Subgroups — #28407431

Order 14464.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299631	C₂	1	2	yes	maximal, complemented
2	#20299629	C₂	1	2	yes	maximal, complemented
3	#20299628	C₂	1	2	yes	maximal, complemented
4	#20299625	C₂	1	2	yes	maximal, complemented
5	#20299621	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299616	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299570	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299569	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16813679	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299566	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268657	C₂²	1	2	yes	complemented
13	C₂×#14268656	C₂²	1	2	yes	complemented
14	C₂×#14268655	C₂²	1	2	yes	complemented
15	C₂²×#11744753	C₂²	1	2	yes	complemented
16	#16813689	C₂²	1	2	yes	complemented
17	#16813689	C₂²	1	2	yes	complemented
18	#16813679	C₂²	1	1	yes	characteristic
19	#16813679	C₂²	1	1	yes	characteristic
20	#16813679	C₂²	1	1	yes	characteristic
21	#16813679	C₂²	1	1	yes	characteristic
22	#16813679	C₂²	1	2	yes	complemented
23	#16813679	C₂²	1	2	yes	complemented
24	C₂²⋊#11744753	C₂²	1	2	yes	complemented
25	C₂²⋊#11744753	C₂²	1	2	yes	complemented
26	#16813677	C₂²	1	2	yes	complemented
27	C₄×#11744753	C₂²	1	2	yes	complemented
28	C₂×D₄×D₁₁₃	C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
31	C₄×D₄₅₂	C₄	1	1	yes	characteristic
32	D₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
33	#16813584	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
36	#16813582	C₂²	1	1	yes	characteristic
37	#16813582	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₁₁₃	C₄	1	1	yes	characteristic
39	C₂²⋊C₄×D₁₁₃	C₄	1	1	yes	characteristic
40	C₂²⋊C₄×D₁₁₃	C₄	1	1	yes	characteristic
41	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₁₁₃	C₄	1	1	yes	characteristic
43	#14268657	—	4	2	no	—
44	#14268656	—	4	2	no	—
45	#14268655	—	4	2	no	—
46	C₂×#11744753	—	4	2	no	—
47	C₂×#11744753	C₂³	1	2	yes	—
48	C₂×#11744753	C₂³	1	2	yes	—
49	C₂×#11744753	C₂³	1	2	yes	—
50	C₂×#11744753	C₂³	1	2	yes	—
51	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
56	D₄×D₁₁₃	—	2	4	no	—
57	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₁₃	—	4	2	no	—
60	C₂×C₄×D₁₁₃	—	4	2	no	—
61	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
71	#14268626	C₂×C₄	1	1	yes	characteristic
72	#14268626	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
75	#14268624	C₂×C₄	1	1	yes	characteristic
76	#14268624	C₂×C₄	1	1	yes	characteristic
77	#14268624	C₂×C₄	1	1	yes	characteristic
78	#14268624	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
88	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
89	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₁₁₃	—	2	2	no	—
92	C₂²×D₁₁₃	—	2	2	no	—
93	C₂²×D₁₁₃	—	2	2	no	—
94	C₂²×D₁₁₃	—	2	2	no	—
95	C₂²×D₁₁₃	—	4	1	no	—
96	C₂²×D₁₁₃	—	4	1	no	—
97	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
100	D₄×C₁₁₃	—	2	2	no	—
101	C₂×C₄×C₁₁₃	—	4	1	no	—
102	C₂×C₄×C₁₁₃	—	4	1	no	—
103	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²⋊D₁₁₃	—	2	2	no	—
109	C₂²⋊D₁₁₃	M₁₆	1	4	yes	complemented
110	C₂×Dic₁₁₃	—	2	2	no	—
111	C₂×Dic₁₁₃	—	2	2	no	—
112	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
117	D₄₅₂	—	2	2	no	—
118	C₄×D₁₁₃	—	2	2	no	—
119	C₄×D₁₁₃	—	2	2	no	—
120	C₄×D₁₁₃	—	2	2	no	—
121	C₄×D₁₁₃	—	2	2	no	—
