Subgroups — #28404450

Order 14464.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299333	C₂	1	4	yes	maximal, complemented
2	#20299269	C₂	1	4	yes	maximal, complemented
3	#20299244	C₂	1	8	yes	maximal, complemented
4	#20299203	C₂	1	4	yes	maximal, complemented
5	#20299162	C₂	1	2	yes	maximal, complemented
6	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299148	C₂	1	4	yes	maximal, complemented
8	C₂×#16813619	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299072	C₂	1	2	yes	maximal, complemented
10	C₄×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#14268647	C₂²	1	8	yes	complemented
12	C₂²×D₄₅₂	C₂²	1	2	yes	—
13	C₂²×D₄₅₂	C₂²	1	4	yes	—
14	#16813622	C₂²	1	2	yes	complemented
15	#16813619	C₂²	1	4	yes	—
16	#16813619	C₂²	1	2	yes	—
17	#16813619	C₂²	1	2	yes	—
18	#16813618	C₂²	1	4	yes	complemented
19	#16813614	C₂²	1	2	yes	complemented
20	#16813611	C₂²	1	4	yes	complemented
21	#16813610	C₂²	1	8	yes	complemented
22	#16813607	C₂²	1	16	yes	complemented
23	#16813601	C₂²	1	4	yes	—
24	#16813596	C₂²	1	8	yes	complemented
25	#16813596	C₂²	1	8	yes	complemented
26	#16813595	C₂²	1	16	yes	complemented
27	#16813590	C₂²	1	8	yes	—
28	C₄×Dic₂₂₆	C₂²	1	4	yes	complemented
29	C₄×D₄₅₂	C₂²	1	8	yes	complemented
30	C₄×D₄₅₂	C₂²	1	8	yes	complemented
31	#16813584	C₂²	1	8	yes	complemented
32	#16813581	C₂²	1	2	yes	—
33	C₂²⋊C₄×D₁₁₃	C₂²	1	8	yes	—
34	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	complemented
36	C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
37	C₄×D₄×C₁₁₃	C₂²	1	2	yes	complemented
38	#117×C₁₁₃	C₂²	1	1	yes	characteristic
39	#117×C₁₁₃	C₂²	1	2	yes	complemented
40	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	complemented
42	C₂³×D₁₁₃	C₂³	1	4	yes	—
43	#14268647	—	4	16	no	—
44	C₂×Dic₂₂₆	C₂³	1	4	yes	—
45	C₂×D₄₅₂	—	2	8	no	—
46	C₂×D₄₅₂	—	4	8	no	—
47	C₂×D₄₅₂	C₂³	1	4	yes	—
48	C₂×D₄₅₂	C₂³	1	4	yes	—
49	C₂×D₄₅₂	C₂³	1	8	yes	—
50	C₂×D₄₅₂	C₂³	1	8	yes	—
51	C₂×C₂²⋊D₁₁₃	C₂³	1	16	yes	—
52	C₂×C₄×D₁₁₃	C₂³	1	16	yes	—
53	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
54	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
55	#14268626	C₂³	1	8	yes	—
56	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
57	#14268624	C₂³	1	16	yes	—
58	#14268624	C₂³	1	16	yes	—
59	C₄×Dic₁₁₃	C₂³	1	4	yes	—
60	#41×C₁₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
63	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
66	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
70	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
72	C₄²×C₁₁₃	C₂³	1	2	yes	—
73	C₄²×C₁₁₃	C₂³	1	4	yes	—
74	C₄²×C₁₁₃	C₂³	1	2	yes	—
75	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
76	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
78	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×D₁₁₃	—	2	8	no	—
80	C₂²×D₁₁₃	—	4	8	no	—
81	C₂²×D₁₁₃	C₂⁴	1	8	yes	—
82	Q₈×C₁₁₃	—	2	2	no	—
83	D₄×C₁₁₃	—	2	4	no	—
84	D₄×C₁₁₃	—	2	2	no	—
85	C₂×C₄×C₁₁₃	—	2	4	no	—
86	C₂×C₄×C₁₁₃	—	2	4	no	—
87	C₂×C₄×C₁₁₃	—	2	8	no	—
