Subgroups — C₂×#20299134 #28403842

Order 14464.

1232 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299134	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813617	C₂	1	2	yes	maximal, complemented
3	C₂×#16813613	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16813611	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16813606	C₂	1	2	yes	maximal, complemented
6	C₂×#16813596	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16813584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16813582	C₂	1	2	yes	maximal, complemented
10	C₂×#16813580	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
15	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
16	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
18	#16813617	C₂²	1	16	yes	complemented
19	#16813613	C₂²	1	8	yes	complemented
20	#16813611	C₂²	1	8	yes	complemented
21	#16813606	C₂²	1	16	yes	complemented
22	#16813596	C₂²	1	8	yes	—
23	Q₈×Dic₁₁₃	C₂²	1	8	yes	complemented
24	#16813584	C₂²	1	8	yes	complemented
25	#16813582	C₂²	1	16	yes	complemented
26	#16813580	C₂²	1	16	yes	complemented
27	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
30	C₂×#14268626	C₂²	1	2	yes	—
31	C₂×#14268624	C₂²	1	2	yes	—
32	C₂×#14268624	C₂²	1	2	yes	—
33	C₂×#14268624	C₂²	1	2	yes	—
34	C₂×#14268624	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₁₃	C₂²	1	8	yes	complemented
41	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₁₃	C₂²	1	8	yes	complemented
44	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
47	C₂³×D₁₁₃	C₂³	1	2	yes	—
48	C₂×D₄₅₂	—	2	4	no	—
49	C₂×D₄₅₂	C₂³	1	4	yes	—
50	C₂×C₂²⋊D₁₁₃	—	2	8	no	—
51	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
52	C₂²×Dic₁₁₃	—	2	4	no	—
53	C₂²×Dic₁₁₃	—	2	4	no	—
54	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
58	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
59	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
60	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₁₃	—	2	8	no	—
62	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
63	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
64	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
65	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
66	#14268626	C₂³	1	8	yes	—
67	#14268624	C₂³	1	8	yes	—
68	#14268624	C₂³	1	8	yes	—
69	#14268624	C₂³	1	8	yes	—
70	#14268624	C₂³	1	8	yes	—
71	C₄×Dic₁₁₃	C₂³	1	4	yes	—
72	C₄×Dic₁₁₃	C₂³	1	4	yes	—
73	C₄×Dic₁₁₃	C₂³	1	4	yes	—
74	C₄×Dic₁₁₃	C₂³	1	4	yes	—
75	C₄×Dic₁₁₃	C₂³	1	8	yes	—
76	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂×Q₈×C₁₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
79	C₂²×C₄×C₁₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
83	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
84	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
86	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
87	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
88	C₂³×C₁₁₃	—	2	2	no	—
89	C₂³×C₁₁₃	—	2	2	no	—
90	C₂³×C₁₁₃	#41	1	4	yes	—
91	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
92	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
93	C₂²×D₁₁₃	—	2	4	no	—
94	C₂²×D₁₁₃	—	2	4	no	—
95	C₂²×D₁₁₃	—	2	4	no	—
96	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
97	Q₈×C₁₁₃	—	2	8	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	2	4	no	—
100	C₂×C₄×C₁₁₃	—	2	4	no	—
101	C₂×C₄×C₁₁₃	—	2	8	no	—
102	C₂×C₄×C₁₁₃	#41	1	4	yes	—
103	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
