Subgroups — #28399099

Order 14464.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813667	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813630	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299078	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298601	C₂	1	2	yes	maximal, complemented
5	#20298600	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298596	C₂	1	2	yes	maximal, complemented
7	#20298592	C₂	1	1	yes	maximal, characteristic, complemented
8	#20298591	C₂	1	2	yes	maximal, complemented
9	C₂×#16813543	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₁₃	C₂	1	2	yes	maximal, complemented
11	#418×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16813667	—	2	4	no	—
13	#16813667	C₄	1	8	yes	complemented
14	C₂×#14268647	C₂²	1	1	yes	characteristic
15	C₂×#14268647	C₂²	1	1	yes	characteristic
16	C₂×#14268646	—	2	1	no	—
17	C₂×#14268646	C₂²	1	2	yes	—
18	C₂²×C₂²⋊D₁₁₃	—	2	2	no	—
19	C₂×D₄×D₁₁₃	—	2	1	no	—
20	C₂×D₄×D₁₁₃	C₂²	1	2	yes	—
21	#16813630	—	2	4	no	—
22	C₂×#14268630	C₂²	1	2	yes	complemented
23	C₂²×#11744751	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₈×D₁₁₃	C₂²	1	2	yes	complemented
25	C₄×Dic₂₂₆	—	2	1	no	—
26	C₄×D₄₅₂	—	2	1	no	—
27	#16813582	—	2	2	no	—
28	#16813580	C₂²	1	2	yes	complemented
29	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₄×D₁₁₃	C₂²	1	2	yes	complemented
31	#16813547	C₂²	1	2	yes	complemented
32	#16813545	C₂²	1	1	yes	characteristic, complemented
33	#16813545	C₂²	1	1	yes	characteristic, complemented
34	#16813544	C₂²	1	2	yes	complemented
35	#16813543	C₂²	1	2	yes	complemented
36	#16813543	C₂²	1	2	yes	complemented
37	#16813542	C₂²	1	1	yes	characteristic, complemented
38	#16813542	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
41	#117×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
42	#102×C₁₁₃	C₂²	1	2	yes	complemented
43	#102×C₁₁₃	C₂²	1	2	yes	complemented
44	C₂³×D₁₁₃	—	2	2	no	—
45	#14268647	—	2	2	no	—
46	#14268647	C₂×C₄	1	4	yes	—
47	#14268647	C₂×C₄	1	4	yes	complemented
48	#14268647	D₄	1	4	yes	—
49	#14268646	—	2	4	no	—
50	#14268646	—	2	8	no	—
51	#14268646	C₂×C₄	1	8	yes	complemented
52	C₂×Dic₂₂₆	D₄	1	1	yes	characteristic
53	C₂×Dic₂₂₆	C₂³	1	1	yes	characteristic
54	C₂×D₄₅₂	D₄	1	1	yes	characteristic
55	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
57	C₂×C₂²⋊D₁₁₃	—	2	2	no	—
58	C₂×C₂²⋊D₁₁₃	—	2	16	no	—
59	C₂×C₂²⋊D₁₁₃	—	2	2	no	—
60	C₂×C₂²⋊D₁₁₃	D₄	1	2	yes	—
61	D₄×D₁₁₃	—	2	8	no	—
62	D₄×D₁₁₃	—	2	4	no	—
63	D₄×D₁₁₃	C₂×C₄	1	8	yes	complemented
64	C₂²×Dic₁₁₃	—	2	2	no	—
65	C₂²×Dic₁₁₃	—	2	1	no	—
66	C₂×C₄×D₁₁₃	—	2	1	no	—
67	C₂×C₄×D₁₁₃	—	2	2	no	—
68	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
69	#14268630	—	2	4	no	—
70	C₂×#11744751	—	2	2	no	—
71	C₂×#11744751	C₂³	1	1	yes	characteristic
72	C₂×#11744751	C₂³	1	1	yes	characteristic
