Subgroups — C₂²×Q₈×#88322 #28248444

Order 13568.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88322	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊#88322	C₂	1	3	yes	maximal
3	C₂²×C₄×#88322	C₂	1	3	yes	maximal
4	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	3	yes	—
6	C₂³×#88322	C₂²	1	1	yes	characteristic
7	Q₈×#88322	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊#88322	C₂²	1	36	yes	—
9	C₂×C₄×#88322	C₂²	1	36	yes	—
10	C₂³×Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₄	1	12	yes	—
12	C₂×C₄×Q₈×C₅₃	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂²×#88322	—	2	12	no	—
16	C₂²×#88322	Q₈	1	8	yes	direct product factor
17	C₂²×#88322	C₂³	1	18	yes	—
18	C₂²×#88322	C₂³	1	6	yes	—
19	C₄⋊#88322	C₂³	1	48	yes	—
20	C₄×#88322	C₂³	1	48	yes	—
21	C₂²×Q₈×C₅₃	C₈	1	16	yes	complemented
22	C₂²×Q₈×C₅₃	C₂×C₄	1	12	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	C₂×C₄	1	48	yes	—
27	C₄×Q₈×C₅₃	C₂³	1	16	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	36	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	18	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	18	yes	—
31	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	C₂×C₈	1	96	yes	complemented
33	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
34	C₂²×C₄×C₅₃	—	2	12	no	—
35	C₂²×C₄×C₅₃	C₂×C₈	1	24	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	18	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	18	yes	—
39	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₅₃	#41	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
42	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
43	C₄²×C₅₃	C₂²×C₄	1	36	yes	—
44	C₄²×C₅₃	C₂⁴	1	12	yes	—
45	C₂×#88322	—	2	36	no	—
46	C₂×#88322	C₂×Q₈	1	24	yes	direct product factor
47	C₂×#88322	C₂⁴	1	12	yes	—
48	C₂×#88322	C₂⁴	1	4	yes	—
49	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
50	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
51	C₂³×C₅₃	#131	1	4	yes	—
52	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
53	Q₈×C₅₃	C₂²×C₈	1	64	yes	complemented
54	C₂×C₄×C₅₃	—	2	36	no	—
55	C₂×C₄×C₅₃	C₄×Q₈	1	12	yes	—
56	C₂×C₄×C₅₃	C₂²×C₈	1	72	yes	—
57	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
58	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
59	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
60	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
61	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
62	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
63	#88322	—	2	12	no	—
64	#88322	C₂²×Q₈	1	8	yes	direct product factor
65	C₂²×C₈×Q₈	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	C₈×Q₈	1	16	yes	complemented
67	C₂²×C₅₃	C₂×C₄×Q₈	1	6	yes	—
68	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
69	C₂²×C₅₃	C₂×#131	1	6	yes	—
70	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
71	C₄×C₅₃	—	2	12	no	—
72	C₄×C₅₃	C₂×C₄×Q₈	1	6	yes	—
73	C₄×C₅₃	C₂³×C₈	1	24	yes	—
74	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
76	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic
77	C₂×C₈×Q₈	—	53	24	no	—
78	C₂²×C₄⋊C₈	—	53	3	no	—
79	C₂²×C₄×C₈	—	53	3	no	—
80	C₂×C₅₃	C₂×C₈×Q₈	1	12	yes	complemented
81	C₂×C₅₃	C₂²×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
83	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
84	C₂³×Q₈	Dic₅₃	1	1	yes	characteristic
85	C₂³×C₈	—	53	1	no	—
86	C₂³×C₈	—	53	3	no	—
87	C₂×C₄×Q₈	Dic₅₃	1	12	yes	—
88	C₂×C₄×Q₈	C₂×D₅₃	1	12	yes	—
