Subgroups — C₂×#20299319 #28404121

Order 14464.

1112 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	#20299319	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813621	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16813617	C₂	1	2	yes	maximal, complemented
4	C₂×#16813615	C₂	1	2	yes	maximal, complemented
5	C₂×#16813608	C₂	1	2	yes	maximal
6	C₂×#16813598	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16813587	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊D₁₁₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#16813581	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#16813580	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
16	#16813621	C₂²	1	8	yes	complemented
17	#16813617	C₂²	1	16	yes	complemented
18	#16813615	C₂²	1	16	yes	complemented
19	#16813608	C₂²	1	16	yes	—
20	#16813598	C₂²	1	8	yes	complemented
21	Q₈×Dic₁₁₃	C₂²	1	8	yes	complemented
22	#16813587	C₂²	1	16	yes	complemented
23	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	—
24	#16813581	C₂²	1	8	yes	complemented
25	#16813580	C₂²	1	8	yes	complemented
26	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
29	C₂×#14268626	C₂²	1	1	yes	characteristic
30	C₂×#14268626	C₂²	1	2	yes	—
31	C₂×#14268626	C₂²	1	2	yes	—
32	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
34	C₂×#14268624	C₂²	1	2	yes	—
35	C₂×#14268624	C₂²	1	1	yes	characteristic
36	C₂×#14268624	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
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	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
47	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₂₆	—	2	8	no	—
49	C₂×Dic₂₂₆	C₂³	1	8	yes	—
50	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
51	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
52	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
53	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
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	4	no	—
58	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
59	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
60	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
61	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
62	#14268626	C₂³	1	4	yes	—
63	#14268626	C₂³	1	8	yes	—
64	#14268626	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
66	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
67	#14268624	C₂³	1	8	yes	—
68	#14268624	C₂³	1	4	yes	—
69	#14268624	C₂³	1	8	yes	—
70	C₄×Dic₁₁₃	C₂³	1	8	yes	—
71	C₄×Dic₁₁₃	C₂³	1	8	yes	—
72	C₄×Dic₁₁₃	C₂³	1	4	yes	—
73	C₄×Dic₁₁₃	C₂³	1	8	yes	—
74	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₁₁₃	—	2	4	no	—
76	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
82	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
84	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
85	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
86	C₂³×C₁₁₃	—	2	2	no	—
87	C₂³×C₁₁₃	—	2	2	no	—
88	C₂³×C₁₁₃	—	2	2	no	—
89	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
90	C₂³×C₁₁₃	C₂⁴	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	C₂⁴	1	2	yes	—
95	Q₈×C₁₁₃	—	2	8	no	—
96	C₂×C₄×C₁₁₃	—	2	2	no	—
97	C₂×C₄×C₁₁₃	—	2	4	no	—
98	C₂×C₄×C₁₁₃	—	2	4	no	—
99	C₂×C₄×C₁₁₃	—	2	8	no	—
100	C₂×C₄×C₁₁₃	#41	1	4	yes	—
