Subgroups — #28393404

Order 14464.

928 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₁₁₃	C₂	1	2	yes	maximal, complemented
2	#20298505	C₂	1	2	yes	maximal, complemented
3	#20298478	C₂	1	4	yes	maximal, complemented
4	#20298461	C₂	1	2	yes	maximal, complemented
5	#20298458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16813529	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₁₁₃	C₂²	1	4	yes	—
10	D₄×Dic₁₁₃	—	2	8	no	—
11	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
16	#16813529	C₂²	1	4	yes	—
17	#16813529	C₂²	1	4	yes	—
18	#16813529	C₂²	1	1	yes	characteristic
19	#16813529	C₂²	1	2	yes	—
20	#16813529	C₂²	1	1	yes	characteristic
21	C₂×#41×C₁₁₃	—	2	8	no	—
22	C₂×#41×C₁₁₃	C₄	1	8	yes	complemented
23	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
24	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
25	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
28	C₂²×Dic₁₁₃	—	2	8	no	—
29	C₂²×Dic₁₁₃	—	2	8	no	—
30	C₂²×Dic₁₁₃	—	4	4	no	—
31	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
32	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
33	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
34	C₄⋊Dic₁₁₃	—	2	4	no	—
35	C₂²⋊Dic₁₁₃	—	2	8	no	—
36	C₂²⋊Dic₁₁₃	—	2	4	no	—
37	C₂²⋊Dic₁₁₃	—	4	2	no	—
38	C₄×Dic₁₁₃	—	2	4	no	—
39	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
41	#41×C₁₁₃	—	2	32	no	—
42	C₂×Q₈×C₁₁₃	—	2	4	no	—
43	C₂×Q₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
44	C₂×D₄×C₁₁₃	—	2	8	no	—
45	C₂×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
46	C₂×D₄×C₁₁₃	C₂×C₄	1	8	yes	complemented
47	C₂×D₄×C₁₁₃	D₄	1	8	yes	—
48	C₂²×C₄×C₁₁₃	—	2	8	no	—
49	C₂²×C₄×C₁₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₁₃	C₂×C₄	1	8	yes	complemented
51	C₂²×C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
52	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
55	C₂³×C₁₁₃	—	2	4	no	—
56	C₂³×C₁₁₃	—	2	2	no	—
57	C₂³×C₁₁₃	—	2	2	no	—
58	C₂³×C₁₁₃	—	2	2	no	—
59	C₂³×C₁₁₃	—	2	2	no	—
60	C₂³×C₁₁₃	—	2	1	no	—
61	C₂³×C₁₁₃	—	2	1	no	—
62	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
63	C₂³×C₁₁₃	C₂²×C₄	1	4	yes	—
64	C₂³×C₁₁₃	C₂×D₄	1	8	yes	—
65	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
66	Q₈×C₁₁₃	—	2	8	no	—
67	D₄×C₁₁₃	—	2	16	no	—
68	D₄×C₁₁₃	C₂²⋊C₄	1	16	yes	complemented
69	C₂×C₄×C₁₁₃	—	2	16	no	—
70	C₂×C₄×C₁₁₃	—	2	2	no	—
71	C₂×C₄×C₁₁₃	—	2	4	no	—
72	C₂×C₄×C₁₁₃	—	2	8	no	—
73	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
77	C₂×Dic₁₁₃	—	2	2	no	—
78	C₂×Dic₁₁₃	—	2	2	no	—
79	C₂×Dic₁₁₃	—	2	2	no	—
80	C₂×Dic₁₁₃	—	2	4	no	—
81	C₂×Dic₁₁₃	—	2	2	no	—
82	C₂×Dic₁₁₃	—	2	4	no	—
83	C₂×Dic₁₁₃	—	2	4	no	—
84	C₂×Dic₁₁₃	—	2	2	no	—
85	C₂×Dic₁₁₃	—	4	16	no	—
86	C₂×Dic₁₁₃	#41	1	4	yes	complemented
87	C₂²×C₁₁₃	—	2	4	no	—
88	C₂²×C₁₁₃	—	2	4	no	—
89	C₂²×C₁₁₃	—	2	8	no	—
90	C₂²×C₁₁₃	—	2	8	no	—
91	C₂²×C₁₁₃	—	2	2	no	—
92	C₂²×C₁₁₃	—	2	2	no	—
93	C₂²×C₁₁₃	—	2	2	no	—
94	C₂²×C₁₁₃	—	4	2	no	—
95	C₂²×C₁₁₃	—	4	2	no	—
96	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
99	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
100	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₄×C₁₁₃	—	2	2	no	—
103	C₄×C₁₁₃	—	2	4	no	—
104	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
105	Dic₁₁₃	—	2	4	no	—
106	Dic₁₁₃	—	4	4	no	—
107	Dic₁₁₃	—	4	2	no	—
108	C₂×C₁₁₃	—	2	8	no	—
109	C₂×C₁₁₃	—	4	2	no	—
110	C₂×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
112	C₂×C₁₁₃	#520	1	1	yes	characteristic
113	C₂×C₁₁₃	#546	1	2	yes	—
114	C₂×C₁₁₃	#547	1	2	yes	—
115	#2339	—	113	1	no	maximal, Sylow
116	C₁₁₃	#2339	1	1	yes	characteristic, complemented, Sylow
117	C₂²×#41	D₁₁₃	1	1	yes	characteristic
118	C₂×C₄×D₄	—	113	2	no	—
119	#391	—	113	2	no	—
120	#386	—	113	4	no	—
121	#381	—	113	2	no	—
122	#380	—	113	1	no	—
123	C₄×C₂²⋊C₄	—	113	1	no	—
124	C₂×#95	—	113	2	no	—
125	C₂×#41	—	2	8	no	—
