Subgroups — #28399530

Order 14464.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14268647	C₂	1	1	yes	maximal, characteristic
2	#20298722	C₂	1	2	yes	maximal, complemented
3	#20298663	C₂	1	2	yes	maximal, complemented
4	#20298661	C₂	1	4	yes	maximal, complemented
5	#20298649	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₈×D₁₁₃	C₂	1	2	yes	maximal, complemented
8	#431×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14268647	C₄	1	1	yes	characteristic
10	C₂²×#11744751	C₂²	1	2	yes	—
11	C₂×C₈×D₁₁₃	C₂²	1	4	yes	—
12	C₄×Dic₂₂₆	C₄	1	1	yes	characteristic
13	C₄×Dic₂₂₆	C₂²	1	1	yes	characteristic
14	C₄×D₄₅₂	C₄	1	1	yes	characteristic
15	C₄×D₄₅₂	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₁₃	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
18	#16813581	C₄	1	1	yes	characteristic
19	#16813579	C₄	1	2	yes	—
20	C₄²×D₁₁₃	C₂²	1	2	yes	—
21	#16813551	C₂²	1	4	yes	—
22	#16813534	C₂²	1	4	yes	—
23	C₄×#11744751	C₂²	1	1	yes	characteristic
24	C₄×#11744751	C₂²	1	1	yes	characteristic
25	C₄×#11744751	C₂²	1	2	yes	—
26	C₈×Dic₁₁₃	C₂²	1	4	yes	—
27	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
30	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
31	#14268647	C₈	1	8	yes	complemented
32	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
38	C₂×#11744751	—	2	4	no	—
39	C₂×#11744751	C₂³	1	2	yes	—
40	C₂×#11744751	C₂³	1	2	yes	—
41	C₈×D₁₁₃	—	2	8	no	—
42	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
44	#14268626	C₂×C₄	1	2	yes	—
45	#14268626	C₂×C₄	1	2	yes	—
46	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
47	#14268624	C₂×C₄	1	2	yes	—
48	#14268624	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₁₃	C₂³	1	4	yes	—
56	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
59	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
60	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₁₃	—	2	2	no	—
63	C₂×C₄×C₁₁₃	—	2	2	no	—
64	C₂×C₄×C₁₁₃	C₂×C₈	1	4	yes	complemented
65	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₁₁₃	C₂×C₈	1	8	yes	complemented
72	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
75	D₄₅₂	C₂×C₈	1	4	yes	complemented
76	C₄×D₁₁₃	—	2	4	no	—
77	C₄×D₁₁₃	—	2	4	no	—
78	C₄×D₁₁₃	C₂×C₈	1	8	yes	complemented
79	Dic₂₂₆	C₂×C₈	1	4	yes	complemented
80	C₈×C₁₁₃	—	2	4	no	—
81	#11744751	—	2	2	no	—
82	#11744751	#41	1	4	yes	complemented
83	C₂²×C₁₁₃	—	2	1	no	—
84	C₂²×C₁₁₃	—	2	1	no	—
85	C₂²×C₁₁₃	C₂²×C₈	1	2	yes	—
86	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₁₃	—	2	2	no	—
88	C₂×D₁₁₃	—	2	2	no	—
89	C₂×D₁₁₃	C₂²×C₈	1	4	yes	—
90	C₄×C₁₁₃	—	2	1	no	—
91	C₄×C₁₁₃	—	2	1	no	—
92	C₄×C₁₁₃	C₂²×C₈	1	2	yes	—
93	C₄×C₁₁₃	C₂²×C₈	1	2	yes	—
94	C₄×C₁₁₃	#131	1	2	yes	—
95	C₄×C₁₁₃	C₂×#41	1	2	yes	—
96	Dic₁₁₃	—	2	2	no	—
97	Dic₁₁₃	—	2	2	no	—
98	Dic₁₁₃	C₂²×C₈	1	4	yes	—
99	C₂×C₁₁₃	—	2	2	no	—
100	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
101	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
102	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
103	D₁₁₃	—	2	4	no	—
104	C₈×#41	—	113	1	no	maximal, Sylow
105	C₁₁₃	C₈×#41	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	—	113	1	no	—
107	C₈×Q₈	—	113	2	no	—
108	C₈×D₄	—	113	2	no	—
109	C₈×D₄	—	113	4	no	—
110	#431	—	113	2	no	—
111	#431	D₁₁₃	1	1	yes	characteristic, complemented
112	C₂×C₄×C₈	—	113	1	no	—
113	C₂×C₄×C₈	—	113	2	no	—
114	C₂×#41	—	113	1	no	—