101	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
104	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
105	C₂²⋊D₁₁₃	—	2	8	no	—
106	C₂×Dic₁₁₃	—	2	2	no	—
107	C₂×Dic₁₁₃	—	2	4	no	—
108	C₂×Dic₁₁₃	—	2	4	no	—
109	C₂×Dic₁₁₃	—	2	2	no	—
110	C₂×Dic₁₁₃	#41	1	4	yes	—
111	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
112	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
113	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
114	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
115	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
116	C₄×D₁₁₃	—	2	8	no	—
117	Dic₂₂₆	—	2	16	no	—
118	C₂²×C₁₁₃	—	2	2	no	—
119	C₂²×C₁₁₃	—	2	2	no	—
120	C₂²×C₁₁₃	—	2	2	no	—
121	C₂²×C₁₁₃	—	4	4	no	—
122	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
123	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
124	C₂²×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
125	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
126	C₂×D₁₁₃	—	2	2	no	—
127	C₂×D₁₁₃	—	2	2	no	—
128	C₂×D₁₁₃	—	2	2	no	—
129	C₂×D₁₁₃	—	4	4	no	—
130	C₄×C₁₁₃	—	2	4	no	—
131	C₄×C₁₁₃	—	2	2	no	—
132	C₄×C₁₁₃	—	2	4	no	—
133	C₄×C₁₁₃	C₂×#41	1	4	yes	—
134	Dic₁₁₃	—	2	4	no	—
135	Dic₁₁₃	—	2	2	no	—
136	Dic₁₁₃	—	2	4	no	—
137	Dic₁₁₃	—	2	2	no	—
138	Dic₁₁₃	C₂×#41	1	4	yes	—
139	C₂×C₁₁₃	—	4	2	no	—
140	C₂×C₁₁₃	#556	1	4	yes	complemented
141	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
143	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
144	D₁₁₃	—	4	2	no	—
145	C₂×#556	—	113	1	no	maximal, Sylow
146	C₁₁₃	C₂×#556	1	1	yes	characteristic, complemented, Sylow
147	#556	—	113	16	no	—
148	C₂×C₄⋊Q₈	—	113	1	no	—
149	C₂×#126	—	113	2	no	—
150	C₂×#126	—	113	2	no	—
151	C₂×#124	—	113	2	no	—
152	C₂×C₂²⋊Q₈	—	113	1	no	—
153	C₂×C₂²⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
154	C₂×C₄×Q₈	—	113	1	no	—
155	C₂×C₄×Q₈	—	113	2	no	—
156	C₂×C₄×D₄	—	113	1	no	—
157	C₂×#117	—	113	1	no	—
158	C₂×#117	—	113	1	no	—
159	C₂²×Q₈	—	113	2	no	—
160	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
161	C₂²×D₄	—	113	1	no	—
162	C₂³×C₄	—	113	1	no	—
163	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	113	8	no	—
165	#126	—	113	16	no	—
166	#126	—	113	16	no	—
167	#124	—	113	16	no	—
168	C₂²⋊Q₈	—	113	8	no	—
169	C₂²⋊Q₈	C₂×D₁₁₃	1	8	yes	complemented
170	C₄×Q₈	—	113	8	no	—
171	C₄×Q₈	—	113	16	no	—
172	C₄×D₄	—	113	8	no	—
173	#117	—	113	8	no	—
174	#117	—	113	8	no	—
175	C₂×C₄⋊C₄	—	113	1	no	—
176	C₂×C₄⋊C₄	—	113	1	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	2	no	—
181	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
183	C₂×C₂²⋊C₄	—	113	2	no	—
184	C₂×C₂²⋊C₄	—	113	1	no	—
185	C₂×C₂²⋊C₄	—	113	2	no	—
186	C₂×C₂²⋊C₄	—	113	1	no	—
187	C₂×C₂²⋊C₄	—	113	2	no	—
188	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
189	C₂×C₄²	—	113	2	no	—
190	C₂×C₄²	—	113	2	no	—
191	C₂×C₄²	—	113	1	no	—
192	C₂×C₄²	—	113	2	no	—
193	C₂⁴	—	113	1	no	—
194	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	113	8	no	—
197	C₂×Q₈	—	226	8	no	—
198	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
199	C₂×D₄	—	113	4	no	—
200	C₂×D₄	—	226	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	113	1	no	—
203	C₂²×C₄	—	113	2	no	—
204	C₂²×C₄	—	113	1	no	—
205	C₂²×C₄	—	113	1	no	—
206	C₂²×C₄	—	113	2	no	—
207	C₂²×C₄	—	113	4	no	—