73	C₈×D₁₁₃	—	2	4	no	—
74	C₄⋊Dic₁₁₃	—	2	1	no	—
75	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
76	#14268626	—	2	2	no	—
77	#14268624	—	2	2	no	—
78	C₄×Dic₁₁₃	—	2	2	no	—
79	C₂⁴×C₁₁₃	—	2	1	no	—
80	C₂×D₄×C₁₁₃	—	2	1	no	—
81	C₂×D₄×C₁₁₃	—	2	2	no	—
82	C₂×D₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
83	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
84	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
85	M₁₆×C₁₁₃	—	2	2	no	—
86	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
87	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
90	C₄²×C₁₁₃	—	2	1	no	—
91	C₂³×C₁₁₃	—	2	2	no	—
92	C₂³×C₁₁₃	—	2	4	no	—
93	C₂³×C₁₁₃	—	8	1	no	—
94	C₂³×C₁₁₃	—	8	1	no	—
95	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×D₁₁₃	—	2	1	no	—
97	C₂²×D₁₁₃	—	2	8	no	—
98	C₂²×D₁₁₃	—	8	2	no	—
99	C₂²×D₁₁₃	—	8	2	no	—
100	C₂²×D₁₁₃	C₂×D₄	1	2	yes	—
101	D₄×C₁₁₃	—	2	4	no	—
102	D₄×C₁₁₃	—	2	2	no	—
103	D₄×C₁₁₃	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₁₁₃	—	2	1	no	—
105	C₂×C₄×C₁₁₃	—	2	1	no	—
106	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₁₃	—	2	4	no	—
111	C₂²⋊D₁₁₃	—	2	8	no	—
112	C₂²⋊D₁₁₃	—	2	8	no	—
113	C₂²⋊D₁₁₃	C₂²⋊C₄	1	8	yes	—
114	C₂×Dic₁₁₃	—	2	1	no	—
115	C₂×Dic₁₁₃	—	2	8	no	—
116	C₂×Dic₁₁₃	—	2	2	no	—
117	C₂×Dic₁₁₃	—	8	1	no	—
118	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
119	D₄₅₂	—	2	1	no	—
120	D₄₅₂	C₂²⋊C₄	1	4	yes	—
121	D₄₅₂	C₂²×C₄	1	2	yes	—
122	C₄×D₁₁₃	—	2	4	no	—
123	C₄×D₁₁₃	—	8	2	no	—
124	C₄×D₁₁₃	C₂²×C₄	1	4	yes	—
125	C₄×D₁₁₃	C₂×D₄	1	4	yes	—
126	Dic₂₂₆	—	2	1	no	—
127	Dic₂₂₆	C₂²⋊C₄	1	4	yes	—
128	Dic₂₂₆	C₂²×C₄	1	2	yes	—
129	C₈×C₁₁₃	—	2	2	no	—
130	#11744751	—	2	1	no	—
131	#11744751	—	2	1	no	—
132	C₂²×C₁₁₃	—	2	1	no	—
133	C₂²×C₁₁₃	—	2	1	no	—
134	C₂²×C₁₁₃	—	2	4	no	—
135	C₂²×C₁₁₃	—	4	2	no	—
136	C₂²×C₁₁₃	—	4	2	no	—
137	C₂²×C₁₁₃	—	8	4	no	—
138	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₁₃	—	2	2	no	—
141	C₂×D₁₁₃	—	2	2	no	—
142	C₂×D₁₁₃	—	2	2	no	—
143	C₂×D₁₁₃	—	4	1	no	—
144	C₂×D₁₁₃	—	4	1	no	—
145	C₂×D₁₁₃	—	8	8	no	—
146	C₂×D₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
147	C₄×C₁₁₃	—	4	1	no	—
148	C₄×C₁₁₃	—	4	1	no	—
149	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
150	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
152	Dic₁₁₃	—	2	2	no	—
153	Dic₁₁₃	—	4	2	no	—
154	Dic₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₁₁₃	—	2	2	no	—
156	C₂×C₁₁₃	—	4	4	no	—
157	C₂×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₁₁₃	#576	1	1	yes	characteristic
159	C₂×C₁₁₃	#577	1	1	yes	characteristic
160	D₁₁₃	—	2	4	no	—
161	D₁₁₃	—	4	2	no	—
162	#2897	—	113	1	no	maximal, Sylow
163	C₁₁₃	#2897	1	1	yes	characteristic, complemented, Sylow
164	C₂×H(5,𝔽₂)	—	113	1	no	—
165	C₂×#131	—	113	1	no	—
166	#520	—	113	1	no	—
167	#418	—	113	2	no	—
168	#418	D₁₁₃	1	1	yes	characteristic, complemented
