Subgroups — C₂×#20298661 #28399244

Order 14464.

616 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	C₂³×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#20298661	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16813551	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16813534	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#11744751	—	2	4	no	—
12	C₂²×#11744751	C₂²	1	1	yes	characteristic
13	C₂²×#11744751	C₂²	1	1	yes	characteristic
14	C₂²×#11744751	C₂²	1	4	yes	complemented
15	C₂×C₈×D₁₁₃	—	2	4	no	—
16	C₂×C₈×D₁₁₃	C₂²	1	4	yes	complemented
17	C₄×C₂²⋊D₁₁₃	C₄	1	4	yes	—
18	C₄×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
19	C₂×#14268626	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₁₁₃	C₄	1	1	yes	characteristic
21	C₂×#14268624	C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
23	#16813551	C₂²	1	4	yes	complemented
24	#16813534	C₂²	1	4	yes	complemented
25	C₈×Dic₁₁₃	C₂²	1	4	yes	complemented
26	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
28	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₁₁₃	C₂²	1	4	yes	complemented
30	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊D₁₁₃	C₈	1	8	yes	complemented
32	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	4	yes	—
33	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₁₃	—	2	2	no	—
37	C₂×C₄×D₁₁₃	—	2	2	no	—
38	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
40	C₂×#11744751	—	2	2	no	—
41	C₂×#11744751	—	2	8	no	—
42	C₂×#11744751	D₄	1	4	yes	complemented
43	C₂×#11744751	C₂³	1	2	yes	—
44	C₂×#11744751	C₂³	1	2	yes	—
45	C₈×D₁₁₃	—	2	8	no	—
46	#14268626	C₂×C₄	1	4	yes	—
47	C₂²⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
48	#14268624	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₁₃	C₂³	1	2	yes	—
51	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₁₃	—	2	2	no	—
59	C₂×C₈×C₁₁₃	—	2	2	no	—
60	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
61	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
62	C₂³×C₁₁₃	—	2	2	no	—
63	C₂³×C₁₁₃	—	2	2	no	—
64	C₂³×C₁₁₃	C₂×C₈	1	4	yes	complemented
65	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₁₁₃	—	2	2	no	—
68	C₂²×D₁₁₃	—	2	2	no	—
69	C₂²×D₁₁₃	C₂×C₈	1	4	yes	complemented
70	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₁₃	—	2	2	no	—
72	C₂×C₄×C₁₁₃	—	2	4	no	—
73	C₂×C₄×C₁₁₃	—	2	2	no	—
74	C₂×C₄×C₁₁₃	—	2	2	no	—
75	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₁₃	#41	1	2	yes	—
79	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₁₁₃	C₂×C₈	1	16	yes	complemented
81	C₂×Dic₁₁₃	—	2	2	no	—
82	C₂×Dic₁₁₃	—	2	2	no	—
83	C₂×Dic₁₁₃	C₂×C₈	1	4	yes	complemented
84	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
87	C₄×D₁₁₃	—	2	4	no	—
88	C₄×D₁₁₃	—	2	2	no	—
89	C₄×D₁₁₃	—	2	2	no	—
90	C₈×C₁₁₃	—	2	2	no	—
91	C₈×C₁₁₃	—	2	2	no	—
92	#11744751	—	2	2	no	—