115	C₂²×C₈	—	113	2	no	—
116	C₂²×C₈	—	113	4	no	—
117	C₄×Q₈	—	113	1	no	—
118	C₄×Q₈	—	113	1	no	—
119	C₄×D₄	—	113	2	no	—
120	C₄×D₄	—	113	2	no	—
121	C₄×D₄	—	113	1	no	—
122	C₄×D₄	—	113	1	no	—
123	#117	—	113	1	no	—
124	#117	—	113	2	no	—
125	C₂×C₄²	—	113	2	no	—
126	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
127	C₄⋊C₈	—	113	4	no	—
128	C₄⋊C₈	C₂×D₁₁₃	1	2	yes	—
129	C₂²⋊C₈	—	113	4	no	—
130	C₂²⋊C₈	C₂×D₁₁₃	1	2	yes	—
131	C₄×C₈	—	113	4	no	—
132	C₄×C₈	—	113	2	no	—
133	C₄×C₈	—	113	1	no	—
134	C₄×C₈	—	113	1	no	—
135	C₄×C₈	C₂×D₁₁₃	1	2	yes	—
136	#41	—	113	8	no	—
137	C₂×Q₈	—	113	1	no	—
138	C₂×D₄	—	113	2	no	—
139	C₂×D₄	—	113	1	no	—
140	C₂²×C₄	—	113	2	no	—
141	C₂²×C₄	—	113	2	no	—
142	C₂²×C₄	—	113	2	no	—
143	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
146	C₂×C₈	—	113	2	no	—
147	C₂×C₈	—	113	2	no	—
148	C₂×C₈	—	226	8	no	—
149	C₂×C₈	—	226	4	no	—
150	C₂×C₈	C₂²×D₁₁₃	1	4	yes	—
151	C₄⋊C₄	—	113	2	no	—
152	C₄⋊C₄	—	113	2	no	—
153	C₄⋊C₄	—	113	1	no	—
154	C₄⋊C₄	—	113	1	no	—
155	C₂²⋊C₄	—	113	2	no	—
156	C₂²⋊C₄	—	113	2	no	—
157	C₂²⋊C₄	—	113	2	no	—
158	C₄²	—	113	2	no	—
159	C₄²	—	113	2	no	—
160	C₄²	—	113	2	no	—
161	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
162	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
163	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
164	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
165	C₂³	—	113	2	no	—
166	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
167	Q₈	—	113	4	no	—
168	D₄	—	113	8	no	—
169	D₄	—	113	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	113	2	no	—
173	C₂×C₄	—	113	8	no	—
174	C₂×C₄	—	113	2	no	—
175	C₂×C₄	—	113	2	no	—
176	C₂×C₄	—	226	4	no	—
177	C₂×C₄	—	226	4	no	—
178	C₂×C₄	C₈×D₁₁₃	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	113	4	no	—
187	C₈	—	226	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	113	4	no	—
191	C₂²	—	226	2	no	—
192	C₂²	—	226	2	no	—
193	C₂²	C₂×C₈×D₁₁₃	1	2	yes	—
194	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	113	4	no	—
198	C₄	—	226	2	no	—
199	C₄	—	226	2	no	—
200	C₄	C₂×C₈×D₁₁₃	1	2	yes	—
201	C₄	C₂×C₈×D₁₁₃	1	2	yes	—
202	C₄	#16813630	1	2	yes	—
203	C₄	#41×D₁₁₃	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	226	4	no	—
206	C₂	#20299066	1	1	yes	characteristic
207	C₂	C₂²×C₈×D₁₁₃	1	1	yes	characteristic
208	C₂	C₂×#16813630	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₁₃
commutator	C₂×C₁₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	#431×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂³×C₄
prime radical	C₂×#14268647	C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#431	D₁₁₃
hypocenter	C₁₁₃	C₈×#41
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #20298722	C₂	semidirect
3. #20298663	C₂	semidirect
4. #20298661	C₂	semidirect
5. #20298649	C₂	semidirect
6. C₂×C₄×#11744751	C₂	semidirect
7. C₄×C₈×D₁₁₃	C₂	semidirect
8. #431×C₁₁₃	C₂	semidirect
31. #14268647	C₈	semidirect
64. C₂×C₄×C₁₁₃	C₂×C₈	semidirect
71. C₂²⋊D₁₁₃	C₂×C₈	semidirect
75. D₄₅₂	C₂×C₈	semidirect
78. C₄×D₁₁₃	C₂×C₈	semidirect
79. Dic₂₂₆	C₂×C₈	semidirect
82. #11744751	#41	semidirect
105. C₁₁₃	C₈×#41	semidirect
111. #431	D₁₁₃	semidirect
178. C₂×C₄	C₈×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14268647	2
#20298722	2
#20298663	2
#20298661	2
#20298649	2
C₂×C₄×#11744751	2
C₄×C₈×D₁₁₃	2
#431×C₁₁₃	2
C₈×#41	113