Subgroups — #533×D₁₁₃ #28403859

Order 14464.

1168 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299154	C₂	1	2	yes	maximal, complemented
3	#20299153	C₂	1	2	yes	maximal, complemented
4	#20299152	C₂	1	4	yes	maximal, complemented
5	#20299151	C₂	1	2	yes	maximal, complemented
6	#20299150	C₂	1	4	yes	maximal, complemented
7	C₄⋊Q₈×D₁₁₃	C₂	1	2	yes	maximal, complemented
8	#125×D₁₁₃	C₂	1	2	yes	maximal, complemented
9	C₂²⋊Q₈×D₁₁₃	C₂	1	4	yes	maximal
10	C₄×Q₈×D₁₁₃	C₂	1	4	yes	maximal, complemented
11	#117×D₁₁₃	C₂	1	2	yes	maximal
12	C₂×C₄²×D₁₁₃	C₂	1	1	yes	maximal, characteristic
13	#533×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₁₃	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
17	#16813622	C₂²	1	2	yes	complemented
18	#16813621	C₂²	1	4	yes	complemented
19	#16813621	C₂²	1	4	yes	complemented
20	#16813620	C₂²	1	4	yes	complemented
21	#16813614	C₂²	1	2	yes	complemented
22	#16813613	C₂²	1	4	yes	complemented
23	#16813612	C₂²	1	8	yes	complemented
24	#16813602	C₂²	1	4	yes	—
25	#16813601	C₂²	1	4	yes	—
26	#16813600	C₂²	1	8	yes	—
27	#16813599	C₂²	1	8	yes	—
28	#16813598	C₂²	1	4	yes	—
29	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
30	C₄×Dic₂₂₆	C₂²	1	8	yes	complemented
31	C₄×Dic₂₂₆	C₂²	1	4	yes	complemented
32	#16813587	C₂²	1	4	yes	complemented
33	#16813587	C₂²	1	8	yes	complemented
34	#16813581	C₂²	1	2	yes	—
35	#16813580	C₂²	1	4	yes	—
36	#16813579	C₂²	1	4	yes	—
37	#16813578	C₂²	1	4	yes	—
38	C₄⋊C₄×D₁₁₃	C₂²	1	4	yes	—
39	C₄⋊C₄×D₁₁₃	C₂²	1	8	yes	—
40	C₄⋊C₄×D₁₁₃	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	—
42	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
43	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
44	C₄²×D₁₁₃	C₂²	1	2	yes	—
45	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
46	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
47	C₄²×D₁₁₃	C₂²	1	2	yes	—
48	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
49	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
50	C₄²×D₁₁₃	C₂²	1	4	yes	—
51	C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	complemented
52	#125×C₁₁₃	C₂²	1	2	yes	complemented
53	C₂²⋊Q₈×C₁₁₃	C₂²	1	4	yes	—
54	C₄×Q₈×C₁₁₃	C₂²	1	4	yes	complemented
55	#117×C₁₁₃	C₂²	1	2	yes	—
56	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
57	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
58	C₂×Dic₂₂₆	C₂³	1	8	yes	—
59	C₂×Dic₂₂₆	C₂³	1	4	yes	—
60	Q₈×D₁₁₃	—	2	16	no	—
61	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
62	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₁₃	—	2	8	no	—
64	C₂×C₄×D₁₁₃	Q₈	1	8	yes	complemented
65	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
66	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
67	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
68	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
69	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
74	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
