Subgroups — C₂²×Q₈×D₂₁₂ #28291009

Order 13568.

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×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×Q₈×D₂₁₂	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133217	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133197	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×D₂₁₂	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Dic₁₀₆	C₂²	1	6	yes	—
8	C₂²×Q₈×D₅₃	—	2	32	no	—
9	C₂²×Q₈×D₅₃	C₂²	1	48	yes	complemented
10	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
11	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
12	Q₈×D₂₁₂	C₂²	1	256	yes	direct product factor
13	C₂×#13133217	C₂²	1	72	yes	complemented
14	C₂×#13133197	C₂²	1	144	yes	complemented
15	C₂×C₄×D₂₁₂	C₂²	1	72	yes	—
16	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	—
17	C₂²×C₄⋊Dic₅₃	C₂²	1	6	yes	—
18	C₂²×#11719921	C₂²	1	6	yes	—
19	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	complemented
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
23	C₂⁴×D₅₃	C₂³	1	2	yes	—
24	C₂²×Dic₁₀₆	—	2	48	no	—
25	C₂²×Dic₁₀₆	C₂³	1	72	yes	—
26	C₂×Q₈×D₅₃	—	2	384	no	—
27	C₂×Q₈×D₅₃	C₂³	1	128	yes	complemented
28	C₂²×D₂₁₂	Q₈	1	16	yes	direct product factor
29	C₂²×D₂₁₂	C₂³	1	12	yes	—
30	C₂³×Dic₅₃	C₂³	1	6	yes	—
31	C₂²×C₄×D₅₃	—	2	48	no	—
32	C₂²×C₄×D₅₃	C₂³	1	72	yes	—
33	#13133217	C₂³	1	192	yes	complemented
34	#13133197	C₂³	1	384	yes	complemented
35	C₄×D₂₁₂	C₂³	1	192	yes	—
36	C₂×C₄⋊Dic₅₃	C₂³	1	36	yes	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	72	yes	—
38	C₂×#11719921	C₂³	1	72	yes	—
39	C₂²×Q₈×C₅₃	D₄	1	16	yes	complemented
40	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₅₃	C₂³	1	64	yes	complemented
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	36	yes	—
47	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂³×D₅₃	—	2	8	no	—
49	C₂³×D₅₃	—	2	8	no	—
50	C₂³×D₅₃	C₂×Q₈	1	16	yes	complemented
51	C₂³×D₅₃	C₂⁴	1	12	yes	—
52	C₂×Q₈×C₅₃	C₂×D₄	1	96	yes	complemented
53	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
54	C₂²×C₄×C₅₃	—	2	12	no	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	complemented
57	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
60	C₂²×Dic₅₃	—	2	24	no	—
61	C₂²×Dic₅₃	C₂⁴	1	36	yes	—
62	Q₈×D₅₃	—	2	512	no	—
63	C₂×D₂₁₂	C₂×Q₈	1	96	yes	direct product factor
64	C₂×D₂₁₂	C₂⁴	1	16	yes	—
65	C₂×C₄×D₅₃	—	2	288	no	—
66	C₂×C₄×D₅₃	C₂⁴	1	96	yes	—
67	C₂×Dic₁₀₆	—	2	288	no	—
68	C₂×Dic₁₀₆	C₂⁴	1	96	yes	—
69	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
70	C₄²×C₅₃	C₂⁴	1	48	yes	—
71	#11719921	C₂⁴	1	96	yes	—
72	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
73	C₄⋊Dic₅₃	C₂⁴	1	96	yes	—
74	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
75	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
76	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
77	C₂³×C₅₃	C₂⁵	1	3	yes	—
78	C₂²×D₅₃	—	2	64	no	—
79	C₂²×D₅₃	—	2	24	no	—
80	C₂²×D₅₃	—	2	24	no	—
81	C₂²×D₅₃	C₂²×Q₈	1	48	yes	complemented
82	C₂²×D₅₃	C₂⁵	1	8	yes	—
83	Q₈×C₅₃	C₂²×D₄	1	64	yes	complemented
84	C₂×C₄×C₅₃	—	2	36	no	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	72	yes	—
86	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	complemented
87	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
88	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
