Subgroups — C₂×#20299677 #28407567

Order 14464.

1460 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299677	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813710	C₂	1	2	yes	maximal, complemented
3	C₂×#16813709	C₂	1	6	yes	maximal, complemented
4	C₂²×#14268657	C₂	1	3	yes	maximal, complemented
5	C₂×D₁₁₃⋊M₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₁₁₃	C₂	1	1	yes	maximal, characteristic
7	#16813710	C₂²	1	16	yes	complemented
8	#16813709	C₂²	1	48	yes	complemented
9	C₂×#14268657	C₂²	1	1	yes	characteristic
10	C₂×#14268657	C₂²	1	3	yes	—
11	C₂×#14268657	C₂²	1	6	yes	complemented
12	C₂×#14268657	C₂²	1	24	yes	complemented
13	D₁₁₃⋊M₁₆	C₂²	1	24	yes	complemented
14	C₂×#14268656	C₂²	1	6	yes	complemented
15	C₂×#14268655	C₂²	1	6	yes	complemented
16	C₂²×#11744753	C₂²	1	6	yes	complemented
17	C₂×#14268648	C₂²	1	1	yes	characteristic
18	C₂×#14268647	C₄	1	3	yes	—
19	C₂×#14268646	C₂²	1	3	yes	—
20	#41×D₁₁₃	C₄	1	8	yes	—
21	#41×D₁₁₃	C₂²	1	8	yes	—
22	C₂×Q₈×D₁₁₃	C₄	1	1	yes	characteristic
23	C₂×D₄×D₁₁₃	C₄	1	3	yes	—
24	C₂²×C₄×D₁₁₃	C₂²	1	3	yes	—
25	C₂×#41×C₁₁₃	C₄	1	1	yes	characteristic
26	#14268657	C₂³	1	4	yes	—
27	#14268657	C₂³	1	12	yes	—
28	#14268657	C₂³	1	24	yes	complemented
29	#14268656	C₂³	1	24	yes	complemented
30	#14268655	C₂³	1	24	yes	complemented
31	C₂×#11744753	C₂³	1	6	yes	—
32	C₂×#11744753	C₂³	1	2	yes	—
33	C₂×#11744753	C₂³	1	24	yes	complemented
34	C₂³×D₁₁₃	C₂×C₄	1	3	yes	—
35	#14268648	C₂×C₄	1	4	yes	—
36	#14268648	C₂³	1	4	yes	—
37	#14268647	C₂×C₄	1	24	yes	—
38	#14268646	C₂×C₄	1	12	yes	—
39	#14268646	C₂³	1	12	yes	—
40	C₂×Dic₂₂₆	C₂×C₄	1	3	yes	—
41	Q₈×D₁₁₃	C₂×C₄	1	8	yes	—
42	C₂×D₄₅₂	C₂×C₄	1	3	yes	—
43	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	6	yes	—
44	D₄×D₁₁₃	C₂×C₄	1	24	yes	—
45	C₂²×Dic₁₁₃	C₂³	1	3	yes	—
46	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
47	C₂×C₄×D₁₁₃	C₂×C₄	1	12	yes	—
48	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
49	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₁₃	C₂³	1	12	yes	—
51	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
52	#41×C₁₁₃	C₂×C₄	1	8	yes	—
53	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×D₄×C₁₁₃	C₂×C₄	1	3	yes	—
55	C₂²×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
56	C₂³×C₁₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×D₁₁₃	—	4	6	no	—
58	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×D₁₁₃	C₂²×C₄	1	3	yes	—
60	C₂²×D₁₁₃	C₂²×C₄	1	12	yes	—
61	Q₈×C₁₁₃	C₂²×C₄	1	4	yes	—
