Subgroups — #28404218

Order 14464.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 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, complemented
2	#20299247	C₂	1	8	yes	maximal, complemented
3	#20299236	C₂	1	8	yes	maximal, complemented
4	#20299172	C₂	1	2	yes	maximal, complemented
5	#20299157	C₂	1	4	yes	maximal, complemented
6	#20299109	C₂	1	2	yes	maximal, complemented
7	C₂×#16813605	C₂	1	4	yes	maximal, complemented
8	#20299073	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#16813667	—	2	8	no	—
11	C₂×#14268647	C₂²	1	2	yes	—
12	C₂×#14268646	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
14	C₂×D₄×D₁₁₃	C₂²	1	4	yes	complemented
15	#16813620	C₂²	1	4	yes	complemented
16	#16813611	C₂²	1	4	yes	complemented
17	#16813605	C₂²	1	4	yes	complemented
18	#16813605	C₂²	1	8	yes	complemented
19	#16813605	C₂²	1	16	yes	complemented
20	#16813603	C₂²	1	8	yes	complemented
21	#16813601	C₂²	1	4	yes	complemented
22	#16813598	C₂²	1	8	yes	complemented
23	#16813596	C₂²	1	4	yes	complemented
24	#16813594	C₂²	1	16	yes	complemented
25	#16813591	C₂²	1	4	yes	complemented
26	#16813590	C₂²	1	8	yes	complemented
27	D₄×Dic₁₁₃	C₂²	1	8	yes	complemented
28	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	complemented
29	#16813581	C₂²	1	2	yes	complemented
30	C₂×#14268626	C₂²	1	8	yes	complemented
31	C₂×C₂²⋊Dic₁₁₃	C₂²	1	4	yes	complemented
32	C₂×#14268624	C₂²	1	8	yes	complemented
33	C₂×#41×C₁₁₃	—	2	8	no	—
34	C₂×#41×C₁₁₃	C₂²	1	8	yes	complemented
35	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
36	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	complemented
38	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	complemented
39	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₁₁₃	C₂³	1	4	yes	—
41	#14268647	—	2	8	no	—
42	#14268646	—	2	16	no	—
43	C₂×Dic₂₂₆	C₂³	1	2	yes	—
44	C₂×D₄₅₂	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₁₁₃	—	2	16	no	—
46	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
48	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
49	D₄×D₁₁₃	—	2	16	no	—
50	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
51	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
52	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
53	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
54	#14268626	C₂³	1	8	yes	—
55	#14268626	C₂³	1	16	yes	—
56	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
57	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
58	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
59	#14268624	C₂³	1	8	yes	—
60	#14268624	C₂³	1	16	yes	—
61	C₄×Dic₁₁₃	C₂³	1	4	yes	—
62	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
63	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
64	#41×C₁₁₃	—	2	32	no	—
65	C₂×Q₈×C₁₁₃	—	2	4	no	—
66	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
67	C₂×D₄×C₁₁₃	—	2	8	no	—
68	C₂×D₄×C₁₁₃	D₄	1	8	yes	complemented
69	C₂×D₄×C₁₁₃	C₂³	1	8	yes	—
70	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₁₃	—	2	8	no	—
72	C₂²×C₄×C₁₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₁₃	C₂³	1	8	yes	—
74	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
75	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
77	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
78	C₂³×C₁₁₃	—	2	2	no	—
79	C₂³×C₁₁₃	—	4	4	no	—
80	C₂³×C₁₁₃	—	4	2	no	—
