Subgroups — C₂×#20299286 #28404084

Order 14464.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299286	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813617	C₂	1	2	yes	maximal, complemented
3	C₂×#16813613	C₂	1	2	yes	maximal, complemented
4	C₂×#16813607	C₂	1	2	yes	maximal
5	C₂×#16813602	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16813602	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic
9	C₂×#16813580	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813580	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
14	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
18	#16813617	C₂²	1	16	yes	complemented
19	#16813613	C₂²	1	16	yes	complemented
20	#16813607	C₂²	1	16	yes	—
21	#16813602	C₂²	1	8	yes	complemented
22	#16813602	C₂²	1	8	yes	complemented
23	C₄×Dic₂₂₆	C₂²	1	8	yes	complemented
24	C₄×D₄₅₂	C₂²	1	8	yes	—
25	#16813580	C₂²	1	8	yes	complemented
26	#16813580	C₂²	1	16	yes	complemented
27	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
31	C₂×#14268626	C₂²	1	2	yes	—
32	C₂×#14268626	C₂²	1	2	yes	—
33	C₄⋊C₄×D₁₁₃	C₂²	1	8	yes	complemented
34	C₂×#14268624	C₂²	1	1	yes	characteristic
35	C₂×#14268624	C₂²	1	2	yes	—
36	C₂×#14268624	C₂²	1	2	yes	—
37	C₂×#14268624	C₂²	1	1	yes	characteristic
38	C₂×#14268624	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
41	#125×C₁₁₃	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
47	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
48	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₂₆	—	2	4	no	—
50	C₂×Dic₂₂₆	C₂³	1	4	yes	—
51	C₂×D₄₅₂	—	2	4	no	—
52	C₂×D₄₅₂	C₂³	1	4	yes	—
53	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
54	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
56	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₁₃	—	2	8	no	—
58	C₂×C₄×D₁₁₃	—	2	4	no	—
59	C₂×C₄×D₁₁₃	—	2	4	no	—
60	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
61	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
62	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
63	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
64	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
65	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
66	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
67	#14268626	C₂³	1	8	yes	—
68	#14268626	C₂³	1	8	yes	—
69	#14268624	C₂³	1	4	yes	—
70	#14268624	C₂³	1	8	yes	—
71	#14268624	C₂³	1	8	yes	—
72	#14268624	C₂³	1	4	yes	—
73	#14268624	C₂³	1	8	yes	—
74	C₄×Dic₁₁₃	C₂³	1	8	yes	—
75	C₄×Dic₁₁₃	C₂³	1	8	yes	—
76	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
82	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
83	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
84	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
85	C₄²×C₁₁₃	C₂³	1	4	yes	—
86	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²×D₁₁₃	—	2	2	no	—
88	C₂²×D₁₁₃	—	2	2	no	—
89	C₂²×D₁₁₃	—	2	2	no	—
90	C₂²×D₁₁₃	—	2	2	no	—
91	C₂²×D₁₁₃	—	2	2	no	—
92	C₂²×D₁₁₃	#41	1	4	yes	—
93	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
94	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
95	C₂×C₄×C₁₁₃	—	2	4	no	—
96	C₂×C₄×C₁₁₃	—	2	2	no	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	4	no	—
99	C₂×C₄×C₁₁₃	#41	1	4	yes	—
100	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