75	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
76	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
77	#14268626	C₂³	1	8	yes	—
78	#14268626	C₂³	1	8	yes	—
79	#14268626	C₂³	1	8	yes	—
80	#14268626	C₂³	1	8	yes	—
81	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
82	#14268624	C₂³	1	8	yes	—
83	C₄×Dic₁₁₃	C₂³	1	2	yes	—
84	C₄×Dic₁₁₃	C₂³	1	2	yes	—
85	C₄×Dic₁₁₃	C₂³	1	2	yes	—
86	C₄×Dic₁₁₃	C₂³	1	2	yes	—
87	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
88	C₄×Dic₁₁₃	C₂³	1	2	yes	—
89	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₁₃	C₂³	1	8	yes	—
91	C₄×Dic₁₁₃	C₂³	1	4	yes	—
92	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
93	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
96	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
97	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
98	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
99	C₄²×C₁₁₃	C₂³	1	2	yes	—
100	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
101	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
102	C₄²×C₁₁₃	C₂³	1	4	yes	—
103	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²×D₁₁₃	—	2	4	no	—
105	C₂²×D₁₁₃	C₂×Q₈	1	4	yes	—
106	C₂²×D₁₁₃	C₂⁴	1	1	yes	characteristic
107	C₂²×D₁₁₃	C₂⁴	1	1	yes	characteristic
108	Q₈×C₁₁₃	—	2	8	no	—
109	C₂×C₄×C₁₁₃	—	2	4	no	—
110	C₂×C₄×C₁₁₃	C₂×Q₈	1	4	yes	complemented
111	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₁₃	C₂⁴	1	8	yes	—
113	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
115	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₁₃	—	2	4	no	—
117	C₂×Dic₁₁₃	C₂×Q₈	1	4	yes	complemented
118	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
119	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
120	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
123	C₄×D₁₁₃	—	2	4	no	—
124	C₄×D₁₁₃	—	2	16	no	—
125	C₄×D₁₁₃	—	2	4	no	—
126	C₄×D₁₁₃	—	2	4	no	—
127	C₄×D₁₁₃	C₂×Q₈	1	4	yes	—
128	C₄×D₁₁₃	C₂×Q₈	1	4	yes	—
129	C₄×D₁₁₃	C₂×Q₈	1	4	yes	complemented
130	C₄×D₁₁₃	C₂×Q₈	1	4	yes	complemented
131	C₄×D₁₁₃	#41	1	8	yes	—
132	Dic₂₂₆	—	2	8	no	—
133	Dic₂₂₆	—	2	16	no	—
134	C₂²×C₁₁₃	—	2	2	no	—
135	C₂²×C₁₁₃	C₂²×Q₈	1	2	yes	—
136	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
137	C₂×D₁₁₃	—	2	2	no	—
138	C₂×D₁₁₃	—	2	8	no	—
139	C₂×D₁₁₃	C₂²×Q₈	1	2	yes	—
140	C₂×D₁₁₃	C₂²×Q₈	1	2	yes	—
141	C₂×D₁₁₃	C₂×#41	1	4	yes	—
142	C₄×C₁₁₃	—	2	8	no	—
143	C₄×C₁₁₃	—	2	2	no	—
144	C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
145	C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
146	C₄×C₁₁₃	C₂×#41	1	4	yes	—
147	Dic₁₁₃	—	2	8	no	—
148	Dic₁₁₃	—	2	2	no	—
149	Dic₁₁₃	C₂²×Q₈	1	2	yes	—
150	Dic₁₁₃	C₂²×Q₈	1	2	yes	—
151	Dic₁₁₃	C₂×#41	1	4	yes	—
152	C₂×C₁₁₃	—	2	2	no	—
153	C₂×C₁₁₃	C₂³×Q₈	1	1	yes	characteristic
154	C₂×C₁₁₃	C₂²×#41	1	2	yes	—
155	D₁₁₃	—	2	2	no	—
156	D₁₁₃	#533	1	4	yes	direct product factor
157	C₂×#533	—	113	1	no	maximal, Sylow
158	C₁₁₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