93	#11744751	C₂×D₄	1	4	yes	complemented
94	C₂²×C₁₁₃	—	2	2	no	—
95	C₂²×C₁₁₃	—	2	2	no	—
96	C₂²×C₁₁₃	—	2	8	no	—
97	C₂²×C₁₁₃	C₄×D₄	1	2	yes	—
98	C₂²×C₁₁₃	C₂²×C₈	1	2	yes	—
99	C₂²×C₁₁₃	C₂²×C₈	1	4	yes	complemented
100	C₂²×C₁₁₃	#131	1	2	yes	—
101	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₁₃	—	2	2	no	—
103	C₂×D₁₁₃	—	2	2	no	—
104	C₂×D₁₁₃	—	2	8	no	—
105	C₂×D₁₁₃	C₂²×C₈	1	4	yes	complemented
106	C₄×C₁₁₃	—	2	2	no	—
107	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
108	C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
110	Dic₁₁₃	—	2	2	no	—
111	Dic₁₁₃	—	2	2	no	—
112	Dic₁₁₃	C₂²×C₈	1	4	yes	complemented
113	C₂×C₁₁₃	—	2	4	no	—
114	C₂×C₁₁₃	C₈×D₄	1	4	yes	complemented
115	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
117	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
118	D₁₁₃	—	2	4	no	—
119	C₂×C₈×D₄	—	113	1	no	maximal, Sylow
120	C₁₁₃	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₈	—	113	1	no	—
122	C₂³×C₈	—	113	1	no	—
123	C₂×C₄×D₄	—	113	1	no	—
124	C₈×D₄	—	113	8	no	—
125	C₂×C₄⋊C₈	—	113	1	no	—
126	C₂×C₂²⋊C₈	—	113	1	no	—
127	C₂×C₂²⋊C₈	D₁₁₃	1	1	yes	characteristic, complemented
128	C₂×C₄×C₈	—	113	1	no	—
129	C₂²×D₄	—	113	1	no	—
130	C₂³×C₄	—	113	1	no	—
131	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
132	C₂²×C₈	—	113	1	no	—
133	C₂²×C₈	—	113	4	no	—
134	C₂²×C₈	—	113	1	no	—
135	C₂²×C₈	—	113	4	no	—
136	C₂²×C₈	—	226	4	no	—
137	C₂²×C₈	—	226	4	no	—
138	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
140	C₄×D₄	—	113	4	no	—
141	C₄×D₄	—	113	4	no	—
142	C₂×C₄⋊C₄	—	113	1	no	—
143	C₂×C₂²⋊C₄	—	113	1	no	—
144	C₂×C₂²⋊C₄	—	113	1	no	—
145	C₂×C₄²	—	113	1	no	—
146	C₄⋊C₈	—	113	4	no	—
147	C₂²⋊C₈	—	113	4	no	—
148	C₂²⋊C₈	C₂×D₁₁₃	1	4	yes	complemented
149	C₄×C₈	—	113	4	no	—
150	C₂⁴	—	113	1	no	—
151	C₂⁴	C₄×D₁₁₃	1	1	yes	characteristic
152	C₂×D₄	—	113	8	no	—
153	C₂×D₄	—	113	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	113	1	no	—
157	C₂²×C₄	—	113	1	no	—
158	C₂²×C₄	—	113	1	no	—
159	C₂²×C₄	—	113	2	no	—
160	C₂²×C₄	—	113	2	no	—
161	C₂²×C₄	—	226	2	no	—
162	C₂²×C₄	—	226	2	no	—
163	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₁₁₃	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
166	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	113	4	no	—
170	C₂×C₈	—	113	2	no	—
171	C₂×C₈	—	113	2	no	—
172	C₂×C₈	—	226	8	no	—
173	C₂×C₈	—	226	8	no	—
174	C₂×C₈	—	226	2	no	—
175	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
176	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
177	C₄⋊C₄	—	113	4	no	—
178	C₂²⋊C₄	—	113	4	no	—
179	C₂²⋊C₄	—	113	4	no	—
180	C₄²	—	113	2	no	—
181	C₄²	—	113	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	113	4	no	—
185	C₂³	—	113	2	no	—
186	C₂³	—	226	2	no	—
187	C₂³	—	226	2	no	—
188	C₂³	C₈×D₁₁₃	1	4	yes	complemented
189	C₂³	C₂×C₄×D₁₁₃	1	2	yes	—
