Subgroups — C₄×#16813578 #28392912

Order 14464.

752 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813578	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#14268626	C₂	1	2	yes	maximal, complemented
4	#20298449	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₄×Dic₁₁₃	C₂	1	2	yes	maximal
6	C₄×C₂²⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
7	#20298443	C₂	1	2	yes	maximal, complemented
8	C₄²×Dic₁₁₃	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
10	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
12	#16813578	C₄	1	16	yes	direct product factor
13	C₂×#14268626	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₁₃	C₄	1	8	yes	complemented
16	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₁₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁₃	C₂²	1	4	yes	—
23	#16813529	C₂²	1	2	yes	—
24	#16813529	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
28	#95×C₁₁₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₁₃	—	2	4	no	—
30	C₂²×Dic₁₁₃	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₁₃	C₂×C₄	1	4	yes	—
32	C₂²×Dic₁₁₃	C₂×C₄	1	8	yes	—
33	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
34	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
38	#14268626	C₂×C₄	1	16	yes	complemented
39	C₂²⋊Dic₁₁₃	C₂×C₄	1	8	yes	—
40	C₄×Dic₁₁₃	—	2	4	no	—
41	C₄×Dic₁₁₃	—	2	8	no	—
42	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
43	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₁₃	C₂×C₄	1	16	yes	complemented
47	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
50	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
54	C₄²×C₁₁₃	—	2	4	no	—
55	C₂³×C₁₁₃	—	2	2	no	—
56	C₂³×C₁₁₃	—	2	2	no	—
57	C₂³×C₁₁₃	C₂²×C₄	1	4	yes	—
58	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
59	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₁₁₃	—	2	4	no	—
61	C₂×C₄×C₁₁₃	—	2	2	no	—
62	C₂×C₄×C₁₁₃	—	2	4	no	—
63	C₂×C₄×C₁₁₃	—	2	8	no	—
64	C₂×C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
65	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₁₃	#41	1	4	yes	—
68	C₂×Dic₁₁₃	—	2	4	no	—
69	C₂×Dic₁₁₃	—	2	4	no	—
70	C₂×Dic₁₁₃	—	2	2	no	—
71	C₂×Dic₁₁₃	—	2	8	no	—
72	C₂×Dic₁₁₃	C₄²	1	16	yes	complemented
73	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₁₃	C₂²×C₄	1	8	yes	—
75	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
78	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
79	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₁₃	#41	1	4	yes	—
81	C₂²×C₁₁₃	—	2	2	no	—
82	C₂²×C₁₁₃	—	2	2	no	—
83	C₂²×C₁₁₃	—	2	8	no	—
84	C₂²×C₁₁₃	C₂×C₄²	1	4	yes	—
85	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₂²×C₁₁₃	C₂³×C₄	1	2	yes	—
87	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
88	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
89	C₄×C₁₁₃	—	2	2	no	—
90	C₄×C₁₁₃	—	2	8	no	—
91	C₄×C₁₁₃	#117	1	4	yes	complemented
92	Dic₁₁₃	—	2	2	no	—
93	Dic₁₁₃	—	2	8	no	—
94	Dic₁₁₃	C₂×C₄²	1	4	yes	—
95	Dic₁₁₃	C₂×C₄²	1	4	yes	—
96	Dic₁₁₃	#117	1	4	yes	complemented
97	C₂×C₁₁₃	—	2	4	no	—
98	C₂×C₁₁₃	C₂²×C₄²	1	1	yes	characteristic
99	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
100	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
101	C₂×C₁₁₃	C₄×#41	1	4	yes	—
102	C₄×#117	—	113	1	no	maximal, Sylow
103	C₁₁₃	C₄×#117	1	1	yes	characteristic, complemented, Sylow
104	C₂×#117	—	113	2	no	—
105	C₂²×C₄²	—	113	1	no	—
106	C₄×C₄⋊C₄	—	113	2	no	—
107	C₄×C₄⋊C₄	—	113	2	no	—
108	C₄×C₂²⋊C₄	—	113	1	no	—
109	C₄×C₂²⋊C₄	—	113	2	no	—
110	C₄×C₂²⋊C₄	D₁₁₃	1	1	yes	characteristic
