Subgroups — C₂×Q₈×Dic₅₂ #16743369

Order 3328.

1830 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₆	C₂	1	2	yes	maximal
2	Q₈×Dic₅₂	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718343	C₂	1	3	yes	maximal
4	C₂×#11718511	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₅₂	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×Q₈×Dic₁₃	—	2	2	no	—
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	Q₈×Dic₂₆	—	2	8	no	—
11	Q₈×Dic₂₆	C₂²	1	16	yes	—
12	C₂×#88142	—	2	6	no	—
13	C₂×#88123	C₂²	1	6	yes	—
14	C₂×C₄×Dic₂₆	C₂²	1	6	yes	—
15	C₈×Q₈×C₁₃	C₂²	1	8	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
18	C₂×#88058	C₂²	1	6	yes	—
19	C₂×#88058	C₂²	1	3	yes	—
20	C₂×#88056	C₂²	1	6	yes	—
21	#11718511	C₂²	1	48	yes	—
22	C₄×Dic₅₂	C₂²	1	24	yes	—
23	#11718343	C₂²	1	24	yes	—
24	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₂₆	—	2	6	no	—
26	C₂²×Dic₂₆	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	D₄	1	4	yes	—
30	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
33	Q₈×Dic₁₃	—	2	16	no	—
34	C₂×C₄⋊Dic₁₃	—	2	6	no	—
35	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
36	C₂×#6101	—	2	6	no	—
37	C₂×C₄×Dic₁₃	—	2	6	no	—
38	C₂×Dic₅₂	Q₈	1	8	yes	direct product factor
39	C₂×Dic₅₂	C₂³	1	4	yes	—
40	#88142	—	2	48	no	—
41	#88123	—	2	12	no	—
42	#88123	C₂³	1	24	yes	—
43	C₄×Dic₂₆	—	2	12	no	—
44	C₄×Dic₂₆	C₂³	1	24	yes	—
45	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
46	C₄×C₈×C₁₃	C₂³	1	12	yes	—
47	#88058	C₂³	1	24	yes	—
48	#88058	C₂³	1	12	yes	—
49	#88056	C₂³	1	24	yes	—
50	C₂×Q₈×Q₁₆	—	13	1	no	maximal, Sylow
51	C₂×Q₈×C₁₃	Q₁₆	1	8	yes	complemented
52	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂²×Dic₁₃	—	2	6	no	—
58	C₂×Dic₂₆	—	2	2	no	—
59	C₂×Dic₂₆	—	2	4	no	—
60	C₂×Dic₂₆	—	2	24	no	—
61	C₂×Dic₂₆	—	8	12	no	—
62	C₂×Dic₂₆	C₂×Q₈	1	8	yes	—
63	C₂×Dic₂₆	C₂⁴	1	4	yes	—
64	C₂×C₈×C₁₃	—	2	6	no	—
65	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
68	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
69	C₄²×C₁₃	C₂×D₄	1	6	yes	—
70	C₄²×C₁₃	C₂⁴	1	6	yes	—
71	C₄⋊Dic₁₃	—	2	6	no	—
72	C₄⋊Dic₁₃	—	2	24	no	—
73	C₄⋊Dic₁₃	C₂⁴	1	12	yes	—
74	#6101	—	2	24	no	—
75	C₄×Dic₁₃	—	2	24	no	—
76	Dic₅₂	C₂×Q₈	1	16	yes	direct product factor
77	C₂×Q₈²	—	13	2	no	—
78	Q₈×Q₁₆	—	13	16	no	—
79	C₂×#500	—	13	3	no	—
80	C₂×#477	—	13	6	no	—
81	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
82	C₂×C₄×Q₁₆	—	13	3	no	—
83	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₁₃	C₂×Q₁₆	1	16	yes	complemented
85	C₂×C₄×C₁₃	—	2	6	no	—
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	12	yes	—
87	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
90	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
92	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₁₃	—	2	8	no	—
94	C₂×Dic₁₃	—	2	4	no	—
95	C₂×Dic₁₃	—	2	12	no	—
96	C₂×Dic₁₃	—	4	12	no	—
97	Dic₂₆	—	2	8	no	—
98	Dic₂₆	—	2	4	no	—
99	Dic₂₆	—	8	24	no	—
100	Dic₂₆	C₂²×Q₈	1	8	yes	—
101	C₈×C₁₃	—	2	6	no	—
102	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
103	C₂²×Q₁₆	—	13	1	no	—
104	Q₈²	—	13	16	no	—
105	Q₈²	—	26	8	no	—
106	C₂×C₄⋊Q₈	—	13	6	no	—
107	C₂×C₄⋊Q₈	—	26	6	no	—
108	C₂×C₄×Q₈	—	13	6	no	—
109	C₂×C₄×Q₈	—	26	2	no	—
110	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
111	#500	—	13	24	no	—
112	#477	—	13	48	no	—
113	C₈×Q₈	C₂×D₁₃	1	8	yes	—
114	C₄×Q₁₆	—	13	24	no	—
115	C₂×#107	—	13	6	no	—
116	C₂×#107	—	13	3	no	—
117	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
118	C₂×#103	—	13	6	no	—
119	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
120	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
121	C₂²×C₁₃	C₂²×Q₁₆	1	2	yes	—
122	C₂²×C₁₃	#576	1	2	yes	—
123	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₁₃	—	2	6	no	—