89	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
90	C₂×Dic₅₃	—	2	72	no	—
91	C₂×Dic₅₃	C₂⁵	1	24	yes	—
92	D₂₁₂	C₂²×Q₈	1	64	yes	direct product factor
93	C₄×D₅₃	—	2	192	no	—
94	Dic₁₀₆	—	2	192	no	—
95	C₂²×D₄×Q₈	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	D₄×Q₈	1	16	yes	complemented
97	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
98	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
99	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
100	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
101	C₂×D₅₃	—	2	8	no	—
102	C₂×D₅₃	—	2	8	no	—
103	C₂×D₅₃	—	2	96	no	—
104	C₂×D₅₃	C₂³×Q₈	1	16	yes	complemented
105	C₄×C₅₃	—	2	12	no	—
106	C₄×C₅₃	C₂³×D₄	1	24	yes	—
107	C₄×C₅₃	C₂³×Q₈	1	8	yes	complemented
108	Dic₅₃	—	2	24	no	—
109	C₂⁴×Q₈	—	53	2	no	—
110	C₂×D₄×Q₈	—	53	48	no	—
111	C₂²×C₄⋊Q₈	—	53	3	no	—
112	C₂²×C₂²⋊Q₈	—	53	6	no	—
113	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
114	C₂²×C₄×D₄	—	53	3	no	—
115	C₂×C₅₃	C₂×D₄×Q₈	1	12	yes	complemented
116	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
117	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
119	D₅₃	—	2	16	no	—
120	C₂³×Q₈	—	53	48	no	—
121	C₂³×Q₈	—	53	6	no	—
122	C₂³×Q₈	—	106	32	no	—
123	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂³×D₄	—	53	1	no	—
125	C₂⁴×C₄	—	53	6	no	—
126	D₄×Q₈	—	53	256	no	—
127	C₂×C₄⋊Q₈	—	53	72	no	—
128	C₂×C₂²⋊Q₈	—	53	144	no	—
129	C₂×C₄×Q₈	C₂×D₅₃	1	24	yes	complemented
130	C₂×C₄×D₄	—	53	72	no	—
131	C₂²×C₄⋊C₄	—	53	3	no	—
132	C₂²×C₄⋊C₄	—	53	6	no	—
133	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
134	C₂²×C₂²⋊C₄	—	53	6	no	—
135	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
136	C₅₃	C₂²×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
137	C₂⁵	—	53	2	no	—
138	C₂²×Q₈	—	53	128	no	—
139	C₂²×Q₈	—	53	72	no	—
140	C₂²×Q₈	—	106	384	no	—
141	C₂²×Q₈	—	106	48	no	—
142	C₂²×Q₈	D₂₁₂	1	16	yes	direct product factor
143	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
144	C₂²×D₄	—	53	16	no	—
145	C₂²×D₄	—	53	12	no	—
146	C₂³×C₄	—	53	6	no	—
147	C₂³×C₄	—	53	72	no	—
148	C₂³×C₄	—	106	48	no	—
149	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
151	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	53	192	no	—
153	C₂²⋊Q₈	—	53	384	no	—
154	C₄×Q₈	C₂²×D₅₃	1	64	yes	complemented
155	C₄×D₄	—	53	192	no	—
156	C₂×C₄⋊C₄	—	53	36	no	—
157	C₂×C₄⋊C₄	—	53	72	no	—
158	C₂×C₄⋊C₄	C₂²×D₅₃	1	36	yes	—
159	C₂×C₂²⋊C₄	—	53	72	no	—
160	C₂×C₄²	C₂²×D₅₃	1	36	yes	—
161	C₂⁴	—	53	16	no	—
162	C₂⁴	—	53	12	no	—
163	C₂⁴	—	106	8	no	—
164	C₂⁴	—	106	8	no	—
165	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
166	C₂×Q₈	—	53	96	no	—
167	C₂×Q₈	—	106	512	no	—
168	C₂×Q₈	—	106	288	no	—
169	C₂×Q₈	C₂×D₂₁₂	1	96	yes	direct product factor
170	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
171	C₂×D₄	—	53	96	no	—
172	C₂×D₄	—	53	16	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	53	36	no	—
175	C₂²×C₄	—	53	96	no	—
176	C₂²×C₄	—	106	24	no	—
177	C₂²×C₄	—	106	288	no	—
178	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
179	C₂²×C₄	Q₈×D₅₃	1	8	yes	complemented
180	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
181	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
182	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
183	C₄⋊C₄	—	53	48	no	—