88	C₂×C₄×C₁₁₃	#41	1	4	yes	complemented
89	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
95	C₂²⋊D₁₁₃	—	4	16	no	—
96	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
97	D₄₅₂	—	2	8	no	—
98	D₄₅₂	—	2	8	no	—
99	D₄₅₂	—	2	16	no	—
100	D₄₅₂	—	4	8	no	—
101	C₄×D₁₁₃	—	4	16	no	—
102	Dic₂₂₆	—	2	8	no	—
103	C₂²×C₁₁₃	—	2	2	no	—
104	C₂²×C₁₁₃	—	2	2	no	—
105	C₂²×C₁₁₃	—	2	4	no	—
106	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
107	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
108	C₂×D₁₁₃	—	2	8	no	—
109	C₂×D₁₁₃	—	2	16	no	—
110	C₂×D₁₁₃	—	8	8	no	—
111	C₄×C₁₁₃	—	2	2	no	—
112	C₄×C₁₁₃	—	2	4	no	—
113	C₄×C₁₁₃	—	2	2	no	—
114	C₄×C₁₁₃	C₂×#41	1	2	yes	—
115	C₄×C₁₁₃	C₂×#41	1	2	yes	—
116	C₄×C₁₁₃	H(5,𝔽₂)	1	4	yes	—
117	Dic₁₁₃	—	2	8	no	—
118	C₂×C₁₁₃	—	2	2	no	—
119	C₂×C₁₁₃	—	4	2	no	—
120	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
121	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
122	D₁₁₃	—	4	8	no	—
123	#3508	—	113	1	no	maximal, Sylow
124	C₁₁₃	#3508	1	1	yes	characteristic, complemented, Sylow
125	#559	—	113	4	no	—
126	#550	—	113	4	no	—
127	#546	—	113	8	no	—
128	#540	—	113	4	no	—
129	#535	—	113	2	no	—
130	#533	—	113	1	no	—
131	#532	—	113	4	no	—
132	C₂×#127	—	113	1	no	—
133	#518	—	113	2	no	—
134	C₄×#41	D₁₁₃	1	1	yes	characteristic, complemented
135	C₂×#41	—	113	8	no	—
136	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
137	C₂²×D₄	—	113	4	no	—
138	C₂²×D₄	—	113	2	no	—
139	C₄⋊Q₈	—	113	2	no	—
140	#127	—	113	4	no	—
141	#127	—	113	2	no	—
142	#127	—	113	2	no	—
143	#127	—	113	4	no	—
144	#125	—	113	2	no	—
145	#124	—	113	4	no	—
146	#124	—	113	8	no	—
147	#123	—	113	16	no	—
148	C₂²⋊Q₈	—	113	4	no	—
149	#121	—	113	8	no	—
150	#121	—	113	8	no	—
151	#121	—	113	16	no	—
152	#120	—	113	8	no	—
153	C₄×Q₈	—	113	4	no	—
154	C₄×Q₈	C₂×D₁₁₃	1	2	yes	complemented
155	C₄×D₄	—	113	8	no	—
156	C₄×D₄	—	113	8	no	—
157	C₄×D₄	—	113	8	no	—
158	C₄×D₄	C₂×D₁₁₃	1	4	yes	—
159	C₄×D₄	C₂×D₁₁₃	1	2	yes	complemented
160	#117	—	113	2	no	—
161	#117	C₂×D₁₁₃	1	1	yes	characteristic
162	#117	C₂×D₁₁₃	1	2	yes	complemented
163	C₂×C₂²⋊C₄	—	113	8	no	—
164	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×D₁₁₃	1	2	yes	complemented
166	C₂⁴	—	113	4	no	—
167	#41	—	2	4	no	—
168	#41	—	452	16	no	—
169	C₂×Q₈	—	113	4	no	—
170	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
171	C₂×D₄	—	113	16	no	—
172	C₂×D₄	—	113	8	no	—
173	C₂×D₄	—	113	4	no	—
174	C₂×D₄	—	113	4	no	—
175	C₂×D₄	—	113	8	no	—
176	C₂×D₄	—	452	8	no	—
177	C₂×D₄	—	226	8	no	—
178	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
179	C₂×D₄	C₂²×D₁₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	113	16	no	—
181	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
182	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
184	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	113	4	no	—
186	C₄⋊C₄	—	113	4	no	—
187	C₄⋊C₄	—	113	8	no	—
188	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
189	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