159	#533	—	113	2	no	—
160	#533	—	113	2	no	—
161	#533	—	113	4	no	—
162	#533	—	113	4	no	—
163	#533	—	113	2	no	—
164	#533	—	113	1	no	—
165	#533	D₁₁₃	1	1	yes	characteristic, direct product factor
166	C₂×C₄⋊Q₈	—	113	2	no	—
167	C₂×#125	—	113	2	no	—
168	C₂×C₂²⋊Q₈	—	113	4	no	—
169	C₂×C₄×Q₈	—	113	4	no	—
170	C₂×#117	—	113	2	no	—
171	C₂²×C₄²	—	113	1	no	—
172	C₂²×Q₈	—	113	4	no	—
173	C₂³×C₄	—	113	1	no	—
174	C₂³×C₄	—	113	2	no	—
175	C₄⋊Q₈	—	113	4	no	—
176	C₄⋊Q₈	—	113	4	no	—
177	C₄⋊Q₈	—	113	4	no	—
178	C₄⋊Q₈	—	113	2	no	—
179	C₄⋊Q₈	C₂×D₁₁₃	1	2	yes	complemented
180	#125	—	113	8	no	—
181	#125	—	113	4	no	—
182	#125	—	113	2	no	—
183	#125	C₂×D₁₁₃	1	2	yes	complemented
184	C₂²⋊Q₈	—	113	4	no	—
185	C₂²⋊Q₈	—	113	4	no	—
186	C₂²⋊Q₈	—	113	8	no	—
187	C₂²⋊Q₈	—	113	4	no	—
188	C₂²⋊Q₈	—	113	8	no	—
189	C₂²⋊Q₈	C₂×D₁₁₃	1	4	yes	—
190	C₄×Q₈	—	113	4	no	—
191	C₄×Q₈	—	113	4	no	—
192	C₄×Q₈	—	113	8	no	—
193	C₄×Q₈	—	113	8	no	—
194	C₄×Q₈	—	113	4	no	—
195	C₄×Q₈	C₂×D₁₁₃	1	4	yes	complemented
196	#117	—	113	2	no	—
197	#117	—	113	4	no	—
198	#117	—	113	4	no	—
199	#117	—	113	4	no	—
200	#117	C₂×D₁₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	—	113	4	no	—
202	C₂×C₄⋊C₄	—	113	8	no	—
203	C₂×C₄⋊C₄	—	113	4	no	—
204	C₂×C₂²⋊C₄	—	113	4	no	—
205	C₂×C₄²	—	113	2	no	—
206	C₂×C₄²	—	113	1	no	—
207	C₂×C₄²	—	113	4	no	—
208	C₂×C₄²	—	113	2	no	—
209	C₂×C₄²	—	113	1	no	—
210	C₂×C₄²	—	113	1	no	—
211	C₂×C₄²	—	113	1	no	—
212	C₂×C₄²	—	113	1	no	—
213	C₂×C₄²	—	113	2	no	—
214	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
215	C₂⁴	—	113	1	no	—
216	C₂×Q₈	—	113	8	no	—
217	C₂×Q₈	—	113	4	no	—
218	C₂×Q₈	—	226	16	no	—
219	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
220	C₂²×C₄	—	113	1	no	—
221	C₂²×C₄	—	113	8	no	—
222	C₂²×C₄	—	113	1	no	—
223	C₂²×C₄	—	113	2	no	—
224	C₂²×C₄	—	113	1	no	—
225	C₂²×C₄	—	113	1	no	—
226	C₂²×C₄	—	113	1	no	—
227	C₂²×C₄	—	113	2	no	—
228	C₂²×C₄	—	113	2	no	—
229	C₂²×C₄	—	113	2	no	—
230	C₂²×C₄	—	113	2	no	—
231	C₂²×C₄	—	113	8	no	—
232	C₂²×C₄	—	226	8	no	—
233	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
235	C₄⋊C₄	—	113	8	no	—
236	C₄⋊C₄	—	113	4	no	—
237	C₄⋊C₄	—	113	8	no	—
238	C₄⋊C₄	—	113	8	no	—
239	C₄⋊C₄	—	113	8	no	—
240	C₄⋊C₄	—	113	8	no	—
241	C₄⋊C₄	—	113	4	no	—
242	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
243	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
244	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
245	C₂²⋊C₄	—	113	4	no	—
246	C₂²⋊C₄	—	113	8	no	—
247	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
248	C₄²	—	113	2	no	—
249	C₄²	—	113	2	no	—
250	C₄²	—	113	2	no	—
251	C₄²	—	113	1	no	—
252	C₄²	—	113	2	no	—
253	C₄²	—	113	1	no	—
254	C₄²	—	113	8	no	—
255	C₄²	—	113	2	no	—
256	C₄²	—	113	4	no	—
257	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
258	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
259	C₄²	C₂²×D₁₁₃	1	2	yes	—
260	C₄²	C₂²×D₁₁₃	1	4	yes	—
261	C₂³	—	113	1	no	—
262	C₂³	—	113	4	no	—
263	C₂³	—	113	1	no	—