122	C₄×D₁₁₃	—	2	2	no	—
123	C₄×D₁₁₃	—	4	1	no	—
124	C₄×D₁₁₃	—	4	1	no	—
125	C₄×D₁₁₃	—	4	1	no	—
126	C₄×D₁₁₃	—	4	1	no	—
127	#11744753	—	2	2	no	—
128	#11744753	—	2	2	no	—
129	#11744753	—	2	2	no	—
130	#11744753	—	2	2	no	—
131	C₂²×C₁₁₃	—	2	1	no	—
132	C₂²×C₁₁₃	—	2	1	no	—
133	C₂²×C₁₁₃	—	2	1	no	—
134	C₂²×C₁₁₃	—	2	1	no	—
135	C₂²×C₁₁₃	—	2	1	no	—
136	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
137	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₁₁₃	—	2	1	no	—
139	C₂×D₁₁₃	—	2	1	no	—
140	C₂×D₁₁₃	—	2	1	no	—
141	C₂×D₁₁₃	—	2	1	no	—
142	C₂×D₁₁₃	—	2	1	no	—
143	C₂×D₁₁₃	—	2	1	no	—
144	C₂×D₁₁₃	—	2	1	no	—
145	C₂×D₁₁₃	—	4	4	no	—
146	C₂×D₁₁₃	—	4	4	no	—
147	C₂×D₁₁₃	C₂×M₁₆	1	2	yes	—
148	C₂×D₁₁₃	#131	1	2	yes	—
149	C₄×C₁₁₃	—	2	1	no	—
150	C₄×C₁₁₃	—	2	1	no	—
151	C₄×C₁₁₃	—	2	1	no	—
152	C₄×C₁₁₃	—	2	1	no	—
153	C₄×C₁₁₃	—	2	1	no	—
154	Dic₁₁₃	—	2	1	no	—
155	Dic₁₁₃	—	2	1	no	—
156	Dic₁₁₃	—	2	1	no	—
157	Dic₁₁₃	C₂×M₁₆	1	2	yes	—
158	Dic₁₁₃	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₁₁₃	—	2	2	no	—
160	C₂×C₁₁₃	—	4	1	no	—
161	C₂×C₁₁₃	#518	1	1	yes	characteristic
162	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
164	D₁₁₃	—	2	2	no	—
165	D₁₁₃	—	2	2	no	—
166	D₁₁₃	—	4	1	no	—
167	#2974	—	113	1	no	maximal, Sylow
168	C₁₁₃	#2974	1	1	yes	characteristic, complemented, Sylow
169	C₂×C₄×D₄	—	113	1	no	—
170	#436	—	113	2	no	—
171	#435	—	113	2	no	—
172	#435	—	113	2	no	—
173	C₈×D₄	—	113	2	no	—
174	#432	—	113	1	no	—
175	#431	—	113	1	no	—
176	#407	—	113	1	no	—
177	#407	—	113	1	no	—
178	C₂×C₂²⋊C₈	—	113	1	no	—
179	C₂×C₂²⋊C₈	—	113	1	no	—
180	C₂²×D₄	—	113	1	no	—
181	C₂³×C₄	—	113	1	no	—
182	C₂³×C₄	—	113	1	no	—
183	C₂×M₁₆	—	113	2	no	—
184	C₂×M₁₆	—	113	2	no	—
185	C₂²×C₈	—	113	2	no	—
186	C₂²×C₈	—	113	2	no	—
187	C₄×D₄	—	113	1	no	—
188	C₄×D₄	—	113	1	no	—
189	C₄×D₄	—	113	1	no	—
190	C₄×D₄	—	113	1	no	—
191	C₄×D₄	—	113	1	no	—
192	C₄×D₄	—	113	1	no	—
193	C₄×D₄	—	113	1	no	—
194	C₄×D₄	#90673	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	113	1	no	—
196	C₂×C₂²⋊C₄	—	113	1	no	—
197	C₂×C₂²⋊C₄	—	113	1	no	—
198	C₂×C₄²	—	113	1	no	—
199	C₄⋊C₈	—	113	2	no	—
200	C₄⋊C₈	—	113	2	no	—
201	C₂²⋊C₈	—	113	2	no	—
202	C₂²⋊C₈	—	113	2	no	—
203	C₂²⋊C₈	—	113	1	no	—
204	C₂²⋊C₈	—	113	1	no	—
205	C₂²⋊C₈	—	113	1	no	—
206	C₂²⋊C₈	—	113	1	no	—
207	C₂²⋊C₈	—	113	2	no	—
208	C₂²⋊C₈	—	113	2	no	—
209	#97	—	113	2	no	—
210	C₄×C₈	—	113	2	no	—
211	C₂⁴	—	113	1	no	—
212	C₂⁴	—	113	1	no	—
213	C₂×D₄	—	113	1	no	—
214	C₂×D₄	—	113	1	no	—
215	C₂×D₄	—	113	1	no	—
216	C₂×D₄	—	226	4	no	—
217	C₂×D₄	C₂×#90673	1	1	yes	characteristic
218	C₂²×C₄	—	113	1	no	—
219	C₂²×C₄	—	113	1	no	—
220	C₂²×C₄	—	113	1	no	—
221	C₂²×C₄	—	113	1	no	—
222	C₂²×C₄	—	113	1	no	—
223	C₂²×C₄	—	113	1	no	—
224	C₂²×C₄	—	113	1	no	—
225	C₂²×C₄	—	113	1	no	—
226	C₂²×C₄	—	113	1	no	—
227	C₂²×C₄	—	113	1	no	—
228	C₂²×C₄	—	113	1	no	—
229	C₂²×C₄	—	452	2	no	—
230	C₂²×C₄	—	452	2	no	—
231	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
232	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
233	M₁₆	—	452	2	no	—
234	M₁₆	—	452	2	no	—
235	C₂×C₈	—	113	2	no	—
236	C₂×C₈	—	113	2	no	—
237	C₂×C₈	—	113	2	no	—
238	C₂×C₈	—	113	2	no	—
239	C₂×C₈	—	452	2	no	—
240	C₂×C₈	—	452	2	no	—
241	C₄⋊C₄	—	113	1	no	—
242	C₄⋊C₄	—	113	1	no	—
243	C₄⋊C₄	—	113	1	no	—
244	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
245	C₂²⋊C₄	—	113	1	no	—
246	C₂²⋊C₄	—	113	1	no	—