190	C₂²⋊C₄	—	113	4	no	—
191	C₂²⋊C₄	—	113	16	no	—
192	C₂²⋊C₄	—	113	16	no	—
193	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
194	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
195	C₄²	—	113	4	no	—
196	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
197	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
198	C₄²	C₂²×D₁₁₃	1	2	yes	—
199	C₄²	C₂²×D₁₁₃	1	2	yes	—
200	C₄²	C₂²×D₁₁₃	1	4	yes	—
201	C₂³	—	113	8	no	—
202	C₂³	—	452	8	no	—
203	C₂³	—	226	8	no	—
204	C₂³	C₂³×D₁₁₃	1	2	yes	—
205	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	226	8	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	2	no	—
210	D₄	—	452	8	no	—
211	D₄	—	452	16	no	—
212	D₄	—	226	8	no	—
213	D₄	—	226	8	no	—
214	D₄	—	226	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	113	8	no	—
219	C₂×C₄	—	452	16	no	—
220	C₂×C₄	#14268647	1	4	yes	complemented
221	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
222	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
223	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
224	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	904	8	no	—
231	C₂²	—	226	8	no	—
232	C₂²	—	226	16	no	—
233	C₂²	C₂×#14268647	1	2	yes	—
234	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	2	2	no	—
238	C₄	—	226	8	no	—
239	C₄	C₂×#14268647	1	2	yes	—
240	C₄	C₂×#14268647	1	2	yes	—
241	C₄	#16813668	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	—	452	8	no	—
245	C₂	C₂²×#14268647	1	1	yes	characteristic
246	C₂	C₂×#16813668	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₄×#41×C₁₁₃	C₂
hypercenter	C₄×#41	D₁₁₃
hypocenter	C₁₁₃	#3508
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #20299333	C₂	semidirect
2. #20299269	C₂	semidirect
3. #20299244	C₂	semidirect
4. #20299203	C₂	semidirect
5. #20299162	C₂	semidirect
6. #20299155	C₂	semidirect
7. #20299148	C₂	semidirect
8. C₂×#16813619	C₂	semidirect
9. #20299072	C₂	semidirect
10. C₄×#41×C₁₁₃	C₂	semidirect
11. C₂×#14268647	C₂²	semidirect
14. #16813622	C₂²	semidirect
18. #16813618	C₂²	semidirect
19. #16813614	C₂²	semidirect
20. #16813611	C₂²	semidirect
21. #16813610	C₂²	semidirect
22. #16813607	C₂²	semidirect
24. #16813596	C₂²	semidirect
25. #16813596	C₂²	semidirect
26. #16813595	C₂²	semidirect
28. C₄×Dic₂₂₆	C₂²	semidirect
29. C₄×D₄₅₂	C₂²	semidirect
30. C₄×D₄₅₂	C₂²	semidirect
31. #16813584	C₂²	semidirect
35. C₄×Q₈×C₁₁₃	C₂²	semidirect
37. C₄×D₄×C₁₁₃	C₂²	semidirect
39. #117×C₁₁₃	C₂²	semidirect
41. C₂×C₄²×C₁₁₃	C₂²	semidirect
88. C₂×C₄×C₁₁₃	#41	semidirect
124. C₁₁₃	#3508	semidirect
134. C₄×#41	D₁₁₃	semidirect
154. C₄×Q₈	C₂×D₁₁₃	semidirect
159. C₄×D₄	C₂×D₁₁₃	semidirect
162. #117	C₂×D₁₁₃	semidirect
165. C₂×C₄²	C₂×D₁₁₃	semidirect
220. C₂×C₄	#14268647	semidirect

Maximal subgroups

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

subgroup	index
#20299333	2
#20299269	2
#20299244	2
#20299203	2
#20299162	2
#20299155	2
#20299148	2
C₂×#16813619	2
#20299072	2
C₄×#41×C₁₁₃	2
#3508	113