Subgroups — C₂×#20299174 #28403893

Order 14464.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268647	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299174	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16813622	C₂	1	2	yes	maximal, complemented
5	C₂×#16813610	C₂	1	2	yes	maximal, complemented
6	C₂×#16813606	C₂	1	4	yes	maximal, complemented
7	C₂×#16813602	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14268647	—	2	8	no	—
10	C₂×#14268647	C₂²	1	8	yes	—
11	C₂²×Dic₂₂₆	—	2	8	no	—
12	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
13	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
14	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₂₆	C₂²	1	8	yes	complemented
16	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
18	C₂³×Dic₁₁₃	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
20	#16813622	C₂²	1	16	yes	complemented
21	#16813610	C₂²	1	16	yes	complemented
22	#16813606	C₂²	1	32	yes	complemented
23	#16813602	C₂²	1	32	yes	complemented
24	C₂×C₄⋊Dic₁₁₃	C₂²	1	8	yes	—
25	C₂×#14268624	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
27	#117×C₁₁₃	C₂²	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
31	C₂³×D₁₁₃	C₂³	1	2	yes	—
32	#14268647	—	2	32	no	—
33	C₂×Dic₂₂₆	—	2	8	no	—
34	C₂×Dic₂₂₆	—	2	8	no	—
35	C₂×Dic₂₂₆	—	2	32	no	—
36	C₂×Dic₂₂₆	—	2	4	no	—
37	C₂×Dic₂₂₆	C₂³	1	8	yes	—
38	C₂×Dic₂₂₆	C₂³	1	8	yes	—
39	C₂×Dic₂₂₆	C₂³	1	4	yes	—
40	C₂×D₄₅₂	—	2	4	no	—
41	C₂×D₄₅₂	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₁₁₃	—	2	8	no	—
43	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
44	C₂²×Dic₁₁₃	—	2	8	no	—
45	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
46	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
48	C₂×C₄×D₁₁₃	—	2	8	no	—
49	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
50	C₄⋊Dic₁₁₃	C₂³	1	32	yes	—
51	#14268624	C₂³	1	32	yes	—
52	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	D₄	1	8	yes	—
54	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
58	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
59	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
60	C₄²×C₁₁₃	C₂³	1	8	yes	—
61	C₂³×C₁₁₃	—	2	4	no	—
62	C₂³×C₁₁₃	C₂×D₄	1	4	yes	—
63	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
64	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×D₁₁₃	—	2	4	no	—
66	C₂²×D₁₁₃	—	2	8	no	—
67	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
68	C₂×C₄×C₁₁₃	—	2	8	no	—
69	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
70	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₁₁₃	C₂×D₄	1	16	yes	—
72	C₂×C₄×C₁₁₃	C₂⁴	1	8	yes	—
73	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
74	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
75	C₂²⋊D₁₁₃	—	2	16	no	—
76	C₂×Dic₁₁₃	—	2	4	no	—
77	C₂×Dic₁₁₃	—	2	16	no	—
78	C₂×Dic₁₁₃	—	2	8	no	—
79	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
80	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
81	D₄₅₂	—	2	8	no	—
82	C₄×D₁₁₃	—	2	16	no	—
83	Dic₂₂₆	—	2	16	no	—
84	Dic₂₂₆	—	2	16	no	—
85	Dic₂₂₆	—	2	8	no	—
86	C₂²×C₁₁₃	—	2	8	no	—
87	C₂²×C₁₁₃	—	2	4	no	—
88	C₂²×C₁₁₃	C₂²×D₄	1	4	yes	—
89	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
90	C₂²×C₁₁₃	2₋¹⁺⁴	1	4	yes	—
91	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
92	C₂×D₁₁₃	—	2	4	no	—
93	C₂×D₁₁₃	—	2	8	no	—
94	C₂×D₁₁₃	—	4	8	no	—
95	C₄×C₁₁₃	—	2	8	no	—
96	C₄×C₁₁₃	C₂²×D₄	1	4	yes	—
97	C₄×C₁₁₃	C₂²×D₄	1	4	yes	—
98	Dic₁₁₃	—	2	8	no	—
99	Dic₁₁₃	—	2	4	no	—
100	C₂×C₁₁₃	—	2	4	no	—
101	C₂×C₁₁₃	#536	1	4	yes	complemented
102	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
103	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
104	D₁₁₃	—	4	4	no	—
105	C₂×#536	—	113	1	no	maximal, Sylow
106	C₁₁₃	C₂×#536	1	1	yes	characteristic, complemented, Sylow
107	C₂²×#41	—	113	1	no	—
108	C₂³×Q₈	—	113	1	no	—
109	#536	—	113	16	no	—
110	C₂×C₄⋊Q₈	—	113	2	no	—
111	C₂×#124	—	113	2	no	—
112	C₂×#123	—	113	4	no	—
