113	1	no	—
245	C₂×C₄	—	113	8	no	—
246	C₂×C₄	—	452	4	no	—
247	C₂×C₄	C₄×#90673	1	1	yes	characteristic
248	C₂×C₄	C₄×#90673	1	1	yes	characteristic
249	C₂×C₄	C₄×#90673	1	1	yes	characteristic
250	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
251	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
252	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
253	C₈	—	113	8	no	—
254	C₈	—	113	4	no	—
255	C₈	C₂²×#90673	1	4	yes	—
256	C₂²	—	4	1	no	—
257	C₂²	—	113	1	no	—
258	C₂²	—	113	1	no	—
259	C₂²	—	113	1	no	—
260	C₂²	—	113	1	no	—
261	C₂²	—	452	1	no	—
262	C₂²	—	452	1	no	—
263	C₂²	—	452	1	no	—
264	C₂²	—	226	2	no	—
265	C₂²	—	226	2	no	—
266	C₂²	—	226	2	no	—
267	C₂²	C₂×C₄×#90673	1	1	yes	characteristic
268	C₂²	C₂×C₄×#90673	1	1	yes	characteristic
269	C₂²	C₂×C₄×#90673	1	1	yes	characteristic
270	C₄	—	113	1	no	—
271	C₄	—	113	1	no	—
272	C₄	—	113	1	no	—
273	C₄	—	113	1	no	—
274	C₄	—	226	8	no	—
275	C₄	C₂×C₄×#90673	1	1	yes	characteristic
276	C₄	C₂×C₄×#90673	1	1	yes	characteristic
277	C₄	C₂×C₄×#90673	1	1	yes	characteristic
278	C₄	C₂³×#90673	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	113	2	no	—
283	C₂	—	226	1	no	—
284	C₂	—	226	1	no	—
285	C₂	—	226	1	no	—
286	C₂	C₂²×C₄×#90673	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#90673
commutator, socle	C₂×C₁₁₃	C₂²×C₄²
Fitting	C₂×M₁₆×C₁₁₃	C₄
prime radical	C₂³×D₁₁₃	C₂×C₄
cosocle	C₄×D₁₁₃	C₂⁴
hypercenter	C₂×M₁₆	#90673
hypocenter	C₁₁₃	#2873
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20299564	C₂	semidirect
4. #20299561	C₂	semidirect
5. M₁₆×#90673	C₂	semidirect
7. C₂×#14268657	C₄	semidirect
8. D₁₁₃⋊M₁₆	C₄	semidirect
9. D₁₁₃⋊M₁₆	C₄	semidirect
12. C₂×#14268656	C₄	semidirect
21. #16813676	C₂²	semidirect
22. C₈×#90673	C₂²	semidirect
24. C₂×#14268631	C₄	semidirect
25. C₂×#14268630	C₄	semidirect
26. M₁₆×D₁₁₃	C₄	semidirect
27. M₁₆×D₁₁₃	C₄	semidirect
31. C₂×M₁₆×C₁₁₃	C₄	semidirect
34. #14268657	C₂×C₄	semidirect
36. #14268656	C₂×C₄	semidirect
37. #14268655	C₂×C₄	semidirect
39. C₂×#11744753	C₂×C₄	semidirect
61. #14268631	C₂×C₄	semidirect
63. #14268630	C₂×C₄	semidirect
64. C₂×#11744751	C₂×C₄	semidirect
65. C₈×D₁₁₃	C₂×C₄	semidirect
69. M₁₆×C₁₁₃	C₂×C₄	semidirect
70. C₂×C₈×C₁₁₃	C₂×C₄	semidirect
144. C₁₁₃	#2873	semidirect
164. C₂×M₁₆	#90673	semidirect
201. M₁₆	C₂×#90673	semidirect
208. 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
C₂×D₁₁₃⋊M₁₆	2
C₂×#16813701	2
#20299564	2
#20299561	2
M₁₆×#90673	2
C₂×M₁₆×D₁₁₃	2
#2873	113