81	C₂³×C₁₁₃	C₂×D₄	1	8	yes	—
82	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
83	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
84	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
85	C₂²×D₁₁₃	—	2	8	no	—
86	C₂²×D₁₁₃	—	4	8	no	—
87	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
88	Q₈×C₁₁₃	—	2	8	no	—
89	D₄×C₁₁₃	—	2	16	no	—
90	D₄×C₁₁₃	C₂×D₄	1	16	yes	—
91	C₂×C₄×C₁₁₃	—	2	2	no	—
92	C₂×C₄×C₁₁₃	—	2	4	no	—
93	C₂×C₄×C₁₁₃	—	2	16	no	—
94	C₂×C₄×C₁₁₃	—	2	8	no	—
95	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
97	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
99	C₂²⋊D₁₁₃	—	2	16	no	—
100	C₂²⋊D₁₁₃	—	2	8	no	—
101	C₂²⋊D₁₁₃	—	4	8	no	—
102	C₂×Dic₁₁₃	—	2	8	no	—
103	C₂×Dic₁₁₃	—	4	8	no	—
104	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
105	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
106	D₄₅₂	—	2	4	no	—
107	C₄×D₁₁₃	—	2	8	no	—
108	Dic₂₂₆	—	2	4	no	—
109	C₂²×C₁₁₃	—	2	8	no	—
110	C₂²×C₁₁₃	—	2	2	no	—
111	C₂²×C₁₁₃	—	2	2	no	—
112	C₂²×C₁₁₃	—	2	4	no	—
113	C₂²×C₁₁₃	—	4	8	no	—
114	C₂²×C₁₁₃	—	4	4	no	—
115	C₂²×C₁₁₃	C₂²×D₄	1	8	yes	—
116	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
117	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₁₃	—	2	4	no	—
119	C₂×D₁₁₃	—	2	8	no	—
120	C₂×D₁₁₃	—	8	8	no	—
121	C₄×C₁₁₃	—	2	2	no	—
122	C₄×C₁₁₃	—	2	4	no	—
123	C₄×C₁₁₃	C₂²×D₄	1	4	yes	—
124	Dic₁₁₃	—	2	4	no	—
125	Dic₁₁₃	—	2	8	no	—
126	C₂×C₁₁₃	—	2	8	no	—
127	C₂×C₁₁₃	—	2	2	no	—
128	C₂×C₁₁₃	—	4	2	no	—
129	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
130	C₂×C₁₁₃	#585	1	2	yes	—
131	D₁₁₃	—	4	4	no	—
132	#3485	—	113	1	no	maximal, Sylow
133	C₁₁₃	#3485	1	1	yes	characteristic, complemented, Sylow
134	C₂×H(5,𝔽₂)	—	113	1	no	—
135	C₂²×#41	D₁₁₃	1	1	yes	characteristic, complemented
136	#547	—	113	8	no	—
137	#546	—	113	8	no	—
138	#536	—	113	2	no	—
139	#534	—	113	4	no	—
140	#525	—	113	2	no	—
141	C₂×#123	—	113	4	no	—
142	#518	—	113	1	no	—
143	H(5,𝔽₂)	—	226	8	no	—
144	C₂×#41	—	2	8	no	—
145	C₂×#41	—	113	4	no	—
146	C₂×#41	—	113	2	no	—
147	C₂×#41	C₂×D₁₁₃	1	8	yes	complemented
148	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic, complemented
149	C₂²×D₄	—	113	4	no	—
150	C₂²×D₄	—	113	4	no	—
151	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
152	C₂²×D₄	C₂×D₁₁₃	1	2	yes	complemented
153	C₂³×C₄	C₂×D₁₁₃	1	2	yes	complemented
154	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic, complemented
155	C₄⋊Q₈	—	113	4	no	—
156	#124	—	113	4	no	—
157	#123	—	113	8	no	—
158	#123	—	113	4	no	—
159	#123	—	113	8	no	—
160	#123	—	113	16	no	—
161	C₂²⋊Q₈	—	113	8	no	—
162	C₂²⋊Q₈	—	113	4	no	—
163	#121	—	113	16	no	—
164	#121	—	113	4	no	—
165	#120	—	113	4	no	—
166	#120	—	113	8	no	—
167	C₄×D₄	—	113	8	no	—
168	C₄×D₄	—	113	8	no	—
169	#117	—	113	2	no	—
170	C₂×C₄⋊C₄	—	113	8	no	—
171	C₂×C₂²⋊C₄	—	113	4	no	—
172	C₂×C₂²⋊C₄	—	113	8	no	—
173	C₂⁴	—	113	4	no	—
174	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
175	C₂⁴	C₂²×D₁₁₃	1	2	yes	—
176	#41	—	2	32	no	—
177	#41	—	226	16	no	—
178	#41	—	226	8	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	113	2	no	—
181	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	113	8	no	—
184	C₂×D₄	—	113	4	no	—
185	C₂×D₄	—	113	8	no	—
186	C₂×D₄	—	113	2	no	—
187	C₂×D₄	—	226	16	no	—
188	C₂×D₄	—	226	16	no	—
189	C₂×D₄	C₂²⋊D₁₁₃	1	8	yes	complemented
190	C₂×D₄	C₂²×D₁₁₃	1	8	yes	—