105	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
107	C₂²⋊D₁₁₃	—	2	16	no	—
108	C₂×Dic₁₁₃	—	2	2	no	—
109	C₂×Dic₁₁₃	—	2	8	no	—
110	C₂×Dic₁₁₃	—	2	2	no	—
111	C₂×Dic₁₁₃	—	2	2	no	—
112	C₂×Dic₁₁₃	—	2	4	no	—
113	C₂×Dic₁₁₃	—	2	8	no	—
114	C₂×Dic₁₁₃	#41	1	4	yes	—
115	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
116	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
117	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
118	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
119	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
120	D₄₅₂	—	2	8	no	—
121	C₄×D₁₁₃	—	2	16	no	—
122	C₂²×C₁₁₃	—	2	8	no	—
123	C₂²×C₁₁₃	—	2	2	no	—
124	C₂²×C₁₁₃	—	2	2	no	—
125	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
126	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
127	C₂²×C₁₁₃	C₂×#41	1	4	yes	—
128	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
129	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
130	C₂×D₁₁₃	—	2	4	no	—
131	C₂×D₁₁₃	—	2	4	no	—
132	C₂×D₁₁₃	—	2	4	no	—
133	C₂×D₁₁₃	—	4	8	no	—
134	C₄×C₁₁₃	—	2	4	no	—
135	C₄×C₁₁₃	—	2	2	no	—
136	C₄×C₁₁₃	—	2	4	no	—
137	C₄×C₁₁₃	C₂×#41	1	4	yes	—
138	Dic₁₁₃	—	2	2	no	—
139	Dic₁₁₃	—	2	2	no	—
140	Dic₁₁₃	—	2	2	no	—
141	Dic₁₁₃	—	2	4	no	—
142	Dic₁₁₃	C₂×#41	1	4	yes	—
143	C₂×C₁₁₃	—	2	4	no	—
144	C₂×C₁₁₃	#529	1	4	yes	complemented
145	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
146	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
147	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
148	D₁₁₃	—	4	4	no	—
149	C₂×#529	—	113	1	no	maximal, Sylow
150	C₁₁₃	C₂×#529	1	1	yes	characteristic, complemented, Sylow
151	#529	—	113	16	no	—
152	C₂×#126	—	113	2	no	—
153	C₂×#125	—	113	1	no	—
154	C₂×#124	—	113	1	no	—
155	C₂×#123	—	113	2	no	—
156	C₂×C₂²⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
157	C₂×#121	—	113	1	no	—
158	C₂×C₄×Q₈	—	113	1	no	—
159	C₂×C₄×D₄	—	113	1	no	—
160	C₂×C₄×D₄	—	113	2	no	—
161	C₂×#117	—	113	2	no	—
162	C₂²×C₄²	—	113	1	no	—
163	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
164	C₂²×D₄	—	113	2	no	—
165	C₂²×D₄	—	113	1	no	—
166	C₂³×C₄	—	113	1	no	—
167	C₂³×C₄	—	113	1	no	—
168	C₂³×C₄	—	113	2	no	—
169	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
170	#126	—	113	16	no	—
171	#125	—	113	8	no	—
172	#124	—	113	8	no	—
173	#123	—	113	16	no	—
174	C₂²⋊Q₈	C₂×D₁₁₃	1	8	yes	complemented
175	#121	—	113	8	no	—
176	C₄×Q₈	—	113	8	no	—
177	C₄×D₄	—	113	8	no	—
178	C₄×D₄	—	113	16	no	—
179	#117	—	113	16	no	—
180	C₂×C₄⋊C₄	—	113	2	no	—
181	C₂×C₄⋊C₄	—	113	2	no	—
182	C₂×C₄⋊C₄	—	113	1	no	—
183	C₂×C₄⋊C₄	—	113	2	no	—
184	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	113	2	no	—
187	C₂×C₂²⋊C₄	—	113	2	no	—
188	C₂×C₂²⋊C₄	—	113	2	no	—
189	C₂×C₂²⋊C₄	—	113	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
191	C₂×C₄²	—	113	1	no	—
192	C₂×C₄²	—	113	2	no	—
193	C₂×C₄²	—	113	1	no	—
194	C₂×C₄²	—	113	1	no	—
195	C₂×C₄²	—	113	1	no	—
196	C₂×C₄²	—	113	8	no	—
197	C₂⁴	—	113	2	no	—
198	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
201	C₂×D₄	—	113	8	no	—
202	C₂×D₄	—	113	4	no	—
203	C₂×D₄	—	226	8	no	—
204	C₂×D₄	—	226	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	113	1	no	—
207	C₂²×C₄	—	113	4	no	—
208	C₂²×C₄	—	113	4	no	—
209	C₂²×C₄	—	113	1	no	—
210	C₂²×C₄	—	113	1	no	—
211	C₂²×C₄	—	113	1	no	—
212	C₂²×C₄	—	113	2	no	—
213	C₂²×C₄	—	113	8	no	—
214	C₂²×C₄	—	226	8	no	—
215	C₂²×C₄	—	226	4	no	—
216	C₂²×C₄	—	226	4	no	—
217	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