247	C₂²⋊C₄	—	113	1	no	—
248	C₂²⋊C₄	—	113	1	no	—
249	C₂²⋊C₄	—	113	1	no	—
250	C₂²⋊C₄	—	113	1	no	—
251	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
252	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
253	C₄²	—	113	1	no	—
254	C₄²	—	113	1	no	—
255	C₄²	—	113	1	no	—
256	C₄²	C₂×#90673	1	1	yes	characteristic
257	C₂³	—	113	1	no	—
258	C₂³	—	113	1	no	—
259	C₂³	—	113	1	no	—
260	C₂³	—	452	1	no	—
261	C₂³	—	452	1	no	—
262	C₂³	—	226	2	no	—
263	C₂³	—	226	2	no	—
264	C₂³	—	226	2	no	—
265	C₂³	—	226	2	no	—
266	C₂³	C₂²×#90673	1	1	yes	characteristic
267	C₂³	C₂²×#90673	1	1	yes	characteristic
268	D₄	—	2	2	no	—
269	D₄	—	113	4	no	—
270	D₄	—	226	2	no	—
271	D₄	—	226	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	113	1	no	—
275	C₂×C₄	—	113	1	no	—
276	C₂×C₄	—	113	1	no	—
277	C₂×C₄	—	113	1	no	—
278	C₂×C₄	—	113	1	no	—
279	C₂×C₄	—	452	1	no	—
280	C₂×C₄	—	452	1	no	—
281	C₂×C₄	—	452	1	no	—
282	C₂×C₄	—	452	1	no	—
283	C₂×C₄	—	226	2	no	—
284	C₂×C₄	—	226	2	no	—
285	C₂×C₄	—	226	2	no	—
286	C₂×C₄	—	226	2	no	—
287	C₂×C₄	—	226	2	no	—
288	C₂×C₄	—	226	2	no	—
289	C₂×C₄	—	226	2	no	—
290	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
291	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
292	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
293	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
294	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
295	C₈	—	226	2	no	—
296	C₈	—	226	2	no	—
297	C₈	—	226	2	no	—
298	C₈	—	226	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	113	2	no	—
305	C₂²	—	113	2	no	—
306	C₂²	—	452	4	no	—
307	C₂²	—	452	4	no	—
308	C₂²	—	226	1	no	—
309	C₂²	—	226	1	no	—
310	C₂²	—	226	1	no	—
311	C₂²	—	226	1	no	—
312	C₂²	—	226	1	no	—
313	C₂²	—	226	1	no	—
314	C₂²	—	226	1	no	—
315	C₂²	D₁₁₃⋊M₁₆	1	2	yes	—
316	C₂²	C₂³×#90673	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	113	2	no	—
323	C₄	—	113	2	no	—
324	C₄	—	226	1	no	—
325	C₄	—	226	1	no	—
326	C₄	—	226	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	452	1	no	—
330	C₂	—	226	2	no	—
331	C₂	—	226	2	no	—
332	C₂	#20299665	1	1	yes	characteristic
333	C₂	C₂×D₁₁₃⋊M₁₆	1	1	yes	characteristic
334	C₂	D₁₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₄×D₄×C₁₁₃	C₄
prime radical	C₂×D₄×D₁₁₃	C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₄×D₄	#90673
hypocenter	C₁₁₃	#2974
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20299631	C₂	semidirect
2. #20299629	C₂	semidirect
3. #20299628	C₂	semidirect
4. #20299625	C₂	semidirect
5. #20299621	C₂	semidirect
6. #20299616	C₂	semidirect
7. #20299570	C₂	semidirect
8. #20299569	C₂	semidirect
9. C₂×#16813679	C₂	semidirect
10. #20299566	C₂	semidirect
12. C₂×#14268657	C₂²	semidirect
13. C₂×#14268656	C₂²	semidirect
14. C₂×#14268655	C₂²	semidirect
15. C₂²×#11744753	C₂²	semidirect
16. #16813689	C₂²	semidirect
17. #16813689	C₂²	semidirect
22. #16813679	C₂²	semidirect
23. #16813679	C₂²	semidirect
24. C₂²⋊#11744753	C₂²	semidirect
25. C₂²⋊#11744753	C₂²	semidirect
26. #16813677	C₂²	semidirect
27. C₄×#11744753	C₂²	semidirect
109. C₂²⋊D₁₁₃	M₁₆	semidirect
168. C₁₁₃	#2974	semidirect

Maximal subgroups

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

subgroup	index
#20299631	2
#20299629	2
#20299628	2
#20299625	2
#20299621	2
#20299616	2
#20299570	2
#20299569	2
C₂×#16813679	2
#20299566	2
C₄×D₄×D₁₁₃	2
#2974	113