62	D₄×C₁₁₃	C₂²×C₄	1	12	yes	—
63	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
65	C₂×C₄×C₁₁₃	C₂²×C₄	1	12	yes	—
66	C₂²⋊D₁₁₃	C₂²×C₄	1	24	yes	—
67	C₂×Dic₁₁₃	C₂²×C₄	1	3	yes	—
68	C₂×Dic₁₁₃	C₂²×C₄	1	6	yes	—
69	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₁₁₃	C₂⁴	1	6	yes	—
71	D₄₅₂	C₂²×C₄	1	12	yes	—
72	C₄×D₁₁₃	C₂²×C₄	1	2	yes	—
73	C₄×D₁₁₃	C₂²×C₄	1	12	yes	—
74	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
75	C₄×D₁₁₃	C₂²×C₄	1	12	yes	—
76	C₄×D₁₁₃	C₂⁴	1	2	yes	—
77	C₄×D₁₁₃	C₂⁴	1	6	yes	—
78	Dic₂₂₆	C₂²×C₄	1	12	yes	—
79	#11744753	C₂⁴	1	12	yes	—
80	#11744753	C₂⁴	1	4	yes	—
81	C₂²×C₁₁₃	—	2	6	no	—
82	C₂²×C₁₁₃	C₂³×C₄	1	6	yes	—
83	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×D₁₁₃	—	2	2	no	—
85	C₂×D₁₁₃	—	2	6	no	—
86	C₂×D₁₁₃	—	4	12	no	—
87	C₂×D₁₁₃	C₂³×C₄	1	2	yes	—
88	C₂×D₁₁₃	C₂³×C₄	1	6	yes	—
89	C₄×C₁₁₃	C₂³×C₄	1	6	yes	—
90	C₄×C₁₁₃	C₂³×C₄	1	2	yes	—
91	Dic₁₁₃	C₂³×C₄	1	6	yes	—
92	Dic₁₁₃	C₂³×C₄	1	1	yes	characteristic
93	Dic₁₁₃	C₂⁵	1	1	yes	characteristic
94	C₂×C₁₁₃	—	2	6	no	—
95	C₂×C₁₁₃	#568	1	2	yes	complemented
96	C₂×C₁₁₃	C₂⁴×C₄	1	1	yes	characteristic
97	D₁₁₃	—	2	2	no	—
98	D₁₁₃	—	2	6	no	—
99	C₂×#568	—	113	1	no	maximal, Sylow
100	C₁₁₃	C₂×#568	1	1	yes	characteristic, complemented, Sylow
101	C₂²×#41	—	113	1	no	—
102	#568	—	113	16	no	—
103	C₂×#131	—	113	6	no	—
104	C₂×#131	—	113	2	no	—
105	C₂²×M₁₆	—	113	3	no	—
106	C₂²×M₁₆	—	113	3	no	—
107	C₂×#41	—	113	8	no	—
108	C₂×#41	—	113	8	no	—
109	C₂×#41	—	113	1	no	—
110	C₂×#41	—	113	3	no	—
111	C₂×#41	—	113	3	no	—
112	C₂×#41	#90673	1	1	yes	characteristic
113	C₂²×Q₈	—	113	1	no	—
114	C₂²×D₄	—	113	3	no	—
115	C₂³×C₄	—	113	3	no	—
116	#131	—	113	48	no	—
117	#131	—	113	16	no	—
118	C₂×M₁₆	—	113	1	no	—
119	C₂×M₁₆	—	113	3	no	—
120	C₂×M₁₆	—	113	6	no	—
121	C₂×M₁₆	—	113	6	no	—
122	C₂×M₁₆	—	113	24	no	—
123	C₂×M₁₆	—	113	24	no	—
124	C₂²×C₈	—	113	6	no	—
125	C₂²×C₈	—	113	6	no	—
126	C₂⁴	—	113	3	no	—
127	#41	—	113	4	no	—
128	#41	—	113	4	no	—
129	#41	—	113	12	no	—
130	#41	—	113	12	no	—
131	#41	—	113	24	no	—
132	#41	C₂×#90673	1	8	yes	—
133	C₂×Q₈	—	113	8	no	—
134	C₂×Q₈	—	113	3	no	—
135	C₂×Q₈	C₂×#90673	1	1	yes	characteristic
136	C₂×D₄	—	113	6	no	—
137	C₂×D₄	—	113	24	no	—
138	C₂×D₄	—	113	3	no	—
139	C₂×D₄	C₂×#90673	1	3	yes	—
140	C₂²×C₄	—	113	3	no	—
141	C₂²×C₄	—	113	3	no	—
142	C₂²×C₄	—	113	1	no	—
143	C₂²×C₄	—	113	12	no	—
144	C₂²×C₄	—	113	3	no	—
145	C₂²×C₄	—	113	12	no	—
146	C₂²×C₄	—	113	3	no	—
147	C₂²×C₄	C₂×#90673	1	3	yes	—
148	M₁₆	—	113	4	no	—
149	M₁₆	—	113	12	no	—