218	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
220	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
222	C₄⋊C₄	—	113	8	no	—
223	C₄⋊C₄	—	113	8	no	—
224	C₄⋊C₄	—	113	4	no	—
225	C₄⋊C₄	—	113	8	no	—
226	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
227	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
228	C₂²⋊C₄	—	113	8	no	—
229	C₂²⋊C₄	—	113	8	no	—
230	C₂²⋊C₄	—	113	8	no	—
231	C₂²⋊C₄	—	113	8	no	—
232	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
233	C₄²	—	113	4	no	—
234	C₄²	—	113	4	no	—
235	C₄²	—	113	4	no	—
236	C₄²	—	113	4	no	—
237	C₄²	—	113	8	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	113	4	no	—
241	C₂³	—	226	4	no	—
242	C₂³	—	226	4	no	—
243	C₂³	—	226	4	no	—
244	C₂³	#14268648	1	4	yes	—
245	C₂³	C₂³×D₁₁₃	1	2	yes	—
246	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	D₄	—	226	16	no	—
249	D₄	—	226	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	113	4	no	—
255	C₂×C₄	—	113	2	no	—
256	C₂×C₄	—	113	2	no	—
257	C₂×C₄	—	113	2	no	—
258	C₂×C₄	—	113	2	no	—
259	C₂×C₄	—	113	4	no	—
260	C₂×C₄	—	226	16	no	—
261	C₂×C₄	—	226	8	no	—
262	C₂×C₄	—	226	2	no	—
263	C₂×C₄	—	226	8	no	—
264	C₂×C₄	—	226	2	no	—
265	C₂×C₄	—	226	2	no	—
266	C₂×C₄	—	226	4	no	—
267	C₂×C₄	#14268646	1	4	yes	—
268	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
269	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
270	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
271	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	452	8	no	—
276	C₂²	—	226	4	no	—
277	C₂²	—	226	4	no	—
278	C₂²	—	226	4	no	—
279	C₂²	#41×D₁₁₃	1	2	yes	—
280	C₂²	C₂×#14268646	1	2	yes	—
281	C₂²	C₂×#14268648	1	4	yes	—
282	C₂²	C₂×#14268648	1	2	yes	—
283	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	113	4	no	—
288	C₄	—	226	2	no	—
289	C₄	—	226	2	no	—
290	C₄	—	226	2	no	—
291	C₄	—	226	4	no	—
292	C₄	C₂×#14268646	1	4	yes	—
293	C₂	—	2	4	no	—
294	C₂	—	452	4	no	—
295	C₂	#20299134	1	4	yes	direct product factor
296	C₂	C₂×#41×D₁₁₃	1	1	yes	characteristic
297	C₂	C₂²×#14268646	1	1	yes	characteristic
298	C₂	C₂²×#14268648	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×C₂²⋊Q₈×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂×#16813596	C₂
hypercenter	C₂×C₂²⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	C₂×#529
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20299134	C₂	direct (N × H)
2. C₂×#16813617	C₂	semidirect
3. C₂×#16813613	C₂	semidirect
4. C₂×#16813611	C₂	semidirect
5. C₂×#16813606	C₂	semidirect
7. C₂×Q₈×Dic₁₁₃	C₂	semidirect
8. C₂×#16813584	C₂	semidirect
9. C₂×#16813582	C₂	semidirect
10. C₂×#16813580	C₂	semidirect
11. C₂²×C₄×Dic₁₁₃	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₁₁₃	C₂	semidirect
18. #16813617	C₂²	semidirect
19. #16813613	C₂²	semidirect
20. #16813611	C₂²	semidirect
21. #16813606	C₂²	semidirect
23. Q₈×Dic₁₁₃	C₂²	semidirect
24. #16813584	C₂²	semidirect
25. #16813582	C₂²	semidirect
26. #16813580	C₂²	semidirect
40. C₂×C₄×Dic₁₁₃	C₂²	semidirect
43. C₂²⋊Q₈×C₁₁₃	C₂²	semidirect
144. C₂×C₁₁₃	#529	semidirect
150. C₁₁₃	C₂×#529	semidirect
156. C₂×C₂²⋊Q₈	D₁₁₃	semidirect
174. C₂²⋊Q₈	C₂×D₁₁₃	semidirect
295. C₂	#20299134	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299134	2
C₂×#16813617	2
C₂×#16813613	2
C₂×#16813611	2
C₂×#16813606	2
C₂×#16813596	2
C₂×Q₈×Dic₁₁₃	2
C₂×#16813584	2
C₂×#16813582	2
C₂×#16813580	2
C₂²×C₄×Dic₁₁₃	2
C₂×C₂²⋊Q₈×C₁₁₃	2
C₂×#529	113

Subgroups — C2×#20299134 #28403842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299134 #28403842