Subgroups — #28399610

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	C₂×C₄×C₂²⋊D₁₁₃	C₂	1	1	yes	maximal, characteristic
2	#20298674	C₂	1	2	yes	maximal, complemented
3	#20298667	C₂	1	2	yes	maximal, complemented
4	#20298666	C₂	1	2	yes	maximal, complemented
5	#20298661	C₂	1	2	yes	maximal, complemented
6	#20298652	C₂	1	1	yes	maximal, characteristic, complemented
7	#20298648	C₂	1	1	yes	maximal, characteristic, complemented
8	#20298560	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊#11744751	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₈×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#407×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×#14268630	C₂²	1	2	yes	complemented
15	C₂²×#11744751	C₂²	1	2	yes	complemented
16	C₂×C₈×D₁₁₃	C₂²	1	2	yes	complemented
17	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
25	C₂×#14268626	C₄	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₁₃	C₄	1	1	yes	characteristic
27	C₂×#14268624	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
29	#16813551	C₂²	1	2	yes	complemented
30	#16813551	C₂²	1	2	yes	complemented
31	C₂²⋊#11744751	C₂²	1	1	yes	characteristic
32	C₂²⋊#11744751	C₂²	1	1	yes	characteristic
33	C₂²⋊#11744751	C₂²	1	2	yes	complemented
34	#16813534	C₂²	1	2	yes	complemented
35	#16813534	C₂²	1	2	yes	complemented
36	#16813532	C₂²	1	2	yes	complemented
37	C₈×Dic₁₁₃	C₂²	1	2	yes	complemented
38	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	complemented
43	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
45	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₁₃	—	4	2	no	—
53	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
57	#14268630	—	4	2	no	—
58	C₂×#11744751	—	4	2	no	—
59	C₂×#11744751	C₂³	1	2	yes	—
60	C₂×#11744751	C₂³	1	2	yes	—
61	C₈×D₁₁₃	—	4	2	no	—
62	#14268626	C₂×C₄	1	1	yes	characteristic
63	#14268626	C₂×C₄	1	1	yes	characteristic
64	#14268626	C₂×C₄	1	1	yes	characteristic
65	#14268626	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
70	#14268624	C₂×C₄	1	1	yes	characteristic
71	#14268624	C₂×C₄	1	1	yes	characteristic
72	#14268624	C₂×C₄	1	1	yes	characteristic
73	#14268624	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
78	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁₃	—	4	2	no	—
80	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
86	M₁₆×C₁₁₃	—	4	2	no	—
87	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
88	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
89	C₂³×C₁₁₃	—	2	2	no	—
90	C₂³×C₁₁₃	—	2	2	no	—
91	C₂³×C₁₁₃	—	4	1	no	—
92	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×D₁₁₃	—	2	2	no	—
96	C₂²×D₁₁₃	—	2	2	no	—
97	C₂²×D₁₁₃	—	4	1	no	—
98	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₁₃	—	2	2	no	—
101	C₂×C₄×C₁₁₃	—	2	2	no	—
102	C₂×C₄×C₁₁₃	—	2	2	no	—
103	C₂×C₄×C₁₁₃	—	4	1	no	—
104	C₂×C₄×C₁₁₃	—	4	1	no	—
105	C₂×C₄×C₁₁₃	—	4	1	no	—
106	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₁₃	—	2	2	no	—
111	C₂²⋊D₁₁₃	—	2	2	no	—
112	C₂²⋊D₁₁₃	—	2	2	no	—
113	C₂²⋊D₁₁₃	M₁₆	1	4	yes	complemented
114	C₂×Dic₁₁₃	—	2	2	no	—
115	C₂×Dic₁₁₃	—	2	2	no	—
116	C₂×Dic₁₁₃	—	4	1	no	—
117	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×D₁₁₃	—	2	2	no	—
124	C₄×D₁₁₃	—	2	2	no	—
125	C₄×D₁₁₃	—	4	1	no	—
126	C₄×D₁₁₃	—	4	1	no	—
127	C₈×C₁₁₃	—	2	2	no	—
128	C₈×C₁₁₃	—	2	2	no	—
129	#11744751	—	2	2	no	—
130	#11744751	—	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	—	2	1	no	—
137	C₂²×C₁₁₃	—	2	1	no	—
138	C₂²×C₁₁₃	—	4	4	no	—
139	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
140	C₂²×C₁₁₃	#131	1	2	yes	—
141	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×D₁₁₃	—	2	1	no	—
143	C₂×D₁₁₃	—	2	1	no	—
144	C₂×D₁₁₃	—	2	1	no	—
145	C₂×D₁₁₃	—	2	1	no	—
146	C₂×D₁₁₃	—	2	1	no	—
147	C₂×D₁₁₃	—	4	4	no	—
148	C₂×D₁₁₃	C₂×M₁₆	1	2	yes	—
149	C₄×C₁₁₃	—	2	1	no	—
150	C₄×C₁₁₃	—	2	1	no	—
151	C₄×C₁₁₃	—	2	1	no	—
152	C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
153	Dic₁₁₃	—	2	1	no	—
154	Dic₁₁₃	—	2	1	no	—
155	Dic₁₁₃	—	2	1	no	—
156	Dic₁₁₃	—	2	1	no	—
157	Dic₁₁₃	—	2	1	no	—
158	Dic₁₁₃	C₂×M₁₆	1	2	yes	—
159	C₂×C₁₁₃	—	2	2	no	—
160	C₂×C₁₁₃	—	2	2	no	—
161	C₂×C₁₁₃	—	4	1	no	—
162	C₂×C₁₁₃	#518	1	1	yes	characteristic
163	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
164	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
