Subgroups — #28404362

Order 14464.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813669	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299255	C₂	1	4	yes	maximal, complemented
3	#20299242	C₂	1	12	yes	maximal, complemented
4	#20299169	C₂	1	3	yes	maximal, complemented
5	#20299165	C₂	1	3	yes	maximal, complemented
6	#20299144	C₂	1	3	yes	maximal, complemented
7	C₂×#16813609	C₂	1	3	yes	maximal, complemented
8	#41×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#16813669	—	2	8	no	—
11	C₂×#14268647	C₂²	1	3	yes	complemented
12	C₂×#14268646	C₂²	1	6	yes	complemented
13	C₂²×Dic₂₂₆	C₂²	1	3	yes	complemented
14	C₂×Q₈×D₁₁₃	C₂²	1	2	yes	complemented
15	#16813620	C₂²	1	3	yes	complemented
16	#16813618	C₂²	1	3	yes	complemented
17	#16813611	C₂²	1	1	yes	characteristic, complemented
18	#16813609	C₂²	1	3	yes	complemented
19	#16813609	C₂²	1	6	yes	complemented
20	#16813609	C₂²	1	12	yes	complemented
21	#16813603	C₂²	1	12	yes	complemented
22	#16813601	C₂²	1	12	yes	complemented
23	#16813594	C₂²	1	12	yes	complemented
24	#16813593	C₂²	1	12	yes	complemented
25	#16813591	C₂²	1	12	yes	complemented
26	Q₈×Dic₁₁₃	C₂²	1	2	yes	complemented
27	D₄×Dic₁₁₃	C₂²	1	6	yes	complemented
28	C₄×C₂²⋊D₁₁₃	C₂²	1	12	yes	complemented
29	#16813581	C₂²	1	3	yes	complemented
30	C₂×C₂²⋊Dic₁₁₃	C₂²	1	12	yes	complemented
31	C₂×C₄×Dic₁₁₃	C₂²	1	3	yes	complemented
32	H(5,𝔽₂)×C₁₁₃	—	2	8	no	—
33	C₂×#41×C₁₁₃	C₂²	1	3	yes	—
34	C₂×#41×C₁₁₃	C₂²	1	2	yes	—
35	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₁₁₃	C₂²	1	3	yes	—
37	C₂²×D₄×C₁₁₃	C₂²	1	6	yes	—
38	#14268647	—	2	12	no	—
39	#14268646	—	2	24	no	—
40	C₂×Dic₂₂₆	—	2	12	no	—
41	C₂×Dic₂₂₆	C₂³	1	6	yes	—
42	C₂×Dic₂₂₆	C₂³	1	3	yes	—
43	Q₈×D₁₁₃	—	2	8	no	—
44	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
46	C₂×C₂²⋊D₁₁₃	C₂³	1	12	yes	—
47	C₂²×Dic₁₁₃	C₂³	1	6	yes	—
48	C₂²×Dic₁₁₃	C₂³	1	6	yes	—
49	C₂×C₄×D₁₁₃	C₂³	1	6	yes	—
50	C₄⋊Dic₁₁₃	C₂³	1	6	yes	—
51	#14268626	C₂³	1	12	yes	—
52	C₂²⋊Dic₁₁₃	C₂³	1	12	yes	—
53	C₂²⋊Dic₁₁₃	C₂³	1	24	yes	—
54	C₂²⋊Dic₁₁₃	C₂³	1	6	yes	—
55	#14268624	C₂³	1	4	yes	—
56	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₄×Dic₁₁₃	C₂³	1	3	yes	—
58	C₄×Dic₁₁₃	C₂³	1	6	yes	—
59	C₂⁴×C₁₁₃	C₂³	1	6	yes	—
60	#41×C₁₁₃	—	2	8	no	—
61	#41×C₁₁₃	—	2	12	no	—
62	#41×C₁₁₃	D₄	1	8	yes	complemented
63	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₁₃	—	2	24	no	—
66	C₂×D₄×C₁₁₃	—	2	12	no	—
67	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
68	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
69	C₂×D₄×C₁₁₃	C₂³	1	3	yes	—
70	C₂×D₄×C₁₁₃	C₂³	1	3	yes	—
71	C₂²×C₄×C₁₁₃	C₂³	1	6	yes	—
72	C₂²×C₄×C₁₁₃	C₂³	1	3	yes	—
73	C₂³×C₁₁₃	—	2	12	no	—
74	C₂³×C₁₁₃	—	4	12	no	—
75	C₂³×C₁₁₃	C₂⁴	1	6	yes	—
76	C₂³×C₁₁₃	C₂⁴	1	3	yes	—
77	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
78	Q₈×C₁₁₃	—	2	2	no	—
79	Q₈×C₁₁₃	C₂×D₄	1	4	yes	—
80	D₄×C₁₁₃	—	2	12	no	—
81	D₄×C₁₁₃	—	2	6	no	—