126	C₂×#41	Dic₁₁₃	1	8	yes	complemented
127	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
128	C₂²×D₄	C₂×D₁₁₃	1	2	yes	—
129	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
130	C₂³×C₄	—	113	4	no	—
131	C₂³×C₄	C₂×D₁₁₃	1	2	yes	—
132	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
133	C₄×D₄	—	226	8	no	—
134	C₂×C₄⋊C₄	—	113	2	no	—
135	C₂×C₂²⋊C₄	—	113	2	no	—
136	C₂×C₂²⋊C₄	—	113	4	no	—
137	C₂×C₂²⋊C₄	—	113	2	no	—
138	C₂×C₄²	—	113	2	no	—
139	#95	—	113	1	no	—
140	#95	—	113	2	no	—
141	#95	—	113	4	no	—
142	#95	—	113	1	no	—
143	#95	—	113	4	no	—
144	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
145	C₂⁴	C₂²×D₁₁₃	1	2	yes	—
146	#41	—	2	32	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	C₂×Dic₁₁₃	1	4	yes	complemented
149	C₂×D₄	—	2	8	no	—
150	C₂×D₄	C₂×Dic₁₁₃	1	4	yes	—
151	C₂×D₄	C₂×Dic₁₁₃	1	8	yes	complemented
152	C₂×D₄	C₂²⋊D₁₁₃	1	8	yes	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	113	2	no	—
156	C₂²×C₄	—	113	2	no	—
157	C₂²×C₄	—	113	4	no	—
158	C₂²×C₄	—	452	4	no	—
159	C₂²×C₄	—	226	8	no	—
160	C₂²×C₄	—	226	8	no	—
161	C₂²×C₄	C₂×Dic₁₁₃	1	8	yes	complemented
162	C₂²×C₄	C₂×Dic₁₁₃	1	4	yes	complemented
163	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
166	C₄⋊C₄	—	226	4	no	—
167	C₂²⋊C₄	—	452	2	no	—
168	C₂²⋊C₄	—	226	4	no	—
169	C₂²⋊C₄	—	226	8	no	—
170	C₄²	—	226	4	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	C₂²×Dic₁₁₃	1	2	yes	—
179	C₂³	C₂²×Dic₁₁₃	1	4	yes	—
180	C₂³	C₂×C₂²⋊D₁₁₃	1	8	yes	—
181	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
182	Q₈	—	2	8	no	—
183	D₄	—	2	16	no	—
184	D₄	C₂²⋊Dic₁₁₃	1	16	yes	complemented
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	113	4	no	—
190	C₂×C₄	—	452	16	no	—
191	C₂×C₄	—	226	2	no	—
192	C₂×C₄	—	226	4	no	—
193	C₂×C₄	—	226	2	no	—
194	C₂×C₄	—	226	4	no	—
195	C₂×C₄	—	226	4	no	—
196	C₂×C₄	—	226	2	no	—
197	C₂×C₄	—	226	2	no	—
198	C₂×C₄	—	226	2	no	—
199	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
200	C₂×C₄	C₂²×Dic₁₁₃	1	4	yes	—
201	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
202	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	4	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	8	no	—
206	C₂²	—	2	8	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂×C₂²⋊Dic₁₁₃	1	8	yes	—
213	C₂²	C₂³×Dic₁₁₃	1	1	yes	characteristic
214	C₂²	C₂²×C₂²⋊D₁₁₃	1	2	yes	—
215	C₂²	#41×D₁₁₃	1	2	yes	—
216	C₂²	#16813668	1	1	yes	characteristic
217	C₂²	#16813669	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	4	no	—
220	C₄	—	452	4	no	—
221	C₄	—	452	2	no	—
222	C₄	—	226	4	no	—
223	C₄	C₂×C₂²⋊Dic₁₁₃	1	4	yes	—
224	C₂	—	2	8	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
227	C₂	#41×Dic₁₁₃	1	1	yes	characteristic
228	C₂	#20299081	1	1	yes	characteristic
229	C₂	#20299236	1	2	yes	—
230	C₂	#20299247	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Fitting, prime radical	C₂²×#41×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
hypercenter	C₂²×#41	D₁₁₃
hypocenter	C₁₁₃	#2339
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₁₁₃	C₂	semidirect
2. #20298505	C₂	semidirect
3. #20298478	C₂	semidirect
4. #20298461	C₂	semidirect
5. #20298458	C₂	semidirect
6. C₄×C₂²⋊Dic₁₁₃	C₂	semidirect
7. C₂×#16813529	C₂	semidirect
22. C₂×#41×C₁₁₃	C₄	semidirect
43. C₂×Q₈×C₁₁₃	C₂×C₄	semidirect
46. C₂×D₄×C₁₁₃	C₂×C₄	semidirect
50. C₂²×C₄×C₁₁₃	C₂×C₄	semidirect
51. C₂²×C₄×C₁₁₃	C₂×C₄	semidirect
68. D₄×C₁₁₃	C₂²⋊C₄	semidirect
86. C₂×Dic₁₁₃	#41	semidirect
116. C₁₁₃	#2339	semidirect
126. C₂×#41	Dic₁₁₃	semidirect
148. C₂×Q₈	C₂×Dic₁₁₃	semidirect
151. C₂×D₄	C₂×Dic₁₁₃	semidirect
161. C₂²×C₄	C₂×Dic₁₁₃	semidirect
162. C₂²×C₄	C₂×Dic₁₁₃	semidirect
184. D₄	C₂²⋊Dic₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₁₃	2
#20298505	2
#20298478	2
#20298461	2
#20298458	2
C₄×C₂²⋊Dic₁₁₃	2
C₂×#16813529	2
C₂²×#41×C₁₁₃	2
#2339	113