125	C₄×C₁₃	D₄×Q₈	1	2	yes	—
126	C₄×C₁₃	C₂²×Q₁₆	1	12	yes	—
127	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
128	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
129	Dic₁₃	—	2	8	no	—
130	Dic₁₃	—	4	12	no	—
131	C₂²×Q₈	—	13	2	no	—
132	C₂²×Q₈	—	26	6	no	—
133	C₂²×Q₈	D₅₂	1	1	yes	characteristic
134	C₂×Q₁₆	—	13	8	no	—
135	C₂×Q₁₆	—	13	4	no	—
136	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
137	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	13	24	no	—
139	C₄⋊Q₈	—	26	48	no	—
140	C₄⋊Q₈	—	26	12	no	—
141	C₄×Q₈	—	13	24	no	—
142	C₄×Q₈	—	26	16	no	—
143	C₄×Q₈	—	26	12	no	—
144	C₄×Q₈	D₅₂	1	4	yes	—
145	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
146	C₂×C₄⋊C₄	—	13	6	no	—
147	C₂×C₄⋊C₄	—	26	6	no	—
148	C₂×C₄⋊C₄	—	26	6	no	—
149	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
150	C₂×C₄²	—	26	6	no	—
151	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
152	#107	—	13	12	no	—
153	#107	—	13	24	no	—
154	C₄⋊C₈	C₂²×D₁₃	1	12	yes	—
155	#103	—	13	24	no	—
156	C₄×C₈	C₂²×D₁₃	1	12	yes	—
157	C₂×C₁₃	Q₈×Q₁₆	1	4	yes	complemented
158	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
159	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#576	1	1	yes	characteristic
161	C₂×Q₈	—	13	8	no	—
162	C₂×Q₈	—	13	4	no	—
163	C₂×Q₈	—	26	2	no	—
164	C₂×Q₈	—	26	24	no	—
165	C₂×Q₈	—	26	4	no	—
166	C₂×Q₈	—	104	12	no	—
167	C₂×Q₈	Dic₅₂	1	8	yes	direct product factor
168	C₂×Q₈	C₂×D₅₂	1	4	yes	—
169	C₂²×C₄	—	26	2	no	—
170	C₂²×C₄	—	26	6	no	—
171	C₂²×C₄	C₂×D₅₂	1	3	yes	—
172	C₂²×C₄	C₂×D₅₂	1	3	yes	—
173	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
174	Q₁₆	—	13	16	no	—
175	C₂×C₈	—	2	6	no	—
176	C₂×C₈	Q₈×D₁₃	1	4	yes	—
177	C₂×C₈	C₂³×D₁₃	1	6	yes	—
178	C₂×C₈	C₂³×D₁₃	1	2	yes	—
179	C₄⋊C₄	—	13	12	no	—
180	C₄⋊C₄	—	26	6	no	—
181	C₄⋊C₄	—	26	24	no	—
182	C₄⋊C₄	—	26	24	no	—
183	C₄⋊C₄	C₂×D₅₂	1	12	yes	—
184	C₄²	—	26	24	no	—
185	C₄²	C₂×D₅₂	1	6	yes	—
186	C₄²	C₂³×D₁₃	1	6	yes	—
187	C₁₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
188	C₂³	C₂²×D₅₂	1	1	yes	characteristic
189	Q₈	—	13	8	no	—
190	Q₈	—	26	8	no	—
191	Q₈	—	26	4	no	—
192	Q₈	—	104	24	no	—
193	Q₈	C₂×Dic₅₂	1	16	yes	direct product factor
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	26	8	no	—
196	C₂×C₄	—	26	4	no	—
197	C₂×C₄	—	26	12	no	—
198	C₂×C₄	—	52	12	no	—
199	C₂×C₄	C₂×Dic₅₂	1	12	yes	—
200	C₂×C₄	C₂²×D₅₂	1	6	yes	—
201	C₂×C₄	C₂²×D₅₂	1	6	yes	—
202	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
203	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
204	C₂×C₄	#88257	1	2	yes	—
205	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂×Q₈×D₁₃	1	4	yes	—
208	C₂²	Q₈×D₅₂	1	2	yes	—
209	C₂²	C₂²×Dic₅₂	1	2	yes	—
210	C₂²	#11719457	1	2	yes	—
211	C₂²	C₂³×D₅₂	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	26	8	no	—
214	C₄	—	52	12	no	—
215	C₄	Q₈×D₅₂	1	2	yes	—
216	C₄	C₂²×Dic₅₂	1	12	yes	—
217	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
218	C₄	C₂×#88257	1	1	yes	characteristic
219	C₂	Q₈×Dic₅₂	1	4	yes	direct product factor
220	C₂	C₂×Q₈×D₅₂	1	1	yes	characteristic
221	C₂	C₂³×Dic₅₂	1	1	yes	characteristic
222	C₂	C₂×#11719457	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₅₂	C₂	direct (N × H)
38. C₂×Dic₅₂	Q₈	direct (N × H)
51. C₂×Q₈×C₁₃	Q₁₆	semidirect
76. Dic₅₂	C₂×Q₈	direct (N × H)
84. Q₈×C₁₃	C₂×Q₁₆	semidirect
157. C₂×C₁₃	Q₈×Q₁₆	semidirect
167. C₂×Q₈	Dic₅₂	direct (N × H)
187. C₁₃	C₂×Q₈×Q₁₆	semidirect
193. Q₈	C₂×Dic₅₂	direct (N × H)
219. C₂	Q₈×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
C₂×Q₈×Dic₂₆	2
Q₈×Dic₅₂	2
C₂×#11718343	2
C₂×#11718511	2
C₂×C₄×Dic₅₂	2
C₂×C₈×Q₈×C₁₃	2
C₂×Q₈×Q₁₆	13

Subgroups — C2×Q8×Dic52 #16743369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₅₂ #16743369