Subgroups — C₂²×C₄×Dic₅₂ #16726936

Order 3328.

2830 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₆	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₅₂	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88056	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂²×#6101	—	2	2	no	—
13	C₂²×C₄×Dic₁₃	—	2	2	no	—
14	C₂²×Dic₅₂	C₄	1	16	yes	direct product factor
15	C₂²×Dic₅₂	C₂²	1	12	yes	—
16	C₂×C₄×Dic₂₆	—	2	12	no	—
17	C₂×C₄×Dic₂₆	C₂²	1	24	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	—
19	C₂×#88058	C₂²	1	12	yes	—
20	C₂×#88056	C₂²	1	24	yes	—
21	C₄×Dic₅₂	C₂²	1	64	yes	direct product factor
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂³×Dic₁₃	—	2	2	no	—
27	C₂²×Dic₂₆	—	2	6	no	—
28	C₂²×Dic₂₆	—	2	8	no	—
29	C₂²×Dic₂₆	C₂×C₄	1	16	yes	—
30	C₂²×Dic₂₆	C₂³	1	12	yes	—
31	C₂²×C₈×C₁₃	—	2	4	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
35	C₂×C₄²×C₁₃	D₄	1	6	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
37	C₂×C₄⋊Dic₁₃	—	2	6	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
39	C₂×#6101	—	2	24	no	—
40	C₂×C₄×Dic₁₃	—	2	24	no	—
41	C₂×Dic₅₂	C₂×C₄	1	96	yes	direct product factor
42	C₂×Dic₅₂	C₂³	1	16	yes	—
43	C₄×Dic₂₆	—	2	48	no	—
44	C₄×Dic₂₆	C₂³	1	32	yes	—
45	C₄×C₈×C₁₃	C₂³	1	16	yes	—
46	#88058	C₂³	1	16	yes	—
47	#88056	C₂³	1	32	yes	—
48	C₂²×C₄×Q₁₆	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	Q₁₆	1	8	yes	complemented
52	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₁₃	#41	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
58	C₂²×Dic₁₃	—	2	16	no	—
59	C₂²×Dic₁₃	—	2	12	no	—
60	C₂²×Dic₁₃	—	2	12	no	—
61	C₂²×Dic₁₃	—	4	8	no	—
62	C₂×Dic₂₆	—	2	48	no	—
63	C₂×Dic₂₆	—	2	12	no	—
64	C₂×Dic₂₆	—	2	24	no	—
65	C₂×Dic₂₆	C₂²×C₄	1	48	yes	—
66	C₂×Dic₂₆	C₂⁴	1	8	yes	—
67	C₂×C₈×C₁₃	—	2	12	no	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
69	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
70	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
71	C₄²×C₁₃	C₂×D₄	1	12	yes	—
72	C₄²×C₁₃	C₂⁴	1	4	yes	—
73	C₄⋊Dic₁₃	—	2	12	no	—
74	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
75	#6101	—	2	32	no	—
76	C₄×Dic₁₃	—	2	32	no	—
77	Dic₅₂	C₂²×C₄	1	64	yes	direct product factor
78	C₂³×Q₁₆	—	13	1	no	—
79	C₂²×C₄×Q₈	—	13	2	no	—
80	C₂×C₄×Q₁₆	—	13	24	no	—
81	C₂²×#107	—	13	1	no	—
82	C₂²×#103	—	13	2	no	—
83	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
84	C₂³×C₁₃	C₄×D₄	1	4	yes	—
85	C₂³×C₁₃	C₂×Q₁₆	1	4	yes	—
86	C₂³×C₁₃	#135	1	4	yes	—
87	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
88	C₂×C₄×C₁₃	—	2	12	no	—
89	C₂×C₄×C₁₃	C₄×D₄	1	12	yes	—
90	C₂×C₄×C₁₃	C₂×Q₁₆	1	24	yes	complemented
91	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