82	D₄×C₁₁₃	—	2	12	no	—
83	D₄×C₁₁₃	C₂×D₄	1	12	yes	—
84	C₂×C₄×C₁₁₃	—	2	12	no	—
85	C₂×C₄×C₁₁₃	C₂×D₄	1	12	yes	—
86	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₁₁₃	C₂⁴	1	3	yes	—
89	C₂²⋊D₁₁₃	—	2	12	no	—
90	C₂²⋊D₁₁₃	—	4	12	no	—
91	C₂×Dic₁₁₃	—	2	12	no	—
92	C₂×Dic₁₁₃	—	4	6	no	—
93	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
94	C₂×Dic₁₁₃	C₂⁴	1	6	yes	—
95	C₂×Dic₁₁₃	C₂⁴	1	6	yes	—
96	D₄₅₂	—	2	2	no	—
97	C₄×D₁₁₃	—	2	12	no	—
98	Dic₂₂₆	—	2	12	no	—
99	Dic₂₂₆	—	2	6	no	—
100	C₂²×C₁₁₃	—	2	6	no	—
101	C₂²×C₁₁₃	—	2	6	no	—
102	C₂²×C₁₁₃	—	2	12	no	—
103	C₂²×C₁₁₃	—	8	12	no	—
104	C₂²×C₁₁₃	C₂²×D₄	1	6	yes	—
105	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
106	C₂×D₁₁₃	—	2	2	no	—
107	C₂×D₁₁₃	—	2	4	no	—
108	C₄×C₁₁₃	—	2	2	no	—
109	C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
110	C₄×C₁₁₃	C₂²×D₄	1	6	yes	—
111	Dic₁₁₃	—	2	6	no	—
112	Dic₁₁₃	—	2	2	no	—
113	Dic₁₁₃	—	2	6	no	—
114	C₂×C₁₁₃	—	2	6	no	—
115	C₂×C₁₁₃	—	4	6	no	—
116	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
117	C₂×C₁₁₃	#585	1	2	yes	—
118	D₁₁₃	—	4	2	no	—
119	#3500	—	113	1	no	maximal, Sylow
120	C₁₁₃	#3500	1	1	yes	characteristic, complemented, Sylow
121	C₂×2₋¹⁺⁴	—	113	1	no	—
122	C₂×H(5,𝔽₂)	D₁₁₃	1	1	yes	characteristic, complemented
123	#548	—	113	4	no	—
124	#546	—	113	12	no	—
125	#536	—	113	3	no	—
126	#535	—	113	3	no	—
127	#532	—	113	3	no	—
128	C₂×#124	—	113	3	no	—
129	C₄×#41	—	113	1	no	—
130	2₋¹⁺⁴	—	226	8	no	—
131	H(5,𝔽₂)	—	2	8	no	—
132	C₂×#41	—	113	6	no	—
133	C₂×#41	—	113	3	no	—
134	C₂×#41	C₂×D₁₁₃	1	3	yes	—
135	C₂×#41	C₂×D₁₁₃	1	2	yes	—
136	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
137	C₂²×Q₈	—	113	2	no	—
138	C₂²×Q₈	—	113	3	no	—
139	C₂²×D₄	C₂×D₁₁₃	1	3	yes	—
140	C₂²×D₄	C₂×D₁₁₃	1	6	yes	—
141	C₄⋊Q₈	—	113	3	no	—
142	#127	—	113	3	no	—
143	#124	—	113	1	no	—
144	#124	—	113	3	no	—
145	#124	—	113	6	no	—
146	#124	—	113	12	no	—
147	#123	—	113	12	no	—
148	C₂²⋊Q₈	—	113	12	no	—
149	#121	—	113	12	no	—
150	#121	—	113	12	no	—
151	#120	—	113	12	no	—
152	C₄×Q₈	—	113	2	no	—
153	C₄×D₄	—	113	6	no	—
154	C₄×D₄	—	113	12	no	—
155	#117	—	113	3	no	—
156	C₂×C₂²⋊C₄	—	113	12	no	—
157	C₂×C₄²	—	113	3	no	—
158	C₂⁴	C₂²×D₁₁₃	1	6	yes	—
159	#41	—	2	8	no	—
160	#41	—	2	12	no	—
161	#41	—	226	24	no	—
162	#41	—	226	12	no	—
163	#41	C₂²⋊D₁₁₃	1	8	yes	complemented
164	C₂×Q₈	—	113	6	no	—
165	C₂×Q₈	—	113	3	no	—
166	C₂×Q₈	—	226	8	no	—
167	C₂×Q₈	—	226	12	no	—
168	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
170	C₂×D₄	—	2	24	no	—
171	C₂×D₄	—	2	12	no	—
172	C₂×D₄	—	113	12	no	—
173	C₂×D₄	—	113	6	no	—
174	C₂×D₄	—	113	1	no	—
175	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
176	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
177	C₂×D₄	C₂²×D₁₁₃	1	3	yes	—
178	C₂×D₄	C₂²×D₁₁₃	1	3	yes	—
179	C₂²×C₄	—	113	6	no	—
180	C₂²×C₄	—	113	6	no	—
181	C₂²×C₄	—	113	6	no	—
182	C₂²×C₄	C₂²×D₁₁₃	1	6	yes	—
183	C₂²×C₄	C₂²×D₁₁₃	1	3	yes	—