208	C₂²×C₄	—	226	4	no	—
209	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
210	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
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	4	no	—
216	C₄⋊C₄	—	113	4	no	—
217	C₄⋊C₄	—	113	8	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	8	yes	—
222	C₂²⋊C₄	—	113	8	no	—
223	C₂²⋊C₄	—	113	4	no	—
224	C₂²⋊C₄	—	113	8	no	—
225	C₂²⋊C₄	—	113	4	no	—
226	C₂²⋊C₄	—	113	8	no	—
227	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
228	C₄²	—	113	8	no	—
229	C₄²	—	113	8	no	—
230	C₄²	—	113	4	no	—
231	C₄²	—	113	8	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	113	2	no	—
236	C₂³	—	226	2	no	—
237	C₂³	—	226	2	no	—
238	C₂³	—	226	2	no	—
239	C₂³	C₂³×D₁₁₃	1	2	yes	—
240	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
241	Q₈	—	2	8	no	—
242	Q₈	—	226	16	no	—
243	D₄	—	226	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	113	4	no	—
249	C₂×C₄	—	113	2	no	—
250	C₂×C₄	—	113	4	no	—
251	C₂×C₄	—	113	2	no	—
252	C₂×C₄	—	113	2	no	—
253	C₂×C₄	—	113	4	no	—
254	C₂×C₄	—	226	4	no	—
255	C₂×C₄	—	226	2	no	—
256	C₂×C₄	—	226	4	no	—
257	C₂×C₄	—	226	2	no	—
258	C₂×C₄	—	226	8	no	—
259	C₂×C₄	#14268646	1	4	yes	—
260	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
261	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
262	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
263	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	452	4	no	—
269	C₂²	—	226	2	no	—
270	C₂²	—	226	2	no	—
271	C₂²	—	226	2	no	—
272	C₂²	#41×D₁₁₃	1	2	yes	—
273	C₂²	C₂×#14268646	1	2	yes	—
274	C₂²	#16813670	1	2	yes	—
275	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	113	4	no	—
280	C₄	—	226	4	no	—
281	C₄	—	226	2	no	—
282	C₄	—	226	4	no	—
283	C₄	—	226	2	no	—
284	C₄	C₂×#14268646	1	4	yes	—
285	C₂	—	4	2	no	—
286	C₂	—	452	2	no	—
287	C₂	#20299319	1	4	yes	direct product factor
288	C₂	C₂×#41×D₁₁₃	1	1	yes	characteristic
289	C₂	C₂²×#14268646	1	1	yes	characteristic
290	C₂	C₂×#16813670	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	C₂²×C₂²⋊D₁₁₃	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	C₂×#556
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. #20299319	C₂	direct (N × H)
2. C₂×#16813621	C₂	semidirect
3. C₂×#16813617	C₂	semidirect
4. C₂×#16813615	C₂	semidirect
6. C₂×#16813598	C₂	semidirect
7. C₂×Q₈×Dic₁₁₃	C₂	semidirect
8. C₂×#16813587	C₂	semidirect
10. C₂×#16813581	C₂	semidirect
11. C₂×#16813580	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₁₁₃	C₂	semidirect
16. #16813621	C₂²	semidirect
17. #16813617	C₂²	semidirect
18. #16813615	C₂²	semidirect
20. #16813598	C₂²	semidirect
21. Q₈×Dic₁₁₃	C₂²	semidirect
22. #16813587	C₂²	semidirect
24. #16813581	C₂²	semidirect
25. #16813580	C₂²	semidirect
43. C₂²⋊Q₈×C₁₁₃	C₂²	semidirect
140. C₂×C₁₁₃	#556	semidirect
146. C₁₁₃	C₂×#556	semidirect
153. C₂×C₂²⋊Q₈	D₁₁₃	semidirect
169. C₂²⋊Q₈	C₂×D₁₁₃	semidirect
287. C₂	#20299319	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299319	2
C₂×#16813621	2
C₂×#16813617	2
C₂×#16813615	2
C₂×#16813608	2
C₂×#16813598	2
C₂×Q₈×Dic₁₁₃	2
C₂×#16813587	2
C₂×C₄×C₂²⋊D₁₁₃	2
C₂×#16813581	2
C₂×#16813580	2
C₂×C₂²⋊Q₈×C₁₁₃	2
C₂×#556	113

Subgroups — C2×#20299319 #28404121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299319 #28404121