Subgroups — C₂²×Q₈×D₅₂ #16769836

Order 3328.

11798 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₁₃	C₂	1	2	yes	maximal, complemented
2	C₂×Q₈×D₅₂	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#88123	C₂	1	3	yes	maximal, complemented
4	C₂²×#88151	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×D₅₂	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×D₁₃	—	2	32	no	—
9	C₂²×Q₈×D₁₃	C₂²	1	48	yes	complemented
10	C₂³×D₅₂	C₂²	1	1	yes	characteristic
11	C₂³×C₄×D₁₃	C₂²	1	6	yes	—
12	C₂³×Dic₂₆	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	24	yes	complemented
14	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
16	C₂²×#6103	C₂²	1	6	yes	—
17	C₂²×C₄⋊Dic₁₃	C₂²	1	3	yes	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
19	Q₈×D₅₂	C₂²	1	256	yes	direct product factor
20	C₂×#88151	C₂²	1	144	yes	complemented
21	C₂×C₄×D₅₂	C₂²	1	72	yes	—
22	C₂×#88123	C₂²	1	72	yes	complemented
23	C₂⁴×D₁₃	C₂³	1	2	yes	—
24	C₂²×Q₈×C₁₃	D₄	1	16	yes	complemented
25	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×Dic₁₃	C₂³	1	6	yes	—
30	C₂×Q₈×D₁₃	—	2	384	no	—
31	C₂×Q₈×D₁₃	C₂³	1	128	yes	complemented
32	C₂²×D₅₂	Q₈	1	16	yes	direct product factor
33	C₂²×D₅₂	C₂³	1	12	yes	—
34	C₂²×C₄×D₁₃	—	2	48	no	—
35	C₂²×C₄×D₁₃	C₂³	1	72	yes	—
36	C₂²×Dic₂₆	—	2	48	no	—
37	C₂²×Dic₂₆	C₂³	1	72	yes	—
38	C₄×Q₈×C₁₃	C₂³	1	64	yes	complemented
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	36	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	36	yes	—
41	C₂×#6103	C₂³	1	72	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	36	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	72	yes	—
44	#88151	C₂³	1	384	yes	complemented
45	C₄×D₅₂	C₂³	1	192	yes	—
46	#88123	C₂³	1	192	yes	complemented
47	C₂²×D₄×Q₈	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂³×D₁₃	—	2	8	no	—
50	C₂³×D₁₃	—	2	8	no	—
51	C₂³×D₁₃	C₂×Q₈	1	16	yes	complemented
52	C₂³×D₁₃	C₂⁴	1	12	yes	—
53	C₂×Q₈×C₁₃	C₂×D₄	1	96	yes	complemented
54	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	—
55	C₂²×C₄×C₁₃	—	2	12	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	complemented
58	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
61	C₂²×Dic₁₃	—	2	24	no	—
62	C₂²×Dic₁₃	C₂⁴	1	36	yes	—
63	Q₈×D₁₃	—	2	512	no	—
64	C₂×D₅₂	C₂×Q₈	1	96	yes	direct product factor
65	C₂×D₅₂	C₂⁴	1	16	yes	—
66	C₂×C₄×D₁₃	—	2	288	no	—
67	C₂×C₄×D₁₃	C₂⁴	1	96	yes	—
68	C₂×Dic₂₆	—	2	288	no	—
69	C₂×Dic₂₆	C₂⁴	1	96	yes	—
70	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	—
71	C₄²×C₁₃	C₂⁴	1	48	yes	—
72	#6103	C₂⁴	1	96	yes	—
73	C₄⋊Dic₁₃	C₂⁴	1	48	yes	—
74	C₄⋊Dic₁₃	C₂⁴	1	96	yes	—
75	C₂⁴×Q₈	—	13	2	no	—
76	C₂×D₄×Q₈	—	13	48	no	—
77	C₂²×C₄⋊Q₈	—	13	3	no	—
78	C₂²×C₂²⋊Q₈	—	13	6	no	—
79	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
80	C₂²×C₄×D₄	—	13	3	no	—
81	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
82	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
83	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
84	C₂³×C₁₃	C₂⁵	1	3	yes	—
85	C₂²×D₁₃	—	2	64	no	—
86	C₂²×D₁₃	—	2	24	no	—
87	C₂²×D₁₃	—	2	24	no	—
88	C₂²×D₁₃	C₂²×Q₈	1	48	yes	complemented
89	C₂²×D₁₃	C₂⁵	1	8	yes	—
90	Q₈×C₁₃	C₂²×D₄	1	64	yes	complemented