111	#376	—	113	2	no	—
112	C₄³	—	113	2	no	—
113	C₂³×C₄	—	113	1	no	—
114	C₂³×C₄	—	113	1	no	—
115	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
116	#117	—	113	16	no	—
117	C₂×C₄⋊C₄	—	113	4	no	—
118	C₂×C₂²⋊C₄	—	113	2	no	—
119	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
120	C₂×C₄²	—	113	8	no	—
121	C₂×C₄²	—	113	4	no	—
122	C₂×C₄²	—	113	1	no	—
123	C₂×C₄²	—	113	1	no	—
124	C₂×C₄²	—	113	1	no	—
125	C₂×C₄²	—	113	1	no	—
126	C₂×C₄²	—	113	4	no	—
127	C₂×C₄²	—	113	2	no	—
128	C₂×C₄²	C₂×D₁₁₃	1	2	yes	—
129	#95	—	113	4	no	—
130	#95	—	113	2	no	—
131	#95	C₂×D₁₁₃	1	2	yes	—
132	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	113	4	no	—
135	C₂²×C₄	—	113	8	no	—
136	C₂²×C₄	—	113	1	no	—
137	C₂²×C₄	—	113	1	no	—
138	C₂²×C₄	—	113	1	no	—
139	C₂²×C₄	—	113	4	no	—
140	C₂²×C₄	—	113	1	no	—
141	C₂²×C₄	—	113	4	no	—
142	C₂²×C₄	—	226	4	no	—
143	C₂²×C₄	C₄×D₁₁₃	1	4	yes	—
144	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
146	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
147	C₄⋊C₄	—	113	16	no	—
148	C₂²⋊C₄	—	113	8	no	—
149	C₂²⋊C₄	C₄×D₁₁₃	1	8	yes	—
150	C₄²	—	2	4	no	—
151	C₄²	—	113	4	no	—
152	C₄²	—	113	4	no	—
153	C₄²	—	113	4	no	—
154	C₄²	—	113	4	no	—
155	C₄²	—	113	16	no	—
156	C₄²	—	226	8	no	—
157	C₄²	—	226	4	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	C₂×C₄×D₁₁₃	1	4	yes	—
161	C₂³	C₂×C₄×D₁₁₃	1	2	yes	—
162	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	113	16	no	—
168	C₂×C₄	—	113	2	no	—
169	C₂×C₄	—	113	2	no	—
170	C₂×C₄	—	113	4	no	—
171	C₂×C₄	—	113	2	no	—
172	C₂×C₄	—	113	4	no	—
173	C₂×C₄	—	113	4	no	—
174	C₂×C₄	—	113	2	no	—
175	C₂×C₄	—	113	8	no	—
176	C₂×C₄	—	226	8	no	—
177	C₂×C₄	—	226	4	no	—
178	C₂×C₄	—	226	2	no	—
179	C₂×C₄	—	226	4	no	—
180	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
181	C₂×C₄	C₂×C₄×D₁₁₃	1	8	yes	—
182	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
183	C₂×C₄	#14268646	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	8	no	—
187	C₂²	C₄²×D₁₁₃	1	4	yes	—
188	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
189	C₂²	C₂²×C₄×D₁₁₃	1	2	yes	—
190	C₂²	#41×D₁₁₃	1	2	yes	—
191	C₂²	C₂×#14268646	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	2	8	no	—
194	C₄	—	113	4	no	—
195	C₄	—	113	4	no	—
196	C₄	—	113	4	no	—
197	C₄	—	226	2	no	—
198	C₄	—	226	8	no	—
199	C₄	#16813578	1	4	yes	direct product factor
200	C₂	—	2	4	no	—
201	C₂	C₂×C₄²×D₁₁₃	1	1	yes	characteristic
202	C₂	#117×D₁₁₃	1	1	yes	characteristic
203	C₂	C₂×#16813578	1	1	yes	characteristic
204	C₂	C₄×#14268646	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₁₃
commutator	C₂×C₁₁₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₁₃
Fitting	C₄×C₂²⋊C₄×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂⁴×C₁₁₃	C₂³
hypercenter	C₄×C₂²⋊C₄	D₁₁₃
hypocenter	C₁₁₃	C₄×#117
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₄×#14268626	C₂	semidirect
4. #20298449	C₂	semidirect
7. #20298443	C₂	semidirect
8. C₄²×Dic₁₁₃	C₂	semidirect
12. #16813578	C₄	direct (N × H)
15. C₂×C₄×Dic₁₁₃	C₄	semidirect
38. #14268626	C₂×C₄	semidirect
46. C₄×Dic₁₁₃	C₂×C₄	semidirect
72. C₂×Dic₁₁₃	C₄²	semidirect
91. C₄×C₁₁₃	#117	semidirect
96. Dic₁₁₃	#117	semidirect
103. C₁₁₃	C₄×#117	semidirect
199. C₄	#16813578	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16813578	2
C₂²×C₄×Dic₁₁₃	2
C₄×#14268626	2
#20298449	2
C₂²⋊C₄×Dic₁₁₃	2
C₄×C₂²⋊Dic₁₁₃	2
#20298443	2
C₄²×Dic₁₁₃	2
C₄×C₂²⋊C₄×C₁₁₃	2
C₄×#117	113

Subgroups — C4×#16813578 #28392912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16813578 #28392912