190	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
191	D₄	—	113	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	113	4	no	—
197	C₂×C₄	—	113	2	no	—
198	C₂×C₄	—	113	2	no	—
199	C₂×C₄	—	113	2	no	—
200	C₂×C₄	—	226	2	no	—
201	C₂×C₄	—	226	2	no	—
202	C₂×C₄	—	226	4	no	—
203	C₂×C₄	—	226	2	no	—
204	C₂×C₄	—	226	2	no	—
205	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
207	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
208	C₂×C₄	#14268646	1	2	yes	—
209	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	113	4	no	—
213	C₈	—	226	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	113	4	no	—
218	C₂²	—	226	2	no	—
219	C₂²	—	226	2	no	—
220	C₂²	—	226	8	no	—
221	C₂²	#16813582	1	2	yes	—
222	C₂²	C₂×C₈×D₁₁₃	1	2	yes	—
223	C₂²	C₂×C₈×D₁₁₃	1	4	yes	complemented
224	C₂²	#16813630	1	2	yes	—
225	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	113	4	no	—
228	C₄	—	226	2	no	—
229	C₄	—	226	2	no	—
230	C₄	#16813582	1	2	yes	—
231	C₄	C₂×D₄×D₁₁₃	1	1	yes	characteristic
232	C₄	C₂×#14268646	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	—	226	4	no	—
235	C₂	#20298661	1	4	yes	direct product factor
236	C₂	C₂×#16813582	1	1	yes	characteristic
237	C₂	C₂²×C₈×D₁₁₃	1	1	yes	characteristic
238	C₂	C₂×#16813630	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₁₃
commutator	C₂×C₁₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	C₂×C₂²⋊C₈×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂²×C₄
prime radical	C₂²×C₂²⋊D₁₁₃	C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₁₁₃
hypocenter	C₁₁₃	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11744751	C₂	semidirect
2. C₂²×C₈×D₁₁₃	C₂	semidirect
4. #20298661	C₂	direct (N × H)
5. C₂×#16813551	C₂	semidirect
6. C₂×#16813534	C₂	semidirect
7. C₂×C₈×Dic₁₁₃	C₂	semidirect
8. C₂×C₂²⋊C₈×C₁₁₃	C₂	semidirect
14. C₂²×#11744751	C₂²	semidirect
16. C₂×C₈×D₁₁₃	C₂²	semidirect
23. #16813551	C₂²	semidirect
24. #16813534	C₂²	semidirect
25. C₈×Dic₁₁₃	C₂²	semidirect
29. C₂²⋊C₈×C₁₁₃	C₂²	semidirect
31. C₂×C₂²⋊D₁₁₃	C₈	semidirect
42. C₂×#11744751	D₄	semidirect
64. C₂³×C₁₁₃	C₂×C₈	semidirect
69. C₂²×D₁₁₃	C₂×C₈	semidirect
80. C₂²⋊D₁₁₃	C₂×C₈	semidirect
83. C₂×Dic₁₁₃	C₂×C₈	semidirect
93. #11744751	C₂×D₄	semidirect
99. C₂²×C₁₁₃	C₂²×C₈	semidirect
105. C₂×D₁₁₃	C₂²×C₈	semidirect
112. Dic₁₁₃	C₂²×C₈	semidirect
114. C₂×C₁₁₃	C₈×D₄	semidirect
120. C₁₁₃	C₂×C₈×D₄	semidirect
127. C₂×C₂²⋊C₈	D₁₁₃	semidirect
148. C₂²⋊C₈	C₂×D₁₁₃	semidirect
188. C₂³	C₈×D₁₁₃	semidirect
223. C₂²	C₂×C₈×D₁₁₃	semidirect
235. C₂	#20298661	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11744751	2
C₂²×C₈×D₁₁₃	2
C₂×C₄×C₂²⋊D₁₁₃	2
#20298661	2
C₂×#16813551	2
C₂×#16813534	2
C₂×C₈×Dic₁₁₃	2
C₂×C₂²⋊C₈×C₁₁₃	2
C₂×C₈×D₄	113

Subgroups — C2×#20298661 #28399244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20298661 #28399244