Subgroups — C₄⋊C₄×D₅₂ #13117739

Order 1664.

776 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₂	C₂	1	2	yes	maximal
3	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718648	C₂	1	2	yes	maximal, complemented
6	#11718596	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718589	C₂	1	2	yes	maximal, complemented
8	#11718329	C₂	1	4	yes	maximal, complemented
9	C₂²×D₅₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₃	C₂²	1	4	yes	—
11	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#6103	C₂²	1	4	yes	—
17	C₂×#6103	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6101	C₂²	1	4	yes	—
23	C₄⋊C₄×D₁₃	—	2	8	no	—
24	C₄×D₅₂	—	2	8	no	—
25	C₄×D₅₂	C₄	1	8	yes	complemented
26	C₁₃×#95	C₂²	1	2	yes	—
27	#88071	C₂²	1	4	yes	—
28	C₂³×D₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂³	1	4	yes	—
35	C₂²×Dic₁₃	C₂³	1	2	yes	—
36	C₂×D₅₂	C₂×C₄	1	4	yes	—
37	C₂×D₅₂	D₄	1	4	yes	—
38	C₂×D₅₂	Q₈	1	4	yes	—
39	C₂×C₄×D₁₃	—	2	4	no	—
40	C₂×C₄×D₁₃	—	2	4	no	—
41	C₂×C₄×D₁₃	—	2	8	no	—
42	C₂×C₄×D₁₃	—	2	8	no	—
43	C₂×C₄×D₁₃	—	4	4	no	—
44	C₂×C₄×D₁₃	C₂×C₄	1	8	yes	complemented
45	C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₁₃×C₄⋊C₄	D₄	1	4	yes	complemented
47	C₄²×C₁₃	—	2	4	no	—
48	C₄²×C₁₃	C₂×C₄	1	4	yes	complemented
49	#6103	—	2	8	no	—
50	#6103	C₂×C₄	1	8	yes	complemented
51	C₄⋊Dic₁₃	—	2	4	no	—
52	C₄⋊Dic₁₃	—	2	4	no	—
53	C₄⋊Dic₁₃	—	4	2	no	—
54	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	complemented
55	#6101	—	2	8	no	—
56	D₄×C₄⋊C₄	—	13	1	no	maximal, Sylow
57	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂²×D₁₃	—	2	2	no	—
59	C₂²×D₁₃	—	2	2	no	—
60	C₂²×D₁₃	—	2	2	no	—
61	C₂²×D₁₃	—	2	2	no	—
62	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
63	C₂²×D₁₃	C₂×D₄	1	4	yes	—
64	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₁₃	#41	1	2	yes	—
80	C₂×Dic₁₃	—	2	2	no	—
81	C₂×Dic₁₃	—	2	2	no	—
82	C₂×Dic₁₃	—	2	4	no	—
83	C₂×Dic₁₃	—	2	4	no	—
84	C₂×Dic₁₃	—	2	4	no	—
85	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
86	D₅₂	C₄⋊C₄	1	16	yes	direct product factor
87	C₄×D₁₃	—	2	16	no	—
88	C₄×D₁₃	—	4	16	no	—
89	C₂×C₄×D₄	—	13	1	no	—
90	C₂×C₄×D₄	—	13	2	no	—
91	C₂²×C₄⋊C₄	—	13	2	no	—
92	#389	—	13	2	no	—
93	#385	—	13	4	no	—
94	#384	—	13	1	no	—
95	#380	—	13	2	no	—
96	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
97	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂×D₁₃	—	2	4	no	—
105	C₂×D₁₃	—	2	4	no	—
106	C₂×D₁₃	—	2	4	no	—
107	C₂×D₁₃	—	2	4	no	—
108	C₂×D₁₃	—	2	4	no	—
109	C₂×D₁₃	—	2	4	no	—
110	C₂×D₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	—	4	2	no	—
114	C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	complemented
115	C₄×C₁₃	C₄×D₄	1	4	yes	complemented
116	Dic₁₃	—	2	4	no	—
117	Dic₁₃	—	4	4	no	—
118	C₂²×D₄	—	13	1	no	—
119	C₂³×C₄	—	13	2	no	—
120	C₂³×C₄	—	13	4	no	—
121	C₄×D₄	—	13	8	no	—
122	C₄×D₄	—	26	8	no	—
123	C₂×C₄⋊C₄	—	13	1	no	—
124	C₂×C₄⋊C₄	—	13	4	no	—
125	C₂×C₄⋊C₄	—	13	2	no	—
126	C₂×C₄⋊C₄	—	13	2	no	—
127	C₂×C₄⋊C₄	—	13	2	no	—
128	C₂×C₄⋊C₄	—	26	8	no	—
129	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	—	13	2	no	—
132	C₂×C₂²⋊C₄	—	13	4	no	—
133	C₂×C₄²	C₂×D₁₃	1	2	yes	—
134	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
135	#95	—	13	4	no	—
136	#95	C₂×D₁₃	1	2	yes	—
137	C₂×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₁₃	#520	1	1	yes	characteristic
140	C₂×C₁₃	D₄²	1	1	yes	characteristic