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	D₁₁₃	1	1	yes	characteristic, complemented
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	C₂×D₁₁₃	1	1	yes	characteristic
183	C₂×M₁₆	—	113	2	no	—
184	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	complemented
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	—	113	1	no	—
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	1	no	—
202	C₂²⋊C₈	—	113	1	no	—
203	C₂²⋊C₈	—	113	2	no	—
204	C₂²⋊C₈	—	113	2	no	—
205	C₂²⋊C₈	—	113	2	no	—
206	C₂²⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
207	C₂²⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
208	C₂²⋊C₈	C₂×D₁₁₃	1	2	yes	complemented
209	#97	—	113	2	no	—
210	C₄×C₈	—	113	2	no	—
211	C₂⁴	—	113	1	no	—
212	C₂⁴	C₄×D₁₁₃	1	1	yes	characteristic
213	C₂×D₄	—	113	1	no	—
214	C₂×D₄	—	113	1	no	—
215	C₂×D₄	—	113	1	no	—
216	C₂×D₄	—	113	1	no	—
217	C₂×D₄	—	226	4	no	—
218	C₂²×C₄	—	4	2	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	—	452	2	no	—
227	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
233	M₁₆	—	4	2	no	—
234	M₁₆	—	452	2	no	—
235	C₂×C₈	—	113	2	no	—
236	C₂×C₈	—	113	2	no	—
237	C₂×C₈	—	452	2	no	—
238	C₂×C₈	—	452	2	no	—
239	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
240	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
241	C₄⋊C₄	—	113	1	no	—
242	C₄⋊C₄	—	113	1	no	—
243	C₄⋊C₄	—	113	1	no	—
244	C₄⋊C₄	—	113	1	no	—
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	—	113	1	no	—
252	C₂²⋊C₄	—	113	1	no	—
253	C₄²	—	113	1	no	—
254	C₄²	—	113	1	no	—
255	C₄²	—	113	1	no	—
256	C₄²	—	113	1	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	113	1	no	—
261	C₂³	—	113	1	no	—
262	C₂³	—	452	1	no	—
263	C₂³	—	226	2	no	—
264	C₂³	—	226	2	no	—
265	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
266	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
267	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
268	D₄	—	113	4	no	—
269	D₄	—	226	2	no	—
270	D₄	—	226	2	no	—
271	D₄	—	226	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	113	1	no	—
279	C₂×C₄	—	113	1	no	—
280	C₂×C₄	—	113	1	no	—
281	C₂×C₄	—	113	1	no	—
282	C₂×C₄	—	113	1	no	—
283	C₂×C₄	—	113	1	no	—
284	C₂×C₄	—	452	1	no	—
285	C₂×C₄	—	452	1	no	—
286	C₂×C₄	—	452	1	no	—
287	C₂×C₄	—	226	2	no	—
288	C₂×C₄	—	226	2	no	—
289	C₂×C₄	—	226	2	no	—
290	C₂×C₄	—	226	2	no	—
291	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	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	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	113	2	no	—
308	C₂²	—	452	4	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	M₁₆×D₁₁₃	1	2	yes	—
315	C₂²	#16813630	1	2	yes	—
316	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	113	2	no	—
321	C₄	—	226	1	no	—
322	C₄	—	226	1	no	—
323	C₄	—	226	1	no	—
324	C₄	—	226	1	no	—
325	C₄	—	226	1	no	—
326	C₄	#16813667	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	452	1	no	—
331	C₂	—	226	2	no	—
332	C₂	#20299071	1	1	yes	characteristic
333	C₂	C₂×M₁₆×D₁₁₃	1	1	yes	characteristic
334	C₂	C₂×#16813630	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#407×C₁₁₃	C₂
prime radical	C₂²×C₂²⋊D₁₁₃	C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#407	D₁₁₃
hypocenter	C₁₁₃	#2974
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #20298674	C₂	semidirect
3. #20298667	C₂	semidirect
4. #20298666	C₂	semidirect
5. #20298661	C₂	semidirect
6. #20298652	C₂	semidirect
7. #20298648	C₂	semidirect
8. #20298560	C₂	semidirect
9. C₂×C₂²⋊#11744751	C₂	semidirect
10. C₂²⋊C₈×D₁₁₃	C₂	semidirect
11. #407×C₁₁₃	C₂	semidirect
14. C₂×#14268630	C₂²	semidirect
15. C₂²×#11744751	C₂²	semidirect
16. C₂×C₈×D₁₁₃	C₂²	semidirect
29. #16813551	C₂²	semidirect
30. #16813551	C₂²	semidirect
33. C₂²⋊#11744751	C₂²	semidirect
34. #16813534	C₂²	semidirect
35. #16813534	C₂²	semidirect
36. #16813532	C₂²	semidirect
37. C₈×Dic₁₁₃	C₂²	semidirect
39. C₂×M₁₆×C₁₁₃	C₂²	semidirect
42. C₂²⋊C₈×C₁₁₃	C₂²	semidirect
113. C₂²⋊D₁₁₃	M₁₆	semidirect
168. C₁₁₃	#2974	semidirect
177. #407	D₁₁₃	semidirect
184. C₂×M₁₆	C₂×D₁₁₃	semidirect
208. C₂²⋊C₈	C₂×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊D₁₁₃	2
#20298674	2
#20298667	2
#20298666	2
#20298661	2
#20298652	2
#20298648	2
#20298560	2
C₂×C₂²⋊#11744751	2
C₂²⋊C₈×D₁₁₃	2
#407×C₁₁₃	2
#2974	113