150	M₁₆	—	113	24	no	—
151	M₁₆	—	113	24	no	—
152	C₂×C₈	—	113	6	no	—
153	C₂×C₈	—	113	2	no	—
154	C₂×C₈	—	113	24	no	—
155	C₂×C₈	—	113	24	no	—
156	C₂³	—	113	12	no	—
157	C₂³	—	113	1	no	—
158	C₂³	—	113	3	no	—
159	C₂³	—	452	6	no	—
160	C₂³	C₂²×#90673	1	3	yes	—
161	Q₈	—	113	12	no	—
162	Q₈	C₂²×#90673	1	4	yes	—
163	D₄	—	113	24	no	—
164	D₄	—	113	12	no	—
165	D₄	C₂²×#90673	1	12	yes	—
166	C₂×C₄	—	113	6	no	—
167	C₂×C₄	—	113	6	no	—
168	C₂×C₄	—	113	1	no	—
169	C₂×C₄	—	113	3	no	—
170	C₂×C₄	—	113	6	no	—
171	C₂×C₄	—	113	2	no	—
172	C₂×C₄	—	113	2	no	—
173	C₂×C₄	—	113	12	no	—
174	C₂×C₄	—	113	6	no	—
175	C₂×C₄	—	113	12	no	—
176	C₂×C₄	C₂²×#90673	1	12	yes	—
177	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
178	C₂×C₄	C₂²×#90673	1	3	yes	—
179	C₈	—	113	12	no	—
180	C₈	—	113	4	no	—
181	C₂²	—	2	6	no	—
182	C₂²	—	113	2	no	—
183	C₂²	—	113	6	no	—
184	C₂²	—	452	12	no	—
185	C₂²	—	226	2	no	—
186	C₂²	—	226	6	no	—
187	C₂²	C₂³×#90673	1	6	yes	—
188	C₂²	C₂³×#90673	1	1	yes	characteristic
189	C₄	—	113	1	no	—
190	C₄	—	113	1	no	—
191	C₄	—	113	6	no	—
192	C₄	C₂³×#90673	1	2	yes	—
193	C₄	C₂³×#90673	1	6	yes	—
194	C₂	—	2	6	no	—
195	C₂	—	226	2	no	—
196	C₂	—	226	6	no	—
197	C₂	#20299677	1	2	yes	direct product factor
198	C₂	C₂⁴×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#90673
commutator	C₂×C₁₁₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#90673
Fitting	C₂×#41×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂³×C₄
prime radical	C₂×#41×D₁₁₃	C₂
cosocle	Dic₁₁₃	C₂⁵
hypercenter	C₂×#41	#90673
hypocenter	C₁₁₃	C₂×#568
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20299677	C₂	direct (N × H)
2. C₂×#16813710	C₂	semidirect
3. C₂×#16813709	C₂	semidirect
4. C₂²×#14268657	C₂	semidirect
5. C₂×D₁₁₃⋊M₁₆	C₂	semidirect
7. #16813710	C₂²	semidirect
8. #16813709	C₂²	semidirect
11. C₂×#14268657	C₂²	semidirect
12. C₂×#14268657	C₂²	semidirect
13. D₁₁₃⋊M₁₆	C₂²	semidirect
14. C₂×#14268656	C₂²	semidirect
15. C₂×#14268655	C₂²	semidirect
16. C₂²×#11744753	C₂²	semidirect
28. #14268657	C₂³	semidirect
29. #14268656	C₂³	semidirect
30. #14268655	C₂³	semidirect
33. C₂×#11744753	C₂³	semidirect
95. C₂×C₁₁₃	#568	semidirect
100. C₁₁₃	C₂×#568	semidirect
197. C₂	#20299677	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299677	2
C₂×#16813710	2
C₂×#16813709	2
C₂²×#14268657	2
C₂×D₁₁₃⋊M₁₆	2
C₂×#41×D₁₁₃	2
C₂×#568	113

Subgroups — C2×#20299677 #28407567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299677 #28407567