141	C₂×C₁₃	#547	1	1	yes	characteristic
142	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
143	C₂×C₁₃	#554	1	1	yes	characteristic
144	D₁₃	—	2	8	no	—
145	C₂⁴	—	13	2	no	—
146	C₂×D₄	—	13	4	no	—
147	C₂×D₄	—	13	4	no	—
148	C₂×D₄	—	13	4	no	—
149	C₂²×C₄	—	13	2	no	—
150	C₂²×C₄	—	13	8	no	—
151	C₂²×C₄	—	13	4	no	—
152	C₂²×C₄	—	26	8	no	—
153	C₂²×C₄	—	26	4	no	—
154	C₂²×C₄	—	26	4	no	—
155	C₂²×C₄	—	26	8	no	—
156	C₂²×C₄	—	52	4	no	—
157	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
158	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	13	4	no	—
164	C₄⋊C₄	—	26	8	no	—
165	C₄⋊C₄	—	26	4	no	—
166	C₄⋊C₄	—	26	4	no	—
167	C₄⋊C₄	—	52	2	no	—
168	C₄⋊C₄	D₅₂	1	4	yes	direct product factor
169	C₂²⋊C₄	—	13	8	no	—
170	C₂²⋊C₄	—	26	8	no	—
171	C₄²	—	2	4	no	—
172	C₄²	C₄×D₁₃	1	4	yes	complemented
173	C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	13	4	no	—
175	C₂³	—	13	4	no	—
176	C₂³	—	13	4	no	—
177	C₂³	—	26	2	no	—
178	C₂³	—	26	2	no	—
179	C₂³	—	26	2	no	—
180	C₂³	—	26	2	no	—
181	C₂³	C₂³×D₁₃	1	1	yes	characteristic
182	D₄	—	13	16	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	13	4	no	—
191	C₂×C₄	—	26	2	no	—
192	C₂×C₄	—	26	2	no	—
193	C₂×C₄	—	26	16	no	—
194	C₂×C₄	—	26	4	no	—
195	C₂×C₄	—	26	4	no	—
196	C₂×C₄	—	26	4	no	—
197	C₂×C₄	—	52	16	no	—
198	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
201	C₂×C₄	C₂×D₅₂	1	4	yes	—
202	C₂×C₄	C₂×D₅₂	1	2	yes	—
203	C₂×C₄	#6127	1	2	yes	—
204	C₂×C₄	D₄×D₁₃	1	2	yes	—
205	C₂×C₄	Q₈×D₁₃	1	2	yes	—
206	C₂²	—	13	8	no	—
207	C₂²	—	26	4	no	—
208	C₂²	—	26	4	no	—
209	C₂²	—	26	4	no	—
210	C₂²	—	26	4	no	—
211	C₂²	—	26	4	no	—
212	C₂²	—	26	4	no	—
213	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
214	C₂²	C₂²×D₅₂	1	1	yes	characteristic
215	C₂²	C₂×#6127	1	1	yes	characteristic
216	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
217	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
218	C₂²	#88256	1	1	yes	characteristic
219	C₂²	#88257	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	2	no	—
223	C₄	—	26	4	no	—
224	C₄	—	52	4	no	—
225	C₄	C₄×D₅₂	1	4	yes	complemented
226	C₄	C₄⋊C₄×D₁₃	1	4	yes	complemented
227	C₂	—	26	8	no	—
228	C₂	C₂×C₄×D₅₂	1	1	yes	characteristic
229	C₂	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
230	C₂	#11719228	1	1	yes	characteristic
231	C₂	D₄×D₅₂	1	1	yes	characteristic
232	C₂	#11719256	1	1	yes	characteristic
233	C₂	Q₈×D₅₂	1	1	yes	characteristic
234	C₂	#11719284	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊C₄×D₁₃	C₂	semidirect
4. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
5. #11718648	C₂	semidirect
6. #11718596	C₂	semidirect
7. #11718589	C₂	semidirect
8. #11718329	C₂	semidirect
25. C₄×D₅₂	C₄	semidirect
44. C₂×C₄×D₁₃	C₂×C₄	semidirect
46. C₁₃×C₄⋊C₄	D₄	semidirect
48. C₄²×C₁₃	C₂×C₄	semidirect
50. #6103	C₂×C₄	semidirect
54. C₄⋊Dic₁₃	C₂×C₄	semidirect
86. D₅₂	C₄⋊C₄	direct (N × H)
96. C₄×C₄⋊C₄	D₁₃	semidirect
110. C₂×D₁₃	C₂×C₄⋊C₄	semidirect
114. C₄×C₁₃	C₂×C₄⋊C₄	semidirect
115. C₄×C₁₃	C₄×D₄	semidirect
168. C₄⋊C₄	D₅₂	direct (N × H)
172. C₄²	C₄×D₁₃	semidirect
173. C₁₃	D₄×C₄⋊C₄	semidirect
225. C₄	C₄×D₅₂	semidirect
226. C₄	C₄⋊C₄×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₁₃	2
C₂×C₄×D₅₂	2
C₄×C₁₃×C₄⋊C₄	2
#11718648	2
#11718596	2
#11718589	2
#11718329	2
D₄×C₄⋊C₄	13

Subgroups — C4⋊C4×D52 #13117739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×D₅₂ #13117739