169	#417	—	113	1	no	—
170	#417	—	113	2	no	—
171	#416	—	113	1	no	—
172	#416	—	113	2	no	—
173	C₂×#102	—	113	1	no	—
174	C₂×#102	—	113	2	no	—
175	H(5,𝔽₂)	—	113	8	no	—
176	H(5,𝔽₂)	—	226	4	no	—
177	C₂×#41	—	113	1	no	—
178	C₂×#41	—	113	2	no	—
179	C₂×#41	—	113	1	no	—
180	C₂×#41	—	226	1	no	—
181	C₂²×D₄	—	113	2	no	—
182	C₂²×D₄	—	226	1	no	—
183	C₂²×D₄	—	226	2	no	—
184	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
185	#131	—	226	4	no	—
186	C₂×M₁₆	—	113	2	no	—
187	C₂×M₁₆	C₂×D₁₁₃	1	1	yes	characteristic, complemented
188	C₂²×C₈	—	113	1	no	—
189	C₂²×C₈	—	113	2	no	—
190	C₄×Q₈	—	226	1	no	—
191	C₄×D₄	—	226	1	no	—
192	C₄×D₄	—	226	2	no	—
193	#117	—	113	2	no	—
194	#117	C₂×D₁₁₃	1	1	yes	characteristic, complemented
195	C₂×C₄⋊C₄	—	113	1	no	—
196	C₂×C₄⋊C₄	—	113	2	no	—
197	#103	—	113	2	no	—
198	#103	—	113	1	no	—
199	#103	—	113	1	no	—
200	#102	—	113	2	no	—
201	#102	—	113	2	no	—
202	#102	—	113	2	no	—
203	#102	—	113	1	no	—
204	#102	—	113	1	no	—
205	#102	C₂×D₁₁₃	1	2	yes	complemented
206	#102	C₂×D₁₁₃	1	2	yes	complemented
207	C₂⁴	—	2	1	no	—
208	C₂⁴	—	226	2	no	—
209	#41	—	113	4	no	—
210	#41	—	113	4	no	—
211	#41	—	113	8	no	—
212	#41	—	113	4	no	—
213	#41	—	226	4	no	—
214	#41	—	226	8	no	—
215	#41	—	226	2	no	—
216	C₂×Q₈	—	113	1	no	—
217	C₂×Q₈	—	113	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	113	2	no	—
221	C₂×D₄	—	113	8	no	—
222	C₂×D₄	—	113	1	no	—
223	C₂×D₄	—	113	1	no	—
224	C₂×D₄	—	226	4	no	—
225	C₂×D₄	—	226	2	no	—
226	C₂×D₄	—	226	8	no	—
227	C₂×D₄	—	226	16	no	—
228	C₂×D₄	—	226	2	no	—
229	C₂×D₄	—	226	1	no	—
230	C₂×D₄	C₄×D₁₁₃	1	4	yes	complemented
231	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
232	C₂²×C₄	—	113	2	no	—
233	C₂²×C₄	—	226	2	no	—
234	C₂²×C₄	—	226	2	no	—
235	C₂²×C₄	—	226	1	no	—
236	C₂²×C₄	—	226	1	no	—
237	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	226	4	no	—
240	C₂×C₈	—	113	1	no	—
241	C₂×C₈	—	113	1	no	—
242	C₂×C₈	—	226	4	no	—
243	C₂×C₈	—	226	2	no	—
244	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
245	C₄⋊C₄	—	113	2	no	—
246	C₄⋊C₄	—	226	2	no	—
247	C₄⋊C₄	—	226	1	no	—
248	C₄⋊C₄	C₂²×D₁₁₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₁₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	226	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	226	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	8	1	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	113	2	no	—
259	C₂³	—	904	2	no	—
260	C₂³	—	904	2	no	—
261	C₂³	—	226	8	no	—
262	C₂³	—	226	1	no	—
263	C₂³	D₄×D₁₁₃	1	1	yes	characteristic
264	Q₈	—	113	4	no	—
265	Q₈	—	113	2	no	—
266	Q₈	—	226	1	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	—	113	8	no	—
270	D₄	—	113	4	no	—
271	D₄	—	113	2	no	—
272	D₄	—	226	8	no	—
273	D₄	—	226	8	no	—
274	D₄	—	226	4	no	—
275	D₄	—	226	1	no	—