105	C₂×Dic₁₁₃	—	2	4	no	—
106	C₂×Dic₁₁₃	—	2	2	no	—
107	C₂×Dic₁₁₃	—	2	2	no	—
108	C₂×Dic₁₁₃	—	2	4	no	—
109	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
110	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
111	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
112	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
113	D₄₅₂	—	2	8	no	—
114	C₄×D₁₁₃	—	2	16	no	—
115	C₄×D₁₁₃	—	2	8	no	—
116	C₄×D₁₁₃	—	2	8	no	—
117	Dic₂₂₆	—	2	8	no	—
118	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
119	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
120	C₂²×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
121	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₁₃	—	2	8	no	—
123	C₂×D₁₁₃	—	2	2	no	—
124	C₂×D₁₁₃	—	2	2	no	—
125	C₂×D₁₁₃	—	2	2	no	—
126	C₂×D₁₁₃	—	2	2	no	—
127	C₂×D₁₁₃	—	2	2	no	—
128	C₂×D₁₁₃	—	4	4	no	—
129	C₂×D₁₁₃	C₂×#41	1	4	yes	—
130	C₄×C₁₁₃	—	2	4	no	—
131	C₄×C₁₁₃	—	2	2	no	—
132	C₄×C₁₁₃	—	2	2	no	—
133	C₄×C₁₁₃	—	2	4	no	—
134	C₄×C₁₁₃	C₂×#41	1	4	yes	—
135	Dic₁₁₃	—	2	4	no	—
136	Dic₁₁₃	—	2	4	no	—
137	Dic₁₁₃	—	2	2	no	—
138	Dic₁₁₃	—	2	2	no	—
139	C₂×C₁₁₃	#552	1	4	yes	complemented
140	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
141	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
143	D₁₁₃	—	2	4	no	—
144	D₁₁₃	—	4	2	no	—
145	C₂×#552	—	113	1	no	maximal, Sylow
146	C₁₁₃	C₂×#552	1	1	yes	characteristic, complemented, Sylow
147	#552	—	113	16	no	—
148	C₂×#126	—	113	2	no	—
149	C₂×#125	—	113	2	no	—
150	C₂×#125	D₁₁₃	1	1	yes	characteristic, complemented
151	C₂×#123	—	113	2	no	—
152	C₂×C₂²⋊Q₈	—	113	1	no	—
153	C₂×C₂²⋊Q₈	—	113	1	no	—
154	C₂×C₄×Q₈	—	113	1	no	—
155	C₂×C₄×D₄	—	113	1	no	—
156	C₂×#117	—	113	1	no	—
157	C₂×#117	—	113	2	no	—
158	C₂²×C₄⋊C₄	—	113	1	no	—
159	C₂²×Q₈	—	113	1	no	—
160	C₂²×D₄	—	113	1	no	—
161	C₂³×C₄	—	113	2	no	—
162	C₂³×C₄	—	113	1	no	—
163	C₂³×C₄	—	113	1	no	—
164	#126	—	113	16	no	—
165	#125	—	113	16	no	—
166	#125	C₂×D₁₁₃	1	8	yes	complemented
167	#123	—	113	16	no	—
168	C₂²⋊Q₈	—	113	8	no	—
169	C₂²⋊Q₈	—	113	8	no	—
170	C₄×Q₈	—	113	8	no	—
171	C₄×D₄	—	113	8	no	—
172	#117	—	113	8	no	—
173	#117	—	113	16	no	—
174	C₂×C₄⋊C₄	—	113	1	no	—
175	C₂×C₄⋊C₄	—	113	2	no	—
176	C₂×C₄⋊C₄	—	113	2	no	—
177	C₂×C₄⋊C₄	—	113	1	no	—
178	C₂×C₄⋊C₄	—	113	2	no	—
179	C₂×C₄⋊C₄	—	113	2	no	—
180	C₂×C₄⋊C₄	—	113	8	no	—
181	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
185	C₂×C₂²⋊C₄	—	113	1	no	—
186	C₂×C₂²⋊C₄	—	113	2	no	—
187	C₂×C₂²⋊C₄	—	113	2	no	—
188	C₂×C₂²⋊C₄	—	113	1	no	—
189	C₂×C₂²⋊C₄	—	113	2	no	—
190	C₂×C₄²	—	113	2	no	—
191	C₂×C₄²	—	113	2	no	—
192	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
193	C₂⁴	—	113	1	no	—
194	C₂⁴	—	113	1	no	—
195	C₂×Q₈	—	113	4	no	—
196	C₂×Q₈	—	226	4	no	—
197	C₂×D₄	—	113	4	no	—
198	C₂×D₄	—	226	4	no	—
199	C₂²×C₄	—	113	2	no	—
200	C₂²×C₄	—	113	1	no	—
201	C₂²×C₄	—	113	2	no	—
202	C₂²×C₄	—	113	1	no	—
203	C₂²×C₄	—	113	8	no	—
204	C₂²×C₄	—	113	4	no	—
205	C₂²×C₄	—	113	4	no	—
206	C₂²×C₄	—	226	8	no	—
207	C₂²×C₄	—	226	4	no	—
208	C₂²×C₄	—	226	4	no	—
209	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
211	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
213	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	113	4	no	—
215	C₄⋊C₄	—	113	8	no	—
216	C₄⋊C₄	—	113	8	no	—
217	C₄⋊C₄	—	113	4	no	—
218	C₄⋊C₄	—	113	8	no	—