89	C₂²×C₄⋊C₄	Dic₅₃	1	3	yes	—
90	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
91	C₈×Q₈	—	53	64	no	—
92	C₂×C₄⋊C₈	—	53	36	no	—
93	C₂×C₄×C₈	—	53	36	no	—
94	C₅₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	#88322	1	16	yes	direct product factor
96	C₂²×Q₈	C₂×Dic₅₃	1	12	yes	—
97	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
98	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
99	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂²×C₈	—	53	8	no	—
101	C₂²×C₈	—	53	6	no	—
102	C₂²×C₈	—	53	18	no	—
103	C₂²×C₈	—	106	12	no	—
104	C₄×Q₈	C₂×Dic₅₃	1	48	yes	—
105	C₄×Q₈	C₂²×D₅₃	1	16	yes	—
106	C₂×C₄⋊C₄	C₂×Dic₅₃	1	36	yes	—
107	C₂×C₄²	C₂×Dic₅₃	1	18	yes	—
108	C₂×C₄²	C₂²×D₅₃	1	18	yes	—
109	C₄⋊C₈	—	53	48	no	—
110	C₄×C₈	—	53	48	no	—
111	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
112	C₂×Q₈	C₂×#88322	1	96	yes	direct product factor
113	C₂×Q₈	C₂²×Dic₅₃	1	16	yes	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	C₂×#88322	1	24	yes	—
116	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
117	C₂²×C₄	#11719949	1	4	yes	—
118	C₂²×C₄	C₂²×Dic₅₃	1	18	yes	—
119	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
120	C₂²×C₄	C₂²×Dic₅₃	1	18	yes	—
121	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
122	C₂×C₈	—	53	24	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	53	12	no	—
125	C₂×C₈	—	106	36	no	—
126	C₄⋊C₄	C₂²×Dic₅₃	1	48	yes	—
127	C₄²	C₂²×Dic₅₃	1	36	yes	—
128	C₄²	C₂³×D₅₃	1	12	yes	—
129	C₂³	Q₈×Dic₅₃	1	4	yes	—
130	C₂³	C₂²×#88322	1	4	yes	—
131	C₂³	#13133224	1	4	yes	—
132	C₂³	C₂³×Dic₅₃	1	3	yes	—
133	Q₈	C₂²×#88322	1	64	yes	direct product factor
134	C₂×C₄	—	2	36	no	—
135	C₂×C₄	Q₈×Dic₅₃	1	12	yes	—
136	C₂×C₄	C₂²×#88322	1	72	yes	—
137	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
138	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
139	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
140	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
141	C₂×C₄	C₂×#11719949	1	6	yes	—
142	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
143	C₈	—	53	8	no	—
144	C₈	—	106	12	no	—
145	C₂²	Q₈×#88322	1	16	yes	direct product factor
146	C₂²	C₂×Q₈×Dic₅₃	1	6	yes	—
147	C₂²	C₂³×#88322	1	6	yes	—
148	C₂²	C₂×#13133224	1	6	yes	—
149	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂×Q₈×Dic₅₃	1	6	yes	—
152	C₄	C₂³×#88322	1	24	yes	—
153	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
154	C₄	C₂²×#11719949	1	1	yes	characteristic
155	C₂	C₂×Q₈×#88322	1	12	yes	direct product factor
156	C₂	C₂²×Q₈×Dic₅₃	1	1	yes	characteristic
157	C₂	C₂⁴×#88322	1	1	yes	characteristic
158	C₂	C₂²×#13133224	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#88322	C₂	direct (N × H)
7. Q₈×#88322	C₂²	direct (N × H)
16. C₂²×#88322	Q₈	direct (N × H)
21. C₂²×Q₈×C₅₃	C₈	semidirect
32. C₂×Q₈×C₅₃	C₂×C₈	semidirect
46. C₂×#88322	C₂×Q₈	direct (N × H)
53. Q₈×C₅₃	C₂²×C₈	semidirect
64. #88322	C₂²×Q₈	direct (N × H)
66. C₂²×C₅₃	C₈×Q₈	semidirect
80. C₂×C₅₃	C₂×C₈×Q₈	semidirect
94. C₅₃	C₂²×C₈×Q₈	semidirect
95. C₂²×Q₈	#88322	direct (N × H)
112. C₂×Q₈	C₂×#88322	direct (N × H)
133. Q₈	C₂²×#88322	direct (N × H)
145. C₂²	Q₈×#88322	direct (N × H)
155. C₂	C₂×Q₈×#88322	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×#88322	2
C₂²×C₄⋊#88322	2
C₂²×C₄×#88322	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×C₈×Q₈	53

Subgroups — C22×Q8×#88322 #28248444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×#88322 #28248444