191	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	113	8	no	—
195	C₂²×C₄	—	113	4	no	—
196	C₂²×C₄	—	113	4	no	—
197	C₂²×C₄	C₂²×D₁₁₃	1	8	yes	—
198	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
199	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	113	8	no	—
203	C₄⋊C₄	—	113	16	no	—
204	C₄⋊C₄	—	113	4	no	—
205	C₂²⋊C₄	—	113	4	no	—
206	C₂²⋊C₄	—	113	8	no	—
207	C₂²⋊C₄	—	113	8	no	—
208	C₂²⋊C₄	—	113	8	no	—
209	C₂²⋊C₄	—	113	16	no	—
210	C₄²	—	113	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	113	4	no	—
215	C₂³	—	452	8	no	—
216	C₂³	—	226	8	no	—
217	C₂³	C₂×C₂²⋊D₁₁₃	1	8	yes	—
218	C₂³	C₂³×D₁₁₃	1	2	yes	—
219	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
220	C₂³	C₂³×D₁₁₃	1	4	yes	—
221	Q₈	—	2	8	no	—
222	Q₈	—	226	4	no	—
223	D₄	—	2	16	no	—
224	D₄	—	452	8	no	—
225	D₄	—	226	8	no	—
226	D₄	—	226	16	no	—
227	D₄	—	226	4	no	—
228	D₄	C₂×C₂²⋊D₁₁₃	1	16	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	113	8	no	—
234	C₂×C₄	—	113	4	no	—
235	C₂×C₄	—	452	8	no	—
236	C₂×C₄	—	226	8	no	—
237	C₂×C₄	—	226	8	no	—
238	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	4	yes	—
239	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
240	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
241	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	904	8	no	—
249	C₂²	—	226	4	no	—
250	C₂²	—	226	8	no	—
251	C₂²	C₂²×C₂²⋊D₁₁₃	1	2	yes	—
252	C₂²	C₂²×C₂²⋊D₁₁₃	1	8	yes	—
253	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	226	4	no	—
257	C₄	—	226	8	no	—
258	C₄	C₂²×C₂²⋊D₁₁₃	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	2	no	—
262	C₂	—	452	4	no	—
263	C₂	C₂³×C₂²⋊D₁₁₃	1	1	yes	characteristic
264	C₂	#20299434	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₂²×#41×C₁₁₃	C₂
hypercenter	C₂²×#41	D₁₁₃
hypocenter	C₁₁₃	#3485
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16813667	C₂	semidirect
2. #20299247	C₂	semidirect
3. #20299236	C₂	semidirect
4. #20299172	C₂	semidirect
5. #20299157	C₂	semidirect
6. #20299109	C₂	semidirect
7. C₂×#16813605	C₂	semidirect
8. #20299073	C₂	semidirect
9. C₂²×#41×C₁₁₃	C₂	semidirect
12. C₂×#14268646	C₂²	semidirect
14. C₂×D₄×D₁₁₃	C₂²	semidirect
15. #16813620	C₂²	semidirect
16. #16813611	C₂²	semidirect
17. #16813605	C₂²	semidirect
18. #16813605	C₂²	semidirect
19. #16813605	C₂²	semidirect
20. #16813603	C₂²	semidirect
21. #16813601	C₂²	semidirect
22. #16813598	C₂²	semidirect
23. #16813596	C₂²	semidirect
24. #16813594	C₂²	semidirect
25. #16813591	C₂²	semidirect
26. #16813590	C₂²	semidirect
27. D₄×Dic₁₁₃	C₂²	semidirect
28. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
29. #16813581	C₂²	semidirect
30. C₂×#14268626	C₂²	semidirect
31. C₂×C₂²⋊Dic₁₁₃	C₂²	semidirect
32. C₂×#14268624	C₂²	semidirect
34. C₂×#41×C₁₁₃	C₂²	semidirect
35. C₂²×Q₈×C₁₁₃	C₂²	semidirect
37. C₂²×D₄×C₁₁₃	C₂²	semidirect
38. C₂³×C₄×C₁₁₃	C₂²	semidirect
39. C₂³×C₄×C₁₁₃	C₂²	semidirect
68. C₂×D₄×C₁₁₃	D₄	semidirect
133. C₁₁₃	#3485	semidirect
135. C₂²×#41	D₁₁₃	semidirect
147. C₂×#41	C₂×D₁₁₃	semidirect
148. C₂²×Q₈	C₂×D₁₁₃	semidirect
152. C₂²×D₄	C₂×D₁₁₃	semidirect
153. C₂³×C₄	C₂×D₁₁₃	semidirect
154. C₂³×C₄	C₂×D₁₁₃	semidirect
189. C₂×D₄	C₂²⋊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
#20299247	2
#20299236	2
#20299172	2
#20299157	2
#20299109	2
C₂×#16813605	2
#20299073	2
C₂²×#41×C₁₁₃	2
#3485	113