184	C₄⋊C₄	—	53	96	no	—
185	C₄⋊C₄	C₂³×D₅₃	1	48	yes	—
186	C₂²⋊C₄	—	53	96	no	—
187	C₄²	C₂³×D₅₃	1	48	yes	—
188	C₂³	—	53	48	no	—
189	C₂³	—	53	8	no	—
190	C₂³	—	106	64	no	—
191	C₂³	—	106	24	no	—
192	C₂³	—	106	24	no	—
193	C₂³	C₂²×D₂₁₂	1	4	yes	—
194	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
195	C₂³	#13133264	1	4	yes	—
196	C₂³	C₂⁴×D₅₃	1	3	yes	—
197	Q₈	—	106	192	no	—
198	Q₈	C₂²×D₂₁₂	1	64	yes	direct product factor
199	D₄	—	53	64	no	—
200	C₂×C₄	—	2	36	no	—
201	C₂×C₄	—	53	24	no	—
202	C₂×C₄	—	106	72	no	—
203	C₂×C₄	—	106	192	no	—
204	C₂×C₄	C₂²×D₂₁₂	1	72	yes	—
205	C₂×C₄	C₂×Q₈×D₅₃	1	24	yes	complemented
206	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
207	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
208	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
209	C₂²	—	53	16	no	—
210	C₂²	—	106	96	no	—
211	C₂²	—	106	8	no	—
212	C₂²	—	106	8	no	—
213	C₂²	Q₈×D₂₁₂	1	16	yes	direct product factor
214	C₂²	C₂³×D₂₁₂	1	6	yes	—
215	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
216	C₂²	C₂×#13133264	1	6	yes	—
217	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
218	C₄	—	2	12	no	—
219	C₄	—	106	24	no	—
220	C₄	C₂³×D₂₁₂	1	24	yes	—
221	C₄	C₂²×Q₈×D₅₃	1	8	yes	complemented
222	C₂	—	106	16	no	—
223	C₂	C₂×Q₈×D₂₁₂	1	12	yes	direct product factor
224	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
225	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
226	C₂	C₂²×#13133264	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₂₁₂	C₂²
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×D₄×Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
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×D₅₃	C₂	semidirect
2. C₂×Q₈×D₂₁₂	C₂	direct (N × H)
3. C₂²×#13133217	C₂	semidirect
4. C₂²×#13133197	C₂	semidirect
6. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
9. C₂²×Q₈×D₅₃	C₂²	semidirect
12. Q₈×D₂₁₂	C₂²	direct (N × H)
13. C₂×#13133217	C₂²	semidirect
14. C₂×#13133197	C₂²	semidirect
20. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
27. C₂×Q₈×D₅₃	C₂³	semidirect
28. C₂²×D₂₁₂	Q₈	direct (N × H)
33. #13133217	C₂³	semidirect
34. #13133197	C₂³	semidirect
39. C₂²×Q₈×C₅₃	D₄	semidirect
44. C₄×Q₈×C₅₃	C₂³	semidirect
50. C₂³×D₅₃	C₂×Q₈	semidirect
52. C₂×Q₈×C₅₃	C₂×D₄	semidirect
56. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
63. C₂×D₂₁₂	C₂×Q₈	direct (N × H)
81. C₂²×D₅₃	C₂²×Q₈	semidirect
83. Q₈×C₅₃	C₂²×D₄	semidirect
86. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
92. D₂₁₂	C₂²×Q₈	direct (N × H)
96. C₂²×C₅₃	D₄×Q₈	semidirect
104. C₂×D₅₃	C₂³×Q₈	semidirect
107. C₄×C₅₃	C₂³×Q₈	semidirect
113. C₂²×C₄×Q₈	D₅₃	semidirect
115. C₂×C₅₃	C₂×D₄×Q₈	semidirect
129. C₂×C₄×Q₈	C₂×D₅₃	semidirect
136. C₅₃	C₂²×D₄×Q₈	semidirect
142. C₂²×Q₈	D₂₁₂	direct (N × H)
154. C₄×Q₈	C₂²×D₅₃	semidirect
169. C₂×Q₈	C₂×D₂₁₂	direct (N × H)
179. C₂²×C₄	Q₈×D₅₃	semidirect
198. Q₈	C₂²×D₂₁₂	direct (N × H)
205. C₂×C₄	C₂×Q₈×D₅₃	semidirect
213. C₂²	Q₈×D₂₁₂	direct (N × H)
221. C₄	C₂²×Q₈×D₅₃	semidirect
223. C₂	C₂×Q₈×D₂₁₂	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×D₅₃	2
C₂×Q₈×D₂₁₂	2
C₂²×#13133217	2
C₂²×#13133197	2
C₂²×C₄×D₂₁₂	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×D₄×Q₈	53

Subgroups — C22×Q8×D212 #28291009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×D₂₁₂ #28291009