Subgroups — C₂²×#16813582 #28403742

Order 14464.

2528 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16813582	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#14268626	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#14268624	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₁₁₃	C₄	1	16	yes	complemented
11	C₂²×C₂²⋊D₁₁₃	C₂²	1	12	yes	—
12	C₂³×Dic₁₁₃	—	2	8	no	—
13	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
16	C₂³×Dic₁₁₃	C₂²	1	12	yes	complemented
17	C₂²×C₄×D₁₁₃	—	2	8	no	—
18	C₂²×C₄×D₁₁₃	C₂²	1	12	yes	complemented
19	#16813582	C₂²	1	64	yes	direct product factor
20	C₂×#14268626	C₂²	1	12	yes	complemented
21	C₂×#14268624	C₂²	1	12	yes	complemented
22	C₂×C₄×Dic₁₁₃	C₂²	1	12	yes	complemented
23	C₂⁵×C₁₁₃	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	12	yes	complemented
27	C₂³×D₁₁₃	—	2	4	no	—
28	C₂³×D₁₁₃	—	2	4	no	—
29	C₂³×D₁₁₃	C₂×C₄	1	8	yes	complemented
30	C₂³×D₁₁₃	C₂³	1	6	yes	—
31	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	96	yes	complemented
32	C₂×C₂²⋊D₁₁₃	C₂³	1	16	yes	—
33	C₂²×Dic₁₁₃	—	2	4	no	—
34	C₂²×Dic₁₁₃	—	2	48	no	—
35	C₂²×Dic₁₁₃	C₂×C₄	1	8	yes	complemented
36	C₂²×Dic₁₁₃	D₄	1	8	yes	complemented
37	C₂²×Dic₁₁₃	C₂³	1	6	yes	—
38	C₂²×Dic₁₁₃	C₂³	1	6	yes	—
39	C₂²×Dic₁₁₃	C₂³	1	6	yes	—
40	C₂²×Dic₁₁₃	C₂³	1	16	yes	complemented
41	C₂×C₄×D₁₁₃	—	2	48	no	—
42	C₂×C₄×D₁₁₃	C₂³	1	16	yes	complemented
43	#14268626	C₂³	1	16	yes	complemented
44	#14268624	C₂³	1	16	yes	complemented
45	C₄×Dic₁₁₃	C₂³	1	16	yes	complemented
46	C₂⁴×C₁₁₃	—	2	4	no	—
47	C₂⁴×C₁₁₃	—	2	4	no	—
48	C₂⁴×C₁₁₃	C₂×C₄	1	8	yes	complemented
49	C₂⁴×C₁₁₃	C₂³	1	6	yes	—
50	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₁₃	C₂³	1	6	yes	—
54	C₂²×C₄×C₁₁₃	C₂³	1	6	yes	—
55	C₂²⋊C₄×C₁₁₃	C₂³	1	16	yes	complemented
56	C₂³×C₁₁₃	—	2	32	no	—
57	C₂³×C₁₁₃	—	2	12	no	—
58	C₂³×C₁₁₃	—	2	12	no	—
59	C₂³×C₁₁₃	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₁₃	C₂²×C₄	1	24	yes	complemented
61	C₂³×C₁₁₃	C₂×D₄	1	4	yes	—
62	C₂³×C₁₁₃	#41	1	4	yes	—
63	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
64	C₂³×C₁₁₃	C₂⁴	1	3	yes	—
65	C₂²×D₁₁₃	—	2	32	no	—
66	C₂²×D₁₁₃	—	2	12	no	—
67	C₂²×D₁₁₃	—	2	12	no	—
68	C₂²×D₁₁₃	C₂²×C₄	1	24	yes	complemented
69	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
70	C₂×C₄×C₁₁₃	—	2	12	no	—
71	C₂×C₄×C₁₁₃	—	2	12	no	—
72	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
73	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
74	C₂²⋊D₁₁₃	C₂²×C₄	1	64	yes	complemented
75	C₂×Dic₁₁₃	—	2	12	no	—
76	C₂×Dic₁₁₃	—	2	32	no	—
77	C₂×Dic₁₁₃	C₂²×C₄	1	24	yes	complemented
78	C₂×Dic₁₁₃	C₂×D₄	1	24	yes	complemented
79	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
80	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
81	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
82	C₄×D₁₁₃	—	2	32	no	—
83	C₂²×C₁₁₃	—	2	48	no	—
84	C₂²×C₁₁₃	—	2	4	no	—
85	C₂²×C₁₁₃	—	2	4	no	—
86	C₂²×C₁₁₃	C₄×D₄	1	16	yes	complemented
