Subgroups — C₂²×Q₈×#1055 #16727271

Order 3328.

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×#1055	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊#1055	C₂	1	3	yes	maximal
3	C₂²×C₄×#1055	C₂	1	3	yes	maximal
4	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	12	yes	—
7	C₂×C₄×Q₈