Subgroups — C₄×Q₈×#168 #12097908

Order 1280.

1070 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#168	C₂	1	2	yes	maximal
3	C₄⋊C₄×#168	C₂	1	6	yes	maximal
4	C₄×C₄⋊#168	C₂	1	3	yes	maximal
5	C₄²×#168	C₂	1	3	yes	maximal
6	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
7	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₄×C₅×C₄⋊C₄	C₄	1	6	yes	—
9	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
10	C₄³×C₅	C₂²	1	3	yes	—
11	Q₈×#168	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊#168	C₂²	1	6	yes	—
13	C₂×C₄×#168	C₂²	1	6	yes	—
14	C₂×C₄×#168	C₂²	1	3	yes	—
15	C₂×C₄×#168	C₂²	1	6	yes	—
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	#64744	C₂²	1	6	yes	—
18	C₄×C₈×Q₈	—	5	1	no	maximal, Sylow
19	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅×Q₈	C₈	1	16	yes	complemented
21	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₅	C₂³	1	3	yes	—
27	C₂×C₄²×C₅	C₂³	1	3	yes	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂²×#168	C₂³	1	6	yes	—
30	C₂²×#168	C₂³	1	2	yes	—
31	C₅×#95	C₂×C₄	1	6	yes	—
32	C₄⋊#168	C₂×C₄	1	24	yes	—
33	C₄×#168	—	2	6	no	—
34	C₄×#168	—	2	12	no	—
35	C₄×#168	C₂×C₄	1	24	yes	—
36	C₄×#168	Q₈	1	4	yes	direct product factor
37	C₂×C₈×Q₈	—	5	2	no	—
38	C₄²×Q₈	D₅	1	1	yes	characteristic
39	C₈×C₄⋊C₄	—	5	6	no	—
40	C₄×C₄⋊C₈	—	5	3	no	—
41	C₄²×C₈	—	5	3	no	—
42	C₂×C₅×Q₈	C₄²	1	4	yes	—
43	C₂×C₅×Q₈	C₂×C₈	1	8	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
49	C₅×C₄⋊C₄	C₄²	1	12	yes	—
50	C₅×C₄⋊C₄	C₂×C₈	1	24	yes	—
51	C₄²×C₅	—	2	12	no	—
52	C₄²×C₅	—	2	6	no	—
53	C₄²×C₅	C₂×C₈	1	24	yes	—
54	C₄²×C₅	C₂²×C₄	1	12	yes	—
55	C₄²×C₅	C₂×Q₈	1	2	yes	—
56	C₄²×C₅	#41	1	2	yes	—
57	C₂×#168	—	2	6	no	—
58	C₂×#168	—	2	6	no	—
59	C₂×#168	C₂²×C₄	1	12	yes	—
60	C₂×#168	C₂²×C₄	1	4	yes	—
61	C₂×#168	C₂×Q₈	1	4	yes	—
62	C₂×#168	#41	1	4	yes	—
63	C₂×C₄×Q₈	Dic₅	1	2	yes	—
64	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
65	C₈×Q₈	—	5	16	no	—
66	C₂×C₄⋊C₈	—	5	6	no	—
67	C₂×C₄×C₈	—	5	6	no	—
68	C₂×C₄×C₈	—	5	1	no	—
69	C₂×C₄×C₈	—	5	6	no	—
70	C₂×C₄×C₈	—	5	3	no	—
71	C₄×C₄⋊C₄	Dic₅	1	6	yes	—
72	C₄×C₄⋊C₄	C₂×D₅	1	3	yes	—
73	C₄³	C₂×D₅	1	3	yes	—
74	#336	—	5	6	no	—
75	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
76	C₅×Q₈	C₄×C₈	1	16	yes	complemented
77	C₂×C₄×C₅	—	2	6	no	—
78	C₂×C₄×C₅	—	2	6	no	—
79	C₂×C₄×C₅	—	2	6	no	—
80	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
81	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
82	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
83	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
84	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
85	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₅	#131	1	4	yes	—
87	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×#41	1	2	yes	—
91	#168	—	2	12	no	—
92	#168	C₄×Q₈	1	8	yes	direct product factor
93	C₂²×Q₈	C₂×Dic₅	1	1	yes	characteristic
94	C₂²×C₈	—	5	2	no	—
95	C₂²×C₈	—	5	6	no	—
96	C₄×Q₈	#168	1	16	yes	direct product factor
97	C₄×Q₈	C₄×D₅	1	8	yes	—
98	C₂×C₄⋊C₄	C₂×Dic₅	1	6	yes	—
99	C₂×C₄⋊C₄	C₂×Dic₅	1	3	yes	—
100	C₂×C₄²	C₂×Dic₅	1	6	yes	—
101	C₂×C₄²	C₂×Dic₅	1	6	yes	—
102	C₂×C₄²	C₂²×D₅	1	3	yes	—
103	C₂×C₄²	C₂²×D₅	1	3	yes	—
104	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
105	C₄⋊C₈	—	5	24	no	—
106	C₄×C₈	—	5	24	no	—
107	C₄×C₈	—	5	4	no	—
108	C₄×C₈	—	10	12	no	—