113	C₂×C₂²⋊Q₈	—	113	4	no	—
114	C₂×#117	D₁₁₃	1	1	yes	characteristic, complemented
115	C₂×#41	—	113	8	no	—
116	C₂×#41	—	226	8	no	—
117	C₂²×Q₈	—	113	2	no	—
118	C₂²×Q₈	—	113	2	no	—
119	C₂²×Q₈	—	113	1	no	—
120	C₂²×Q₈	—	113	8	no	—
121	C₂²×Q₈	—	226	8	no	—
122	C₂²×D₄	—	113	2	no	—
123	C₂²×D₄	—	113	1	no	—
124	C₂³×C₄	—	113	2	no	—
125	C₂³×C₄	—	113	2	no	—
126	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
127	C₄⋊Q₈	—	113	16	no	—
128	#124	—	113	16	no	—
129	#123	—	113	32	no	—
130	C₂²⋊Q₈	—	113	32	no	—
131	#117	C₂×D₁₁₃	1	8	yes	complemented
132	C₂×C₄⋊C₄	—	113	8	no	—
133	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
134	C₂×C₂²⋊C₄	—	113	8	no	—
135	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
136	C₂×C₄²	C₂×D₁₁₃	1	2	yes	—
137	C₂⁴	—	113	2	no	—
138	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
139	#41	—	226	32	no	—
140	C₂×Q₈	—	113	8	no	—
141	C₂×Q₈	—	113	8	no	—
142	C₂×Q₈	—	113	4	no	—
143	C₂×Q₈	—	226	8	no	—
144	C₂×Q₈	—	226	8	no	—
145	C₂×Q₈	—	226	32	no	—
146	C₂×Q₈	—	226	4	no	—
147	C₂×D₄	—	113	8	no	—
148	C₂×D₄	—	113	4	no	—
149	C₂×D₄	—	226	8	no	—
150	C₂×D₄	—	226	4	no	—
151	C₂²×C₄	—	113	8	no	—
152	C₂²×C₄	—	113	4	no	—
153	C₂²×C₄	—	113	2	no	—
154	C₂²×C₄	—	113	8	no	—
155	C₂²×C₄	—	226	8	no	—
156	C₂²×C₄	—	226	8	no	—
157	C₂²×C₄	D₄₅₂	1	8	yes	—
158	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
159	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
160	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
162	C₄⋊C₄	—	113	32	no	—
163	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
164	C₂²⋊C₄	—	113	32	no	—
165	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
166	C₄²	C₂²×D₁₁₃	1	8	yes	—
167	C₂³	—	2	4	no	—
168	C₂³	—	113	4	no	—
169	C₂³	—	226	4	no	—
170	C₂³	—	226	8	no	—
171	C₂³	C₂×D₄₅₂	1	4	yes	—
172	C₂³	C₂³×D₁₁₃	1	2	yes	—
173	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
174	Q₈	—	226	16	no	—
175	Q₈	—	226	16	no	—
176	Q₈	—	226	8	no	—
177	D₄	—	226	16	no	—
178	D₄	—	226	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	113	8	no	—
181	C₂×C₄	—	113	4	no	—
182	C₂×C₄	—	226	4	no	—
183	C₂×C₄	—	226	16	no	—
184	C₂×C₄	—	226	8	no	—
185	C₂×C₄	—	226	16	no	—
186	C₂×C₄	C₂×D₄₅₂	1	4	yes	—
187	C₂×C₄	C₂×D₄₅₂	1	4	yes	—
188	C₂×C₄	C₂×D₄₅₂	1	16	yes	—
189	C₂×C₄	C₂³×D₁₁₃	1	8	yes	—
190	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
191	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	452	8	no	—
195	C₂²	—	226	4	no	—
196	C₂²	—	226	8	no	—
197	C₂²	C₂²×D₄₅₂	1	4	yes	—
198	C₂²	C₂²×D₄₅₂	1	2	yes	—
199	C₂²	#16813669	1	4	yes	—
200	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
201	C₄	—	2	8	no	—
202	C₄	—	226	4	no	—
203	C₄	—	226	8	no	—
204	C₄	C₂²×D₄₅₂	1	4	yes	—
205	C₄	C₂²×D₄₅₂	1	4	yes	—
206	C₂	—	2	4	no	—
207	C₂	—	452	4	no	—
208	C₂	#20299174	1	4	yes	direct product factor
209	C₂	C₂³×D₄₅₂	1	1	yes	characteristic
210	C₂	C₂×#16813669	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×#117×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×#14268647	C₂
hypercenter	C₂×#117	D₁₁₃
hypocenter	C₁₁₃	C₂×#536
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₂₂₆	C₂	semidirect
3. #20299174	C₂	direct (N × H)
4. C₂×#16813622	C₂	semidirect
5. C₂×#16813610	C₂	semidirect
6. C₂×#16813606	C₂	semidirect
7. C₂×#16813602	C₂	semidirect
8. C₂×#117×C₁₁₃	C₂	semidirect
15. C₂²×Dic₂₂₆	C₂²	semidirect
20. #16813622	C₂²	semidirect
21. #16813610	C₂²	semidirect
22. #16813606	C₂²	semidirect
23. #16813602	C₂²	semidirect
27. #117×C₁₁₃	C₂²	semidirect
101. C₂×C₁₁₃	#536	semidirect
106. C₁₁₃	C₂×#536	semidirect
114. C₂×#117	D₁₁₃	semidirect
131. #117	C₂×D₁₁₃	semidirect
208. C₂	#20299174	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14268647	2
C₂³×Dic₂₂₆	2
#20299174	2
C₂×#16813622	2
C₂×#16813610	2
C₂×#16813606	2
C₂×#16813602	2
C₂×#117×C₁₁₃	2
C₂×#536	113

Subgroups — C2×#20299174 #28403893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299174 #28403893