91	C₂×C₄×C₁₃	—	2	36	no	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	72	yes	—
93	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	complemented
94	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
95	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
96	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
97	C₂×Dic₁₃	—	2	72	no	—
98	C₂×Dic₁₃	C₂⁵	1	24	yes	—
99	D₅₂	C₂²×Q₈	1	64	yes	direct product factor
100	C₄×D₁₃	—	2	192	no	—
101	Dic₂₆	—	2	192	no	—
102	C₂³×Q₈	—	13	48	no	—
103	C₂³×Q₈	—	13	6	no	—
104	C₂³×Q₈	—	26	32	no	—
105	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂³×D₄	—	13	1	no	—
107	C₂⁴×C₄	—	13	6	no	—
108	D₄×Q₈	—	13	256	no	—
109	C₂×C₄⋊Q₈	—	13	72	no	—
110	C₂×C₂²⋊Q₈	—	13	144	no	—
111	C₂×C₄×Q₈	C₂×D₁₃	1	24	yes	complemented
112	C₂×C₄×D₄	—	13	72	no	—
113	C₂²×C₄⋊C₄	—	13	3	no	—
114	C₂²×C₄⋊C₄	—	13	6	no	—
115	C₂²×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
116	C₂²×C₂²⋊C₄	—	13	6	no	—
117	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
118	C₂²×C₁₃	D₄×Q₈	1	16	yes	complemented
119	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
120	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
121	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
122	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
123	C₂×D₁₃	—	2	8	no	—
124	C₂×D₁₃	—	2	8	no	—
125	C₂×D₁₃	—	2	96	no	—
126	C₂×D₁₃	C₂³×Q₈	1	16	yes	complemented
127	C₄×C₁₃	—	2	12	no	—
128	C₄×C₁₃	C₂³×D₄	1	24	yes	—
129	C₄×C₁₃	C₂³×Q₈	1	8	yes	complemented
130	Dic₁₃	—	2	24	no	—
131	C₂⁵	—	13	2	no	—
132	C₂²×Q₈	—	13	128	no	—
133	C₂²×Q₈	—	13	72	no	—
134	C₂²×Q₈	—	26	384	no	—
135	C₂²×Q₈	—	26	48	no	—
136	C₂²×Q₈	D₅₂	1	16	yes	direct product factor
137	C₂²×Q₈	C₂²×D₁₃	1	12	yes	—
138	C₂²×D₄	—	13	16	no	—
139	C₂²×D₄	—	13	12	no	—
140	C₂³×C₄	—	13	6	no	—
141	C₂³×C₄	—	13	72	no	—
142	C₂³×C₄	—	26	48	no	—
143	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
144	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
145	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
146	C₄⋊Q₈	—	13	192	no	—
147	C₂²⋊Q₈	—	13	384	no	—
148	C₄×Q₈	C₂²×D₁₃	1	64	yes	complemented
149	C₄×D₄	—	13	192	no	—
150	C₂×C₄⋊C₄	—	13	36	no	—
151	C₂×C₄⋊C₄	—	13	72	no	—
152	C₂×C₄⋊C₄	C₂²×D₁₃	1	36	yes	—
153	C₂×C₂²⋊C₄	—	13	72	no	—
154	C₂×C₄²	C₂²×D₁₃	1	36	yes	—
155	C₂×C₁₃	C₂×D₄×Q₈	1	12	yes	complemented
156	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
157	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
158	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
159	D₁₃	—	2	16	no	—
160	C₂⁴	—	13	16	no	—
161	C₂⁴	—	13	12	no	—
162	C₂⁴	—	26	8	no	—
163	C₂⁴	—	26	8	no	—
164	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
165	C₂×Q₈	—	13	96	no	—
166	C₂×Q₈	—	26	512	no	—
167	C₂×Q₈	—	26	288	no	—
168	C₂×Q₈	C₂×D₅₂	1	96	yes	direct product factor
169	C₂×Q₈	C₂³×D₁₃	1	16	yes	—
170	C₂×D₄	—	13	96	no	—
171	C₂×D₄	—	13	16	no	—
172	C₂²×C₄	—	2	12	no	—
173	C₂²×C₄	—	13	36	no	—
174	C₂²×C₄	—	13	96	no	—
175	C₂²×C₄	—	26	24	no	—
176	C₂²×C₄	—	26	288	no	—
177	C₂²×C₄	C₂×D₅₂	1	24	yes	—
178	C₂²×C₄	Q₈×D₁₃	1	8	yes	complemented
179	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
180	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