109	C₄×C₈	—	10	6	no	—
110	#95	C₂×Dic₅	1	6	yes	—
111	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
112	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂²×C₅	C₄×#41	1	1	yes	characteristic
114	C₂²×C₅	C₂³×C₈	1	2	yes	—
115	C₂²×C₅	C₂×#131	1	2	yes	—
116	C₄×C₅	—	2	6	no	—
117	C₄×C₅	—	2	12	no	—
118	C₄×C₅	C₂×C₄×C₈	1	12	yes	—
119	C₄×C₅	C₈×Q₈	1	8	yes	complemented
120	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
121	C₄×C₅	C₄×#41	1	2	yes	—
122	C₂×Q₈	C₂×#168	1	8	yes	—
123	C₂×Q₈	C₄×Dic₅	1	4	yes	—
124	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
125	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
126	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
127	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
128	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
129	C₂×C₈	—	5	4	no	—
130	C₂×C₈	—	5	4	no	—
131	C₂×C₈	—	5	4	no	—
132	C₂×C₈	—	5	12	no	—
133	C₂×C₈	—	10	6	no	—
134	C₂×C₈	—	10	6	no	—
135	C₄⋊C₄	C₂×#168	1	24	yes	—
136	C₄⋊C₄	C₄×Dic₅	1	12	yes	—
137	C₄²	—	2	6	no	—
138	C₄²	—	2	12	no	—
139	C₄²	C₂×#168	1	24	yes	—
140	C₄²	C₂×C₄×D₅	1	12	yes	—
141	C₄²	Q₈×D₅	1	2	yes	—
142	C₄²	#713	1	2	yes	—
143	C₂×C₅	C₄²×Q₈	1	1	yes	characteristic
144	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
145	C₂×C₅	C₄×#131	1	1	yes	characteristic
146	C₂×C₅	C₂×C₈×Q₈	1	2	yes	—
147	C₂×C₅	C₈×#41	1	2	yes	—
148	C₂³	C₂³×Dic₅	1	1	yes	characteristic
149	Q₈	C₄×#168	1	16	yes	direct product factor
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	C₂²×#168	1	12	yes	—
154	C₂×C₄	C₂²×#168	1	4	yes	—
155	C₂×C₄	C₂²×#168	1	12	yes	—
156	C₂×C₄	C₂×C₄×Dic₅	1	6	yes	—
157	C₂×C₄	C₂×C₄×Dic₅	1	6	yes	—
158	C₂×C₄	Q₈×Dic₅	1	4	yes	—
159	C₂×C₄	#4094	1	4	yes	—
160	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
161	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
162	C₂×C₄	C₂×#713	1	1	yes	characteristic
163	C₂×C₄	D₅×#41	1	2	yes	—
164	C₈	—	5	8	no	—
165	C₈	—	10	12	no	—
166	C₅	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
167	C₂²	C₂³×#168	1	2	yes	—
168	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
169	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
170	C₂²	C₂×#4094	1	2	yes	—
171	C₂²	#41×Dic₅	1	1	yes	characteristic
172	C₄	—	2	6	no	—
173	C₄	—	2	12	no	—
174	C₄	C₂×C₄×#168	1	12	yes	—
175	C₄	Q₈×#168	1	8	yes	direct product factor
176	C₄	C₄×Q₈×D₅	1	2	yes	—
177	C₄	C₄×#713	1	2	yes	—
178	C₂	C₂²×C₄×#168	1	1	yes	characteristic
179	C₂	C₂×Q₈×#168	1	2	yes	—
180	C₂	C₄×Q₈×Dic₅	1	1	yes	characteristic
181	C₂	C₄×#4094	1	1	yes	characteristic
182	C₂	#41×#168	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₄²×C₅×Q₈	C₂
socle, prime radical	C₂³×C₅	C₂³×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₄²×Q₈	D₅
hypocenter	C₅	C₄×C₈×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
11. Q₈×#168	C₄	direct (N × H)
20. C₄×C₅×Q₈	C₈	semidirect
36. C₄×#168	Q₈	direct (N × H)
76. C₅×Q₈	C₄×C₈	semidirect
92. #168	C₄×Q₈	direct (N × H)
96. C₄×Q₈	#168	direct (N × H)
119. C₄×C₅	C₈×Q₈	semidirect
149. Q₈	C₄×#168	direct (N × H)
166. C₅	C₄×C₈×Q₈	semidirect
175. C₄	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
C₄²×C₅×Q₈	2
C₂×Q₈×#168	2
C₄⋊C₄×#168	2
C₄×C₄⋊#168	2
C₄²×#168	2
C₄×C₈×Q₈	5

Subgroups — C4×Q8×#168 #12097908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×#168 #12097908