87	C₂²×C₁₁₃	C₂³×C₄	1	6	yes	—
88	C₂²×C₁₁₃	C₂³×C₄	1	8	yes	complemented
89	C₂²×C₁₁₃	C₂²×D₄	1	6	yes	—
90	C₂²×C₁₁₃	C₂×#41	1	6	yes	—
91	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
92	C₂×D₁₁₃	—	2	48	no	—
93	C₂×D₁₁₃	—	2	4	no	—
94	C₂×D₁₁₃	—	2	4	no	—
95	C₂×D₁₁₃	C₂³×C₄	1	8	yes	complemented
96	C₄×C₁₁₃	—	2	4	no	—
97	C₄×C₁₁₃	—	2	4	no	—
98	Dic₁₁₃	—	2	4	no	—
99	Dic₁₁₃	C₂³×C₄	1	8	yes	complemented
100	Dic₁₁₃	C₂²×D₄	1	8	yes	complemented
101	C₂×C₁₁₃	—	2	8	no	—
102	C₂×C₁₁₃	C₂×C₄×D₄	1	12	yes	complemented
103	C₂×C₁₁₃	C₂⁴×C₄	1	1	yes	characteristic
104	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
105	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
106	D₁₁₃	—	2	8	no	—
107	C₂²×C₄×D₄	—	113	1	no	maximal, Sylow
108	C₁₁₃	C₂²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
109	C₂³×D₄	—	113	1	no	—
110	C₂⁴×C₄	—	113	1	no	—
111	C₂⁴×C₄	—	113	1	no	—
112	C₂×C₄×D₄	—	113	24	no	—
113	C₂²×C₄⋊C₄	—	113	1	no	—
114	C₂²×C₂²⋊C₄	—	113	1	no	—
115	C₂²×C₂²⋊C₄	D₁₁₃	1	1	yes	characteristic, complemented
116	C₂²×C₄²	—	113	1	no	—
117	C₂⁵	—	113	1	no	—
118	C₂⁵	C₂×D₁₁₃	1	1	yes	characteristic
119	C₂²×D₄	—	113	16	no	—
120	C₂²×D₄	—	113	12	no	—
121	C₂³×C₄	—	113	1	no	—
122	C₂³×C₄	—	113	1	no	—
123	C₂³×C₄	—	113	12	no	—
124	C₂³×C₄	—	113	1	no	—
125	C₂³×C₄	—	113	12	no	—
126	C₂³×C₄	—	226	8	no	—
127	C₂³×C₄	—	226	8	no	—
128	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
130	C₄×D₄	—	113	64	no	—
131	C₂×C₄⋊C₄	—	113	12	no	—
132	C₂×C₂²⋊C₄	—	113	12	no	—
133	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	12	yes	complemented
134	C₂×C₄²	—	113	12	no	—
135	C₂⁴	—	2	4	no	—
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	113	8	no	—
138	C₂⁴	—	113	6	no	—
139	C₂⁴	—	226	4	no	—
140	C₂⁴	—	226	4	no	—
141	C₂⁴	C₄×D₁₁₃	1	8	yes	complemented
142	C₂⁴	C₂²×D₁₁₃	1	6	yes	—
143	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
144	C₂×D₄	—	113	96	no	—
145	C₂×D₄	—	113	16	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	113	8	no	—
149	C₂²×C₄	—	113	8	no	—
150	C₂²×C₄	—	113	6	no	—
151	C₂²×C₄	—	113	6	no	—
152	C₂²×C₄	—	113	6	no	—
153	C₂²×C₄	—	113	16	no	—
154	C₂²×C₄	—	113	16	no	—
155	C₂²×C₄	—	226	48	no	—
156	C₂²×C₄	—	226	48	no	—
157	C₂²×C₄	—	226	4	no	—
158	C₂²×C₄	C₂²×D₁₁₃	1	6	yes	—
159	C₂²×C₄	C₂²×D₁₁₃	1	6	yes	—
160	C₄⋊C₄	—	113	16	no	—
161	C₂²⋊C₄	—	113	16	no	—
162	C₂²⋊C₄	C₂²×D₁₁₃	1	16	yes	complemented
163	C₄²	—	113	16	no	—
164	C₂³	—	2	32	no	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	12	no	—
167	C₂³	—	113	24	no	—
168	C₂³	—	113	4	no	—
169	C₂³	—	226	32	no	—
170	C₂³	—	226	12	no	—
171	C₂³	—	226	12	no	—
172	C₂³	C₂×C₄×D₁₁₃	1	4	yes	—
173	C₂³	C₂×C₄×D₁₁₃	1	24	yes	complemented
174	C₂³	D₄×D₁₁₃	1	4	yes	—
175	C₂³	#14268646	1	4	yes	—
176	C₂³	C₂³×D₁₁₃	1	4	yes	—
177	C₂³	C₂³×D₁₁₃	1	3	yes	—
178	D₄	—	113	64	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	113	24	no	—
182	C₂×C₄	—	113	24	no	—
183	C₂×C₄	—	113	4	no	—
184	C₂×C₄	—	113	4	no	—
185	C₂×C₄	—	113	4	no	—
186	C₂×C₄	—	226	32	no	—