219	C₄⋊C₄	—	113	8	no	—
220	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
221	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
222	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
223	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
224	C₂²⋊C₄	—	113	4	no	—
225	C₂²⋊C₄	—	113	8	no	—
226	C₂²⋊C₄	—	113	8	no	—
227	C₂²⋊C₄	—	113	4	no	—
228	C₂²⋊C₄	—	113	8	no	—
229	C₄²	—	113	8	no	—
230	C₄²	—	113	8	no	—
231	C₄²	C₂²×D₁₁₃	1	4	yes	—
232	C₂³	—	113	4	no	—
233	C₂³	—	113	2	no	—
234	C₂³	—	113	2	no	—
235	C₂³	—	226	2	no	—
236	C₂³	—	226	2	no	—
237	C₂³	—	226	2	no	—
238	C₂³	—	226	2	no	—
239	C₂³	—	226	2	no	—
240	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
241	Q₈	—	226	8	no	—
242	D₄	—	226	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	113	4	no	—
248	C₂×C₄	—	113	2	no	—
249	C₂×C₄	—	113	4	no	—
250	C₂×C₄	—	113	2	no	—
251	C₂×C₄	—	226	16	no	—
252	C₂×C₄	—	226	4	no	—
253	C₂×C₄	—	226	2	no	—
254	C₂×C₄	—	226	2	no	—
255	C₂×C₄	—	226	4	no	—
256	C₂×C₄	—	226	8	no	—
257	C₂×C₄	—	226	8	no	—
258	C₂×C₄	#14268648	1	4	yes	—
259	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
260	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
261	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
262	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
263	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
264	C₂²	—	113	4	no	—
265	C₂²	—	452	4	no	—
266	C₂²	—	226	2	no	—
267	C₂²	—	226	2	no	—
268	C₂²	—	226	2	no	—
269	C₂²	—	226	8	no	—
270	C₂²	—	226	2	no	—
271	C₂²	—	226	2	no	—
272	C₂²	#41×D₁₁₃	1	2	yes	—
273	C₂²	C₂×#14268648	1	2	yes	—
274	C₂²	#16813669	1	2	yes	—
275	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	226	4	no	—
281	C₄	—	226	2	no	—
282	C₄	—	226	2	no	—
283	C₄	—	226	4	no	—
284	C₄	C₂×#14268648	1	4	yes	—
285	C₂	—	452	2	no	—
286	C₂	—	226	4	no	—
287	C₂	#20299286	1	4	yes	direct product factor
288	C₂	C₂×#41×D₁₁₃	1	1	yes	characteristic
289	C₂	C₂²×#14268648	1	1	yes	characteristic
290	C₂	C₂×#16813669	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×#125×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×D₄₅₂	C₂²
hypercenter	C₂×#125	D₁₁₃
hypocenter	C₁₁₃	C₂×#552
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #20299286	C₂	direct (N × H)
2. C₂×#16813617	C₂	semidirect
3. C₂×#16813613	C₂	semidirect
5. C₂×#16813602	C₂	semidirect
6. C₂×#16813602	C₂	semidirect
7. C₂×C₄×Dic₂₂₆	C₂	semidirect
9. C₂×#16813580	C₂	semidirect
10. C₂×#16813580	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₁₃	C₂	semidirect
12. C₂×#125×C₁₁₃	C₂	semidirect
18. #16813617	C₂²	semidirect
19. #16813613	C₂²	semidirect
21. #16813602	C₂²	semidirect
22. #16813602	C₂²	semidirect
23. C₄×Dic₂₂₆	C₂²	semidirect
25. #16813580	C₂²	semidirect
26. #16813580	C₂²	semidirect
33. C₄⋊C₄×D₁₁₃	C₂²	semidirect
41. #125×C₁₁₃	C₂²	semidirect
139. C₂×C₁₁₃	#552	semidirect
146. C₁₁₃	C₂×#552	semidirect
150. C₂×#125	D₁₁₃	semidirect
166. #125	C₂×D₁₁₃	semidirect
287. C₂	#20299286	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299286	2
C₂×#16813617	2
C₂×#16813613	2
C₂×#16813607	2
C₂×#16813602	2
C₂×C₄×Dic₂₂₆	2
C₂×C₄×D₄₅₂	2
C₂×#16813580	2
C₂×C₄⋊C₄×D₁₁₃	2
C₂×#125×C₁₁₃	2
C₂×#552	113

Subgroups — C2×#20299286 #28404084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299286 #28404084