264	C₂³	—	226	4	no	—
265	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
266	Q₈	—	2	8	no	—
267	Q₈	—	226	16	no	—
268	Q₈	—	226	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	113	4	no	—
271	C₂×C₄	—	113	1	no	—
272	C₂×C₄	—	113	1	no	—
273	C₂×C₄	—	113	2	no	—
274	C₂×C₄	—	113	4	no	—
275	C₂×C₄	—	113	4	no	—
276	C₂×C₄	—	113	4	no	—
277	C₂×C₄	—	113	4	no	—
278	C₂×C₄	—	113	8	no	—
279	C₂×C₄	—	113	2	no	—
280	C₂×C₄	—	113	8	no	—
281	C₂×C₄	—	226	4	no	—
282	C₂×C₄	—	226	4	no	—
283	C₂×C₄	—	226	16	no	—
284	C₂×C₄	—	226	4	no	—
285	C₂×C₄	—	226	4	no	—
286	C₂×C₄	Q₈×D₁₁₃	1	4	yes	complemented
287	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
288	C₂×C₄	C₂³×D₁₁₃	1	8	yes	—
289	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
291	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	113	2	no	—
294	C₂²	—	113	4	no	—
295	C₂²	—	113	2	no	—
296	C₂²	—	226	8	no	—
297	C₂²	—	226	2	no	—
298	C₂²	C₂×Q₈×D₁₁₃	1	2	yes	—
299	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	8	no	—
302	C₄	—	113	2	no	—
303	C₄	—	113	2	no	—
304	C₄	—	113	4	no	—
305	C₄	—	226	2	no	—
306	C₄	—	226	8	no	—
307	C₄	C₂×Q₈×D₁₁₃	1	2	yes	—
308	C₄	C₂×Q₈×D₁₁₃	1	2	yes	—
309	C₄	#41×D₁₁₃	1	4	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	113	4	no	—
312	C₂	—	226	2	no	—
313	C₂	C₂²×Q₈×D₁₁₃	1	1	yes	characteristic
314	C₂	C₂×#41×D₁₁₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	#533×C₁₁₃	C₂
prime radical	C₂³×D₁₁₃	C₂³
hypercenter	#533	D₁₁₃
hypocenter	C₁₁₃	C₂×#533
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #20299155	C₂	semidirect
2. #20299154	C₂	semidirect
3. #20299153	C₂	semidirect
4. #20299152	C₂	semidirect
5. #20299151	C₂	semidirect
6. #20299150	C₂	semidirect
7. C₄⋊Q₈×D₁₁₃	C₂	semidirect
8. #125×D₁₁₃	C₂	semidirect
10. C₄×Q₈×D₁₁₃	C₂	semidirect
13. #533×C₁₁₃	C₂	semidirect
17. #16813622	C₂²	semidirect
18. #16813621	C₂²	semidirect
19. #16813621	C₂²	semidirect
20. #16813620	C₂²	semidirect
21. #16813614	C₂²	semidirect
22. #16813613	C₂²	semidirect
23. #16813612	C₂²	semidirect
29. Q₈×Dic₁₁₃	C₂²	semidirect
30. C₄×Dic₂₂₆	C₂²	semidirect
31. C₄×Dic₂₂₆	C₂²	semidirect
32. #16813587	C₂²	semidirect
33. #16813587	C₂²	semidirect
51. C₄⋊Q₈×C₁₁₃	C₂²	semidirect
52. #125×C₁₁₃	C₂²	semidirect
54. C₄×Q₈×C₁₁₃	C₂²	semidirect
64. C₂×C₄×D₁₁₃	Q₈	semidirect
110. C₂×C₄×C₁₁₃	C₂×Q₈	semidirect
117. C₂×Dic₁₁₃	C₂×Q₈	semidirect
129. C₄×D₁₁₃	C₂×Q₈	semidirect
130. C₄×D₁₁₃	C₂×Q₈	semidirect
156. D₁₁₃	#533	direct (N × H)
158. C₁₁₃	C₂×#533	semidirect
165. #533	D₁₁₃	direct (N × H)
179. C₄⋊Q₈	C₂×D₁₁₃	semidirect
183. #125	C₂×D₁₁₃	semidirect
195. C₄×Q₈	C₂×D₁₁₃	semidirect
286. C₂×C₄	Q₈×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299155	2
#20299154	2
#20299153	2
#20299152	2
#20299151	2
#20299150	2
C₄⋊Q₈×D₁₁₃	2
#125×D₁₁₃	2
C₂²⋊Q₈×D₁₁₃	2
C₄×Q₈×D₁₁₃	2
#117×D₁₁₃	2
C₂×C₄²×D₁₁₃	2
#533×C₁₁₃	2
C₂×#533	113

Subgroups — #533×D113 #28403859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #533×D₁₁₃ #28403859