Subgroups — C₂²×Q₈×#168 #12963652

Order 1280.

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	C₂²×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#168	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#168	C₂	1	3	yes	maximal
4	C₂²×C₄×#168	C₂	1	3	yes	maximal
5	C₂³×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₄	1	12	yes	—
7	C₂×C₄×C₅×Q₈	C₂²	1	12	yes	—
8	C₂²×C₅×C₄⋊C₄	C₄	1	3	yes	—
9	C₂²×C₄²×C₅	C₂²	1	3	yes	—
10	C₂³×#168	C₂²	1	3	yes	—
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	Q₈×#168	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊#168	C₂²	1	36	yes	—
14	C₂×C₄×#168	C₂²	1	36	yes	—
15	C₂²×C₈×Q₈	—	5	1	no	maximal, Sylow
16	C₂²×C₅×Q₈	C₈	1	16	yes	complemented
17	C₂²×C₅×Q₈	C₂×C₄	1	12	yes	—
18	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₄×C₅×Q₈	C₂×C₄	1	48	yes	—
22	C₄×C₅×Q₈	C₂³	1	16	yes	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	36	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	18	yes	—
25	C₂×C₄²×C₅	C₂³	1	18	yes	—
26	C₂²×#168	—	2	12	no	—
27	C₂²×#168	Q₈	1	8	yes	direct product factor
28	C₂²×#168	C₂³	1	18	yes	—
29	C₂²×#168	C₂³	1	6	yes	—
30	C₄⋊#168	C₂³	1	48	yes	—
31	C₄×#168	C₂³	1	48	yes	—
32	C₂²×C₄×Q₈	D₅	1	1	yes	characteristic
33	C₂×C₈×Q₈	—	5	24	no	—
34	C₂²×C₄⋊C₈	—	5	3	no	—
35	C₂²×C₄×C₈	—	5	3	no	—
36	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₅×Q₈	C₂×C₈	1	96	yes	complemented
38	C₂×C₅×Q₈	C₂²×C₄	1	16	yes	—
39	C₂²×C₄×C₅	—	2	12	no	—
40	C₂²×C₄×C₅	C₂×C₈	1	24	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	18	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	18	yes	—
44	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₅	#41	1	4	yes	—
46	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
47	C₅×C₄⋊C₄	C₂²×C₄	1	48	yes	—
48	C₄²×C₅	C₂²×C₄	1	36	yes	—
49	C₄²×C₅	C₂⁴	1	12	yes	—
50	C₂×#168	—	2	36	no	—
51	C₂×#168	C₂×Q₈	1	24	yes	direct product factor
52	C₂×#168	C₂⁴	1	12	yes	—
53	C₂×#168	C₂⁴	1	4	yes	—
54	C₂³×Q₈	Dic₅	1	1	yes	characteristic
55	C₂³×C₈	—	5	1	no	—
56	C₂³×C₈	—	5	3	no	—
57	C₂×C₄×Q₈	Dic₅	1	12	yes	—
58	C₂×C₄×Q₈	C₂×D₅	1	12	yes	—
59	C₂²×C₄⋊C₄	Dic₅	1	3	yes	—
60	C₂²×C₄²	C₂×D₅	1	3	yes	—
61	C₈×Q₈	—	5	64	no	—
62	C₂×C₄⋊C₈	—	5	36	no	—
63	C₂×C₄×C₈	—	5	36	no	—
64	C₂³×C₅	C₄×Q₈	1	4	yes	—
65	C₂³×C₅	C₂²×C₈	1	4	yes	—
66	C₂³×C₅	#131	1	4	yes	—
67	C₂³×C₅	C₂³×C₄	1	3	yes	—
68	C₅×Q₈	C₂²×C₈	1	64	yes	complemented
69	C₂×C₄×C₅	—	2	36	no	—
70	C₂×C₄×C₅	C₄×Q₈	1	12	yes	—
71	C₂×C₄×C₅	C₂²×C₈	1	72	yes	—
72	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
73	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
74	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
75	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
76	C₂×C₄×C₅	C₂×#41	1	6	yes	—
77	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
78	#168	—	2	12	no	—
79	#168	C₂²×Q₈	1	8	yes	direct product factor
80	C₂²×Q₈	#168	1	16	yes	direct product factor
81	C₂²×Q₈	C₂×Dic₅	1	12	yes	—
82	C₂³×C₄	C₂×Dic₅	1	3	yes	—
83	C₂³×C₄	C₂×Dic₅	1	3	yes	—
84	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
85	C₂²×C₈	—	5	8	no	—
86	C₂²×C₈	—	5	6	no	—
87	C₂²×C₈	—	5	18	no	—
88	C₂²×C₈	—	10	12	no	—
89	C₄×Q₈	C₂×Dic₅	1	48	yes	—
90	C₄×Q₈	C₂²×D₅	1	16	yes	—
91	C₂×C₄⋊C₄	C₂×Dic₅	1	36	yes	—
92	C₂×C₄²	C₂×Dic₅	1	18	yes	—
