Subgroups — C₂²×C₈×D₅₂ #16726752

Order 3328.

3694 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₁₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₅₂	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88060	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₅₂	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₁₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6103	C₄	1	2	yes	—
13	C₂²×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
14	C₂³×#1055	C₂²	1	2	yes	—
15	C₂²×C₈×D₁₃	—	2	16	no	—
16	C₂²×C₈×D₁₃	C₂²	1	24	yes	complemented
17	C₂×C₄×D₅₂	C₄	1	12	yes	—
18	C₂×C₄×D₅₂	C₂²	1	12	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
20	C₂×#88060	C₂²	1	24	yes	complemented
21	C₂×C₄⋊#1055	C₂²	1	12	yes	complemented
22	C₈×D₅₂	C₂²	1	64	yes	direct product factor
23	C₂⁴×D₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
28	C₂²×D₅₂	C₈	1	16	yes	direct product factor
29	C₂²×D₅₂	C₂×C₄	1	12	yes	—
30	C₂²×C₄×D₁₃	—	2	8	no	—
31	C₂²×C₄×D₁₃	—	2	8	no	—
32	C₂²×C₄×D₁₃	C₂×C₄	1	12	yes	—
33	C₂²×C₄×D₁₃	C₂³	1	12	yes	—
34	C₂²×C₈×C₁₃	—	2	4	no	—
35	C₂²×C₈×C₁₃	D₄	1	8	yes	complemented
36	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
40	C₂×#6103	C₂×C₄	1	24	yes	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	12	yes	—
42	C₂²×#1055	—	2	8	no	—
43	C₂²×#1055	C₂³	1	12	yes	—
44	C₂×C₈×D₁₃	—	2	96	no	—
45	C₂×C₈×D₁₃	C₂³	1	32	yes	complemented
46	C₄×D₅₂	C₂×C₄	1	48	yes	—
47	C₄×D₅₂	C₂³	1	16	yes	—
48	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
49	#88060	C₂³	1	32	yes	complemented
50	C₄⋊#1055	C₂³	1	16	yes	complemented
51	C₂²×C₈×D₄	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂³×D₁₃	—	2	8	no	—
54	C₂³×D₁₃	—	2	8	no	—
55	C₂³×D₁₃	C₂×C₈	1	16	yes	complemented
56	C₂³×D₁₃	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	complemented
59	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₃	#41	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
65	C₂²×Dic₁₃	—	2	8	no	—
66	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
67	C₂×D₅₂	C₂×C₈	1	96	yes	direct product factor
68	C₂×D₅₂	C₂²×C₄	1	16	yes	—
69	C₂×C₄×D₁₃	—	2	48	no	—
70	C₂×C₄×D₁₃	—	2	24	no	—
71	C₂×C₄×D₁₃	—	2	24	no	—
72	C₂×C₄×D₁₃	C₂²×C₄	1	24	yes	—
73	C₂×C₄×D₁₃	C₂⁴	1	8	yes	—
74	C₂×C₈×C₁₃	—	2	12	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	24	yes	complemented
76	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
79	C₄²×C₁₃	C₂⁴	1	4	yes	—
80	#6103	C₂²×C₄	1	32	yes	—
81	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
82	C₂×#1055	—	2	24	no	—
83	C₂×#1055	C₂⁴	1	8	yes	—
84	C₈×D₁₃	—	2	64	no	—
85	C₂⁴×C₈	—	13	2	no	—
86	C₂²×C₄×D₄	—	13	1	no	—
87	C₂×C₈×D₄	—	13	24	no	—
88	C₂²×C₄⋊C₈	—	13	1	no	—
89	C₂²×C₂²⋊C₈	—	13	2	no	—
90	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
91	C₂³×C₁₃	C₄×D₄	1	4	yes	—
92	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
93	C₂³×C₁₃	#131	1	4	yes	—
94	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
95	C₂²×D₁₃	—	2	24	no	—
96	C₂²×D₁₃	—	2	24	no	—
97	C₂²×D₁₃	—	2	64	no	—
98	C₂²×D₁₃	C₂²×C₈	1	48	yes	complemented
99	C₂²×D₁₃	C₂³×C₄	1	8	yes	—
100	C₂×C₄×C₁₃	—	2	12	no	—
101	C₂×C₄×C₁₃	C₄×D₄	1	12	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	24	yes	complemented
103	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
107	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