184	C₄⋊C₄	—	113	6	no	—
185	C₄⋊C₄	—	113	12	no	—
186	C₂²⋊C₄	—	113	12	no	—
187	C₂²⋊C₄	—	113	24	no	—
188	C₂²⋊C₄	—	113	6	no	—
189	C₂²⋊C₄	—	113	4	no	—
190	C₄²	—	113	1	no	—
191	C₄²	—	113	3	no	—
192	C₄²	—	113	6	no	—
193	C₂³	—	2	12	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	113	2	no	—
196	C₂³	C₂³×D₁₁₃	1	6	yes	—
197	C₂³	C₂³×D₁₁₃	1	3	yes	—
198	Q₈	—	2	2	no	—
199	Q₈	—	226	12	no	—
200	Q₈	—	226	6	no	—
201	Q₈	C₂×C₂²⋊D₁₁₃	1	4	yes	—
202	D₄	—	2	12	no	—
203	D₄	—	2	12	no	—
204	D₄	—	2	6	no	—
205	D₄	—	452	12	no	—
206	D₄	—	226	12	no	—
207	D₄	—	226	2	no	—
208	D₄	C₂×C₂²⋊D₁₁₃	1	12	yes	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	113	2	no	—
211	C₂×C₄	—	113	6	no	—
212	C₂×C₄	—	113	6	no	—
213	C₂×C₄	—	452	6	no	—
214	C₂×C₄	—	226	12	no	—
215	C₂×C₄	—	226	12	no	—
216	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	12	yes	—
217	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
218	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₁₁₃	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	6	no	—
222	C₂²	—	2	12	no	—
223	C₂²	—	8	12	no	—
224	C₂²	—	226	2	no	—
225	C₂²	—	226	4	no	—
226	C₂²	C₂²×C₂²⋊D₁₁₃	1	6	yes	—
227	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	226	6	no	—
230	C₄	—	226	2	no	—
231	C₄	—	226	6	no	—
232	C₄	C₂²×C₂²⋊D₁₁₃	1	6	yes	—
233	C₄	C₂²×C₂²⋊D₁₁₃	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	4	6	no	—
236	C₂	—	452	2	no	—
237	C₂	C₂³×C₂²⋊D₁₁₃	1	1	yes	characteristic
238	C₂	#20299433	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₁₁₃	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₁₃
hypocenter	C₁₁₃	#3500
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#16813669	C₂	semidirect
2. #20299255	C₂	semidirect
3. #20299242	C₂	semidirect
4. #20299169	C₂	semidirect
5. #20299165	C₂	semidirect
6. #20299144	C₂	semidirect
7. C₂×#16813609	C₂	semidirect
8. #41×Dic₁₁₃	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₁₁₃	C₂	semidirect
11. C₂×#14268647	C₂²	semidirect
12. C₂×#14268646	C₂²	semidirect
13. C₂²×Dic₂₂₆	C₂²	semidirect
14. C₂×Q₈×D₁₁₃	C₂²	semidirect
15. #16813620	C₂²	semidirect
16. #16813618	C₂²	semidirect
17. #16813611	C₂²	semidirect
18. #16813609	C₂²	semidirect
19. #16813609	C₂²	semidirect
20. #16813609	C₂²	semidirect
21. #16813603	C₂²	semidirect
22. #16813601	C₂²	semidirect
23. #16813594	C₂²	semidirect
24. #16813593	C₂²	semidirect
25. #16813591	C₂²	semidirect
26. Q₈×Dic₁₁₃	C₂²	semidirect
27. D₄×Dic₁₁₃	C₂²	semidirect
28. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
29. #16813581	C₂²	semidirect
30. C₂×C₂²⋊Dic₁₁₃	C₂²	semidirect
31. C₂×C₄×Dic₁₁₃	C₂²	semidirect
62. #41×C₁₁₃	D₄	semidirect
120. C₁₁₃	#3500	semidirect
122. C₂×H(5,𝔽₂)	D₁₁₃	semidirect
163. #41	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×#16813669	2
#20299255	2
#20299242	2
#20299169	2
#20299165	2
#20299144	2
C₂×#16813609	2
#41×Dic₁₁₃	2
C₂×H(5,𝔽₂)×C₁₁₃	2
#3500	113