93	C₂×C₄²	C₂²×D₅	1	18	yes	—
94	C₄⋊C₈	—	5	48	no	—
95	C₄×C₈	—	5	48	no	—
96	C₂²×C₅	C₈×Q₈	1	16	yes	complemented
97	C₂²×C₅	C₂×C₄×Q₈	1	6	yes	—
98	C₂²×C₅	C₂³×C₈	1	6	yes	—
99	C₂²×C₅	C₂×#131	1	6	yes	—
100	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₅	—	2	12	no	—
102	C₄×C₅	C₂×C₄×Q₈	1	6	yes	—
103	C₄×C₅	C₂³×C₈	1	24	yes	—
104	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
105	C₄×C₅	C₂²×#41	1	1	yes	characteristic
106	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
107	C₂×Q₈	C₂×#168	1	96	yes	direct product factor
108	C₂×Q₈	C₂²×Dic₅	1	16	yes	—
109	C₂²×C₄	—	2	12	no	—
110	C₂²×C₄	C₂×#168	1	24	yes	—
111	C₂²×C₄	Q₈×D₅	1	4	yes	—
112	C₂²×C₄	#713	1	4	yes	—
113	C₂²×C₄	C₂²×Dic₅	1	18	yes	—
114	C₂²×C₄	C₂²×Dic₅	1	18	yes	—
115	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
116	C₂²×C₄	C₂³×D₅	1	3	yes	—
117	C₂×C₈	—	5	24	no	—
118	C₂×C₈	—	5	4	no	—
119	C₂×C₈	—	5	12	no	—
120	C₂×C₈	—	10	36	no	—
121	C₄⋊C₄	C₂²×Dic₅	1	48	yes	—
122	C₄²	C₂²×Dic₅	1	36	yes	—
123	C₄²	C₂³×D₅	1	12	yes	—
124	C₂×C₅	C₂×C₈×Q₈	1	12	yes	complemented
125	C₂×C₅	C₂²×C₄×Q₈	1	1	yes	characteristic
126	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
127	C₂×C₅	C₂²×#131	1	1	yes	characteristic
128	C₂³	C₂²×#168	1	4	yes	—
129	C₂³	Q₈×Dic₅	1	4	yes	—
130	C₂³	#4094	1	4	yes	—
131	C₂³	C₂³×Dic₅	1	3	yes	—
132	Q₈	C₂²×#168	1	64	yes	direct product factor
133	C₂×C₄	—	2	36	no	—
134	C₂×C₄	C₂²×#168	1	72	yes	—
135	C₂×C₄	Q₈×Dic₅	1	12	yes	—
136	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
137	C₂×C₄	C₂×#713	1	6	yes	—
138	C₂×C₄	C₂³×Dic₅	1	12	yes	—
139	C₂×C₄	C₂³×Dic₅	1	12	yes	—
140	C₂×C₄	C₂³×Dic₅	1	3	yes	—
141	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
142	C₈	—	5	8	no	—
143	C₈	—	10	12	no	—
144	C₅	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
145	C₂²	Q₈×#168	1	16	yes	direct product factor
146	C₂²	C₂³×#168	1	6	yes	—
147	C₂²	C₂×Q₈×Dic₅	1	6	yes	—
148	C₂²	C₂×#4094	1	6	yes	—
149	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂³×#168	1	24	yes	—
152	C₄	C₂×Q₈×Dic₅	1	6	yes	—
153	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
154	C₄	C₂²×#713	1	1	yes	characteristic
155	C₂	C₂×Q₈×#168	1	12	yes	direct product factor
156	C₂	C₂⁴×#168	1	1	yes	characteristic
157	C₂	C₂²×Q₈×Dic₅	1	1	yes	characteristic
158	C₂	C₂²×#4094	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂×C₅	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×Q₈	C₂
socle, prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₅
hypocenter	C₅	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×#168	C₂	direct (N × H)
12. Q₈×#168	C₂²	direct (N × H)
16. C₂²×C₅×Q₈	C₈	semidirect
27. C₂²×#168	Q₈	direct (N × H)
37. C₂×C₅×Q₈	C₂×C₈	semidirect
51. C₂×#168	C₂×Q₈	direct (N × H)
68. C₅×Q₈	C₂²×C₈	semidirect
79. #168	C₂²×Q₈	direct (N × H)
80. C₂²×Q₈	#168	direct (N × H)
96. C₂²×C₅	C₈×Q₈	semidirect
107. C₂×Q₈	C₂×#168	direct (N × H)
124. C₂×C₅	C₂×C₈×Q₈	semidirect
132. Q₈	C₂²×#168	direct (N × H)
144. C₅	C₂²×C₈×Q₈	semidirect
145. C₂²	Q₈×#168	direct (N × H)
155. C₂	C₂×Q₈×#168	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₅×Q₈	2
C₂×Q₈×#168	2
C₂²×C₄⋊#168	2
C₂²×C₄×#168	2
C₂²×C₈×Q₈	5

Subgroups — C22×Q8×#168 #12963652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×#168 #12963652