108	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	24	no	—
110	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
111	D₅₂	C₂²×C₈	1	64	yes	direct product factor
112	C₄×D₁₃	—	2	48	no	—
113	C₄×D₁₃	—	2	8	no	—
114	C₄×D₁₃	—	2	8	no	—
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	C₂²×D₄	1	8	yes	complemented
117	#1055	—	2	8	no	—
118	C₂³×D₄	—	13	1	no	—
119	C₂⁴×C₄	—	13	2	no	—
120	C₂³×C₈	—	13	2	no	—
121	C₂³×C₈	—	13	24	no	—
122	C₂³×C₈	—	26	16	no	—
123	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
125	C₂×C₄×D₄	—	13	12	no	—
126	C₂×C₄×D₄	—	13	12	no	—
127	C₂²×C₄⋊C₄	—	13	1	no	—
128	C₂²×C₂²⋊C₄	—	13	2	no	—
129	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
130	C₈×D₄	—	13	64	no	—
131	C₂×C₄⋊C₈	—	13	12	no	—
132	C₂×C₂²⋊C₈	—	13	24	no	—
133	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	complemented
134	C₂²×C₁₃	C₈×D₄	1	16	yes	complemented
135	C₂²×C₁₃	C₂×C₄×D₄	1	6	yes	—
136	C₂²×C₁₃	C₂³×C₈	1	6	yes	—
137	C₂²×C₁₃	C₂×#131	1	6	yes	—
138	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
139	C₂×D₁₃	—	2	8	no	—
140	C₂×D₁₃	—	2	96	no	—
141	C₂×D₁₃	—	2	8	no	—
142	C₂×D₁₃	C₂³×C₈	1	16	yes	complemented
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	C₂×C₄×D₄	1	6	yes	—
145	C₄×C₁₃	C₂³×C₈	1	8	yes	complemented
146	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
147	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
148	Dic₁₃	—	2	8	no	—
149	C₂⁵	—	13	2	no	—
150	C₂²×D₄	—	13	16	no	—
151	C₂²×D₄	—	13	12	no	—
152	C₂³×C₄	—	13	2	no	—
153	C₂³×C₄	—	13	12	no	—
154	C₂³×C₄	—	13	12	no	—
155	C₂³×C₄	—	26	8	no	—
156	C₂³×C₄	—	26	8	no	—
157	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
158	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
160	C₂²×C₈	—	2	4	no	—
161	C₂²×C₈	—	13	12	no	—
162	C₂²×C₈	—	13	32	no	—
163	C₂²×C₈	—	26	8	no	—
164	C₂²×C₈	—	26	96	no	—
165	C₂²×C₈	D₅₂	1	8	yes	direct product factor
166	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
167	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
168	C₄×D₄	—	13	48	no	—
169	C₄×D₄	—	13	16	no	—
170	C₂×C₄⋊C₄	—	13	12	no	—
171	C₂×C₂²⋊C₄	—	13	24	no	—
172	C₂×C₄²	C₄×D₁₃	1	6	yes	—
173	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
174	C₄⋊C₈	—	13	16	no	—
175	C₂²⋊C₈	—	13	32	no	—
176	C₄×C₈	C₂²×D₁₃	1	16	yes	complemented
177	C₂×C₁₃	C₂×C₈×D₄	1	12	yes	complemented
178	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
179	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
180	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
181	D₁₃	—	2	16	no	—
182	C₂⁴	—	13	16	no	—
183	C₂⁴	—	13	12	no	—
184	C₂⁴	—	26	8	no	—
185	C₂⁴	—	26	8	no	—
186	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂×D₄	—	13	96	no	—
188	C₂×D₄	—	13	16	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	13	12	no	—
191	C₂²×C₄	—	13	24	no	—
192	C₂²×C₄	—	13	8	no	—
193	C₂²×C₄	—	26	8	no	—
194	C₂²×C₄	—	26	48	no	—
195	C₂²×C₄	—	26	24	no	—
196	C₂²×C₄	—	26	24	no	—
197	C₂²×C₄	C₈×D₁₃	1	8	yes	complemented
198	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
199	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
201	C₂²×C₄	C₂×D₅₂	1	4	yes	—
202	C₂²×C₄	#6127	1	4	yes	—
203	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
204	C₂×C₈	—	2	12	no	—
205	C₂×C₈	—	13	8	no	—
206	C₂×C₈	—	26	24	no	—
207	C₂×C₈	—	26	64	no	—
208	C₂×C₈	C₂×D₅₂	1	24	yes	direct product factor