93	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
94	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
95	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
96	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
97	C₂×Dic₁₃	—	2	48	no	—
98	C₂×Dic₁₃	—	2	8	no	—
99	C₂×Dic₁₃	—	2	8	no	—
100	C₂×Dic₁₃	—	4	24	no	—
101	Dic₂₆	—	2	48	no	—
102	Dic₂₆	—	2	8	no	—
103	Dic₂₆	C₂³×C₄	1	16	yes	—
104	C₈×C₁₃	—	2	4	no	—
105	C₈×C₁₃	C₂³×C₄	1	8	yes	—
106	C₂³×Q₈	—	13	2	no	—
107	C₂²×Q₁₆	—	13	16	no	—
108	C₂²×Q₁₆	—	13	12	no	—
109	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
110	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	13	24	no	—
112	C₂×C₄×Q₈	—	26	12	no	—
113	C₂²×C₄⋊C₄	—	13	2	no	—
114	C₂²×C₄⋊C₄	—	26	2	no	—
115	C₂²×C₄²	—	26	2	no	—
116	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
117	C₄×Q₁₆	—	13	64	no	—
118	C₂×#107	—	13	12	no	—
119	C₂×#103	—	13	24	no	—
120	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	—
121	C₂²×C₁₃	C₄×Q₁₆	1	16	yes	complemented
122	C₂²×C₁₃	C₂×C₄×D₄	1	6	yes	—
123	C₂²×C₁₃	C₂²×Q₁₆	1	6	yes	—
124	C₂²×C₁₃	C₂×#135	1	6	yes	—
125	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
126	C₄×C₁₃	—	2	4	no	—
127	C₄×C₁₃	C₂×C₄×D₄	1	6	yes	—
128	C₄×C₁₃	C₂²×Q₁₆	1	8	yes	complemented
129	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
131	Dic₁₃	—	2	16	no	—
132	Dic₁₃	—	4	8	no	—
133	C₂²×Q₈	—	13	16	no	—
134	C₂²×Q₈	—	13	12	no	—
135	C₂²×Q₈	—	26	6	no	—
136	C₂²×Q₈	—	26	8	no	—
137	C₂³×C₄	—	26	2	no	—
138	C₂³×C₄	—	26	2	no	—
139	C₂³×C₄	D₅₂	1	1	yes	characteristic
140	C₂³×C₄	D₅₂	1	1	yes	characteristic
141	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
142	C₂×Q₁₆	—	13	96	no	—
143	C₂×Q₁₆	—	13	16	no	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	C₄×D₁₃	1	8	yes	—
146	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
147	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
148	C₄×Q₈	—	13	32	no	—
149	C₄×Q₈	—	26	48	no	—
150	C₂×C₄⋊C₄	—	13	12	no	—
151	C₂×C₄⋊C₄	—	26	6	no	—
152	C₂×C₄⋊C₄	—	26	24	no	—
153	C₂×C₄²	—	26	24	no	—
154	C₂×C₄²	D₅₂	1	6	yes	—
155	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
156	#107	—	13	16	no	—
157	#103	—	13	32	no	—
158	C₄×C₈	C₂²×D₁₃	1	16	yes	—
159	C₂×C₁₃	C₂×C₄×Q₁₆	1	12	yes	complemented
160	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
161	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
162	C₂×C₁₃	C₂²×#135	1	1	yes	characteristic
163	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
164	C₂×Q₈	—	13	48	no	—
165	C₂×Q₈	—	13	8	no	—
166	C₂×Q₈	—	26	48	no	—
167	C₂×Q₈	—	26	12	no	—
168	C₂×Q₈	—	26	24	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	26	16	no	—
171	C₂²×C₄	—	26	12	no	—
172	C₂²×C₄	—	26	12	no	—
173	C₂²×C₄	—	52	8	no	—