276	D₄	C₂×C₄×D₁₁₃	1	4	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	113	2	no	—
280	C₂×C₄	—	113	4	no	—
281	C₂×C₄	—	113	4	no	—
282	C₂×C₄	—	904	2	no	—
283	C₂×C₄	—	904	1	no	—
284	C₂×C₄	—	226	8	no	—
285	C₂×C₄	—	226	2	no	—
286	C₂×C₄	—	226	1	no	—
287	C₂×C₄	—	226	4	no	—
288	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
289	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
291	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	226	1	no	—
294	C₈	—	226	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	113	4	no	—
302	C₂²	—	904	8	no	—
303	C₂²	—	452	1	no	—
304	C₂²	—	452	1	no	—
305	C₂²	—	226	2	no	—
306	C₂²	—	226	2	no	—
307	C₂²	—	226	2	no	—
308	C₂²	C₂²⋊C₄×D₁₁₃	1	2	yes	—
309	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	113	4	no	—
313	C₄	—	452	2	no	—
314	C₄	—	226	2	no	—
315	C₄	C₂²⋊C₄×D₁₁₃	1	2	yes	—
316	C₄	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
317	C₄	C₂×D₄×D₁₁₃	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	4	no	—
320	C₂	—	452	2	no	—
321	C₂	—	226	4	no	—
322	C₂	C₂×C₂²⋊C₄×D₁₁₃	1	1	yes	characteristic
323	C₂	#20299402	1	1	yes	characteristic
324	C₂	#20299408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	#418×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
prime radical	C₂×#16813667	C₂
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#418	D₁₁₃
hypocenter	C₁₁₃	#2897
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
2. C₂×#16813630	C₂	semidirect
3. #20299078	C₂	semidirect
4. #20298601	C₂	semidirect
5. #20298600	C₂	semidirect
6. #20298596	C₂	semidirect
7. #20298592	C₂	semidirect
8. #20298591	C₂	semidirect
9. C₂×#16813543	C₂	semidirect
10. #102×D₁₁₃	C₂	semidirect
11. #418×C₁₁₃	C₂	semidirect
13. #16813667	C₄	semidirect
22. C₂×#14268630	C₂²	semidirect
23. C₂²×#11744751	C₂²	semidirect
24. C₂×C₈×D₁₁₃	C₂²	semidirect
28. #16813580	C₂²	semidirect
29. C₂×C₄⋊Dic₁₁₃	C₂²	semidirect
30. C₄⋊C₄×D₁₁₃	C₂²	semidirect
31. #16813547	C₂²	semidirect
32. #16813545	C₂²	semidirect
33. #16813545	C₂²	semidirect
34. #16813544	C₂²	semidirect
35. #16813543	C₂²	semidirect
36. #16813543	C₂²	semidirect
37. #16813542	C₂²	semidirect
38. #16813542	C₂²	semidirect
40. C₂×M₁₆×C₁₁₃	C₂²	semidirect
41. #117×C₁₁₃	C₂²	semidirect
42. #102×C₁₁₃	C₂²	semidirect
43. #102×C₁₁₃	C₂²	semidirect
47. #14268647	C₂×C₄	semidirect
51. #14268646	C₂×C₄	semidirect
63. D₄×D₁₁₃	C₂×C₄	semidirect
82. C₂×D₄×C₁₁₃	C₂×C₄	semidirect
163. C₁₁₃	#2897	semidirect
168. #418	D₁₁₃	semidirect
187. C₂×M₁₆	C₂×D₁₁₃	semidirect
194. #117	C₂×D₁₁₃	semidirect
205. #102	C₂×D₁₁₃	semidirect
206. #102	C₂×D₁₁₃	semidirect
230. C₂×D₄	C₄×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813667	2
C₂×#16813630	2
#20299078	2
#20298601	2
#20298600	2
#20298596	2
#20298592	2
#20298591	2
C₂×#16813543	2
#102×D₁₁₃	2
#418×C₁₁₃	2
#2897	113