209	C₂×C₈	C₂³×D₁₃	1	4	yes	—
210	C₂×C₈	C₂³×D₁₃	1	4	yes	—
211	C₄⋊C₄	—	13	16	no	—
212	C₂²⋊C₄	—	13	32	no	—
213	C₄²	C₂×C₄×D₁₃	1	12	yes	—
214	C₄²	C₂³×D₁₃	1	4	yes	—
215	C₁₃	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	13	48	no	—
217	C₂³	—	13	8	no	—
218	C₂³	—	26	24	no	—
219	C₂³	—	26	24	no	—
220	C₂³	—	26	64	no	—
221	C₂³	C₄×D₅₂	1	4	yes	—
222	C₂³	C₂×C₈×D₁₃	1	4	yes	—
223	C₂³	#88155	1	4	yes	—
224	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
225	D₄	—	13	64	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	13	8	no	—
228	C₂×C₄	—	26	24	no	—
229	C₂×C₄	—	26	48	no	—
230	C₂×C₄	—	26	8	no	—
231	C₂×C₄	—	26	8	no	—
232	C₂×C₄	C₄×D₅₂	1	12	yes	—
233	C₂×C₄	C₂×C₈×D₁₃	1	24	yes	complemented
234	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
235	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
236	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
237	C₂×C₄	C₂²×D₅₂	1	6	yes	—
238	C₂×C₄	C₂×#6127	1	6	yes	—
239	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	26	8	no	—
242	C₈	C₂²×D₅₂	1	8	yes	direct product factor
243	C₂²	—	13	16	no	—
244	C₂²	—	26	8	no	—
245	C₂²	—	26	8	no	—
246	C₂²	—	26	96	no	—
247	C₂²	C₈×D₅₂	1	16	yes	direct product factor
248	C₂²	C₂×C₄×D₅₂	1	6	yes	—
249	C₂²	C₂²×C₈×D₁₃	1	6	yes	—
250	C₂²	C₂×#88155	1	6	yes	—
251	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	26	8	no	—
254	C₄	C₂×C₄×D₅₂	1	6	yes	—
255	C₄	C₂²×C₈×D₁₃	1	8	yes	complemented
256	C₄	C₂³×D₅₂	1	1	yes	characteristic
257	C₄	C₂²×#6127	1	1	yes	characteristic
258	C₂	—	26	16	no	—
259	C₂	C₂×C₈×D₅₂	1	12	yes	direct product factor
260	C₂	C₂²×C₄×D₅₂	1	1	yes	characteristic
261	C₂	C₂³×C₈×D₁₃	1	1	yes	characteristic
262	C₂	C₂²×#88155	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂³×D₅₂	C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×C₈×D₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×D₁₃	C₂	semidirect
3. C₂×C₈×D₅₂	C₂	direct (N × H)
4. C₂²×C₄⋊#1055	C₂	semidirect
5. C₂²×#88060	C₂	semidirect
6. C₂²×C₄×C₈×C₁₃	C₂	semidirect
16. C₂²×C₈×D₁₃	C₂²	semidirect
19. C₂×C₄×C₈×C₁₃	C₂²	semidirect
20. C₂×#88060	C₂²	semidirect
21. C₂×C₄⋊#1055	C₂²	semidirect
22. C₈×D₅₂	C₂²	direct (N × H)
28. C₂²×D₅₂	C₈	direct (N × H)
35. C₂²×C₈×C₁₃	D₄	semidirect
45. C₂×C₈×D₁₃	C₂³	semidirect
48. C₄×C₈×C₁₃	C₂³	semidirect
49. #88060	C₂³	semidirect
50. C₄⋊#1055	C₂³	semidirect
55. C₂³×D₁₃	C₂×C₈	semidirect
58. C₂²×C₄×C₁₃	C₂×C₈	semidirect
67. C₂×D₅₂	C₂×C₈	direct (N × H)
75. C₂×C₈×C₁₃	C₂×D₄	semidirect
90. C₂²×C₄×C₈	D₁₃	semidirect
98. C₂²×D₁₃	C₂²×C₈	semidirect
102. C₂×C₄×C₁₃	C₂²×C₈	semidirect
111. D₅₂	C₂²×C₈	direct (N × H)
116. C₈×C₁₃	C₂²×D₄	semidirect
133. C₂×C₄×C₈	C₂×D₁₃	semidirect
134. C₂²×C₁₃	C₈×D₄	semidirect
142. C₂×D₁₃	C₂³×C₈	semidirect
145. C₄×C₁₃	C₂³×C₈	semidirect
165. C₂²×C₈	D₅₂	direct (N × H)
176. C₄×C₈	C₂²×D₁₃	semidirect
177. C₂×C₁₃	C₂×C₈×D₄	semidirect
197. C₂²×C₄	C₈×D₁₃	semidirect
208. C₂×C₈	C₂×D₅₂	direct (N × H)
215. C₁₃	C₂²×C₈×D₄	semidirect
233. C₂×C₄	C₂×C₈×D₁₃	semidirect
242. C₈	C₂²×D₅₂	direct (N × H)
247. C₂²	C₈×D₅₂	direct (N × H)
255. C₄	C₂²×C₈×D₁₃	semidirect
259. C₂	C₂×C₈×D₅₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₁₃	2
C₂²×C₄×D₅₂	2
C₂×C₈×D₅₂	2
C₂²×C₄⋊#1055	2
C₂²×#88060	2
C₂²×C₄×C₈×C₁₃	2
C₂²×C₈×D₄	13

Subgroups — C22×C8×D52 #16726752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×D₅₂ #16726752