187	C₂×C₄	—	226	32	no	—
188	C₂×C₄	—	226	12	no	—
189	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
190	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
191	C₂²	—	2	48	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	113	8	no	—
195	C₂²	—	226	48	no	—
196	C₂²	—	226	4	no	—
197	C₂²	—	226	4	no	—
198	C₂²	#16813582	1	16	yes	direct product factor
199	C₂²	C₂²×C₄×D₁₁₃	1	6	yes	—
200	C₂²	C₂²×C₄×D₁₁₃	1	8	yes	complemented
201	C₂²	C₂×D₄×D₁₁₃	1	6	yes	—
202	C₂²	C₂×#14268646	1	6	yes	—
203	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	113	8	no	—
207	C₄	—	113	8	no	—
208	C₄	—	226	4	no	—
209	C₂	—	2	8	no	—
210	C₂	—	226	8	no	—
211	C₂	C₂×#16813582	1	12	yes	direct product factor
212	C₂	C₂³×C₄×D₁₁₃	1	1	yes	characteristic
213	C₂	C₂²×D₄×D₁₁₃	1	1	yes	characteristic
214	C₂	C₂²×#14268646	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₁₃
commutator	C₂×C₁₁₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂²×C₂²⋊C₄×C₁₁₃	C₂
socle	C₂⁴×C₁₁₃	C₂³
prime radical	C₂³×C₂²⋊D₁₁₃	C₂
cosocle	C₂²×C₁₁₃	C₂⁵
hypercenter	C₂²×C₂²⋊C₄	D₁₁₃
hypocenter	C₁₁₃	C₂²×C₄×D₄
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₁₁₃	C₂	semidirect
3. C₂³×C₄×D₁₁₃	C₂	semidirect
4. C₂×#16813582	C₂	direct (N × H)
5. C₂²×#14268626	C₂	semidirect
6. C₂²×#14268624	C₂	semidirect
7. C₂²×C₄×Dic₁₁₃	C₂	semidirect
8. C₂²×C₂²⋊C₄×C₁₁₃	C₂	semidirect
10. C₂²×C₂²⋊D₁₁₃	C₄	semidirect
16. C₂³×Dic₁₁₃	C₂²	semidirect
18. C₂²×C₄×D₁₁₃	C₂²	semidirect
19. #16813582	C₂²	direct (N × H)
20. C₂×#14268626	C₂²	semidirect
21. C₂×#14268624	C₂²	semidirect
22. C₂×C₄×Dic₁₁₃	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₁₁₃	C₂²	semidirect
29. C₂³×D₁₁₃	C₂×C₄	semidirect
31. C₂×C₂²⋊D₁₁₃	C₂×C₄	semidirect
35. C₂²×Dic₁₁₃	C₂×C₄	semidirect
36. C₂²×Dic₁₁₃	D₄	semidirect
40. C₂²×Dic₁₁₃	C₂³	semidirect
42. C₂×C₄×D₁₁₃	C₂³	semidirect
43. #14268626	C₂³	semidirect
44. #14268624	C₂³	semidirect
45. C₄×Dic₁₁₃	C₂³	semidirect
48. C₂⁴×C₁₁₃	C₂×C₄	semidirect
55. C₂²⋊C₄×C₁₁₃	C₂³	semidirect
60. C₂³×C₁₁₃	C₂²×C₄	semidirect
68. C₂²×D₁₁₃	C₂²×C₄	semidirect
74. C₂²⋊D₁₁₃	C₂²×C₄	semidirect
77. C₂×Dic₁₁₃	C₂²×C₄	semidirect
78. C₂×Dic₁₁₃	C₂×D₄	semidirect
86. C₂²×C₁₁₃	C₄×D₄	semidirect
88. C₂²×C₁₁₃	C₂³×C₄	semidirect
95. C₂×D₁₁₃	C₂³×C₄	semidirect
99. Dic₁₁₃	C₂³×C₄	semidirect
100. Dic₁₁₃	C₂²×D₄	semidirect
102. C₂×C₁₁₃	C₂×C₄×D₄	semidirect
108. C₁₁₃	C₂²×C₄×D₄	semidirect
115. C₂²×C₂²⋊C₄	D₁₁₃	semidirect
133. C₂×C₂²⋊C₄	C₂×D₁₁₃	semidirect
141. C₂⁴	C₄×D₁₁₃	semidirect
162. C₂²⋊C₄	C₂²×D₁₁₃	semidirect
173. C₂³	C₂×C₄×D₁₁₃	semidirect
198. C₂²	#16813582	direct (N × H)
200. C₂²	C₂²×C₄×D₁₁₃	semidirect
211. C₂	C₂×#16813582	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×C₂²⋊D₁₁₃	2
C₂⁴×Dic₁₁₃	2
C₂³×C₄×D₁₁₃	2
C₂×#16813582	2
C₂²×#14268626	2
C₂²×#14268624	2
C₂²×C₄×Dic₁₁₃	2
C₂²×C₂²⋊C₄×C₁₁₃	2
C₂²×C₄×D₄	113

Subgroups — C22×#16813582 #28403742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16813582 #28403742