174	C₂²×C₄	Dic₅₂	1	8	yes	direct product factor
175	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
176	C₂²×C₄	C₂×D₅₂	1	6	yes	—
177	C₂²×C₄	C₂×D₅₂	1	3	yes	—
178	C₂²×C₄	C₂×D₅₂	1	6	yes	—
179	C₂²×C₄	#6127	1	4	yes	—
180	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
181	Q₁₆	—	13	64	no	—
182	C₂×C₈	—	2	12	no	—
183	C₂×C₈	C₂×C₄×D₁₃	1	24	yes	—
184	C₂×C₈	C₂³×D₁₃	1	4	yes	—
185	C₂×C₈	C₂³×D₁₃	1	4	yes	—
186	C₄⋊C₄	—	13	8	no	—
187	C₄⋊C₄	—	26	12	no	—
188	C₄⋊C₄	—	26	32	no	—
189	C₄²	—	26	32	no	—
190	C₄²	C₂×D₅₂	1	12	yes	—
191	C₄²	C₂³×D₁₃	1	4	yes	—
192	C₁₃	C₂²×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₄×D₅₂	1	4	yes	—
194	C₂³	#88158	1	4	yes	—
195	C₂³	C₂×Dic₅₂	1	4	yes	—
196	C₂³	C₂²×D₅₂	1	3	yes	—
197	Q₈	—	13	16	no	—
198	Q₈	—	26	48	no	—
199	Q₈	—	26	8	no	—
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	—	26	48	no	—
202	C₂×C₄	—	26	8	no	—
203	C₂×C₄	—	26	8	no	—
204	C₂×C₄	—	52	24	no	—
205	C₂×C₄	C₄×D₅₂	1	12	yes	—
206	C₂×C₄	C₂×Dic₅₂	1	24	yes	direct product factor
207	C₂×C₄	C₂²×C₄×D₁₃	1	6	yes	—
208	C₂×C₄	C₂²×D₅₂	1	4	yes	—
209	C₂×C₄	C₂²×D₅₂	1	3	yes	—
210	C₂×C₄	C₂²×D₅₂	1	4	yes	—
211	C₂×C₄	C₂×#6127	1	6	yes	—
212	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂²×C₄×D₁₃	1	8	yes	—
215	C₂²	C₄×Dic₅₂	1	16	yes	direct product factor
216	C₂²	C₂×C₄×D₅₂	1	6	yes	—
217	C₂²	C₂×#88158	1	6	yes	—
218	C₂²	C₂²×Dic₅₂	1	6	yes	—
219	C₂²	C₂³×D₅₂	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	26	16	no	—
222	C₄	—	52	8	no	—
223	C₄	C₂×C₄×D₅₂	1	6	yes	—
224	C₄	C₂²×Dic₅₂	1	8	yes	direct product factor
225	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
226	C₄	C₂²×#6127	1	1	yes	characteristic
227	C₂	C₂×C₄×Dic₅₂	1	12	yes	direct product factor
228	C₂	C₂²×C₄×D₅₂	1	1	yes	characteristic
229	C₂	C₂³×Dic₅₂	1	1	yes	characteristic
230	C₂	C₂²×#88158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×C₄×Q₁₆
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₅₂	C₂	direct (N × H)
14. C₂²×Dic₅₂	C₄	direct (N × H)
21. C₄×Dic₅₂	C₂²	direct (N × H)
41. C₂×Dic₅₂	C₂×C₄	direct (N × H)
51. C₂²×C₄×C₁₃	Q₁₆	semidirect
77. Dic₅₂	C₂²×C₄	direct (N × H)
90. C₂×C₄×C₁₃	C₂×Q₁₆	semidirect
121. C₂²×C₁₃	C₄×Q₁₆	semidirect
128. C₄×C₁₃	C₂²×Q₁₆	semidirect
159. C₂×C₁₃	C₂×C₄×Q₁₆	semidirect
174. C₂²×C₄	Dic₅₂	direct (N × H)
192. C₁₃	C₂²×C₄×Q₁₆	semidirect
206. C₂×C₄	C₂×Dic₅₂	direct (N × H)
215. C₂²	C₄×Dic₅₂	direct (N × H)
224. C₄	C₂²×Dic₅₂	direct (N × H)
227. C₂	C₂×C₄×Dic₅₂	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×Dic₅₂	2
C₂²×C₄×Dic₂₆	2
C₂×C₄×Dic₅₂	2
C₂²×#88058	2
C₂²×#88056	2
C₂²×C₄×C₈×C₁₃	2
C₂²×C₄×Q₁₆	13

Subgroups — C22×C4×Dic52 #16726936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Dic₅₂ #16726936