181	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
182	C₄⋊C₄	—	13	48	no	—
183	C₄⋊C₄	—	13	96	no	—
184	C₄⋊C₄	C₂³×D₁₃	1	48	yes	—
185	C₂²⋊C₄	—	13	96	no	—
186	C₄²	C₂³×D₁₃	1	48	yes	—
187	C₁₃	C₂²×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	13	48	no	—
189	C₂³	—	13	8	no	—
190	C₂³	—	26	64	no	—
191	C₂³	—	26	24	no	—
192	C₂³	—	26	24	no	—
193	C₂³	C₂²×D₅₂	1	4	yes	—
194	C₂³	C₂×Q₈×D₁₃	1	4	yes	—
195	C₂³	#88257	1	4	yes	—
196	C₂³	C₂⁴×D₁₃	1	3	yes	—
197	Q₈	—	26	192	no	—
198	Q₈	C₂²×D₅₂	1	64	yes	direct product factor
199	D₄	—	13	64	no	—
200	C₂×C₄	—	2	36	no	—
201	C₂×C₄	—	13	24	no	—
202	C₂×C₄	—	26	72	no	—
203	C₂×C₄	—	26	192	no	—
204	C₂×C₄	C₂²×D₅₂	1	72	yes	—
205	C₂×C₄	C₂×Q₈×D₁₃	1	24	yes	complemented
206	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
207	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
208	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
209	C₂²	—	13	16	no	—
210	C₂²	—	26	96	no	—
211	C₂²	—	26	8	no	—
212	C₂²	—	26	8	no	—
213	C₂²	Q₈×D₅₂	1	16	yes	direct product factor
214	C₂²	C₂³×D₅₂	1	6	yes	—
215	C₂²	C₂²×Q₈×D₁₃	1	6	yes	—
216	C₂²	C₂×#88257	1	6	yes	—
217	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
218	C₄	—	2	12	no	—
219	C₄	—	26	24	no	—
220	C₄	C₂³×D₅₂	1	24	yes	—
221	C₄	C₂²×Q₈×D₁₃	1	8	yes	complemented
222	C₂	—	26	16	no	—
223	C₂	C₂×Q₈×D₅₂	1	12	yes	direct product factor
224	C₂	C₂⁴×D₅₂	1	1	yes	characteristic
225	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
226	C₂	C₂²×#88257	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂³×D₅₂	C₂²
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂²×D₄×Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₁₃	C₂	semidirect
2. C₂×Q₈×D₅₂	C₂	direct (N × H)
3. C₂²×#88123	C₂	semidirect
4. C₂²×#88151	C₂	semidirect
6. C₂²×C₄×Q₈×C₁₃	C₂	semidirect
9. C₂²×Q₈×D₁₃	C₂²	semidirect
13. C₂×C₄×Q₈×C₁₃	C₂²	semidirect
19. Q₈×D₅₂	C₂²	direct (N × H)
20. C₂×#88151	C₂²	semidirect
22. C₂×#88123	C₂²	semidirect
24. C₂²×Q₈×C₁₃	D₄	semidirect
31. C₂×Q₈×D₁₃	C₂³	semidirect
32. C₂²×D₅₂	Q₈	direct (N × H)
38. C₄×Q₈×C₁₃	C₂³	semidirect
44. #88151	C₂³	semidirect
46. #88123	C₂³	semidirect
51. C₂³×D₁₃	C₂×Q₈	semidirect
53. C₂×Q₈×C₁₃	C₂×D₄	semidirect
57. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
64. C₂×D₅₂	C₂×Q₈	direct (N × H)
79. C₂²×C₄×Q₈	D₁₃	semidirect
88. C₂²×D₁₃	C₂²×Q₈	semidirect
90. Q₈×C₁₃	C₂²×D₄	semidirect
93. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
99. D₅₂	C₂²×Q₈	direct (N × H)
111. C₂×C₄×Q₈	C₂×D₁₃	semidirect
118. C₂²×C₁₃	D₄×Q₈	semidirect
126. C₂×D₁₃	C₂³×Q₈	semidirect
129. C₄×C₁₃	C₂³×Q₈	semidirect
136. C₂²×Q₈	D₅₂	direct (N × H)
148. C₄×Q₈	C₂²×D₁₃	semidirect
155. C₂×C₁₃	C₂×D₄×Q₈	semidirect
168. C₂×Q₈	C₂×D₅₂	direct (N × H)
178. C₂²×C₄	Q₈×D₁₃	semidirect
187. C₁₃	C₂²×D₄×Q₈	semidirect
198. Q₈	C₂²×D₅₂	direct (N × H)
205. C₂×C₄	C₂×Q₈×D₁₃	semidirect
213. C₂²	Q₈×D₅₂	direct (N × H)
221. C₄	C₂²×Q₈×D₁₃	semidirect
223. C₂	C₂×Q₈×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×Q₈×D₁₃	2
C₂×Q₈×D₅₂	2
C₂²×#88123	2
C₂²×#88151	2
C₂²×C₄×D₅₂	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×D₄×Q₈	13

Subgroups — C22×Q8×D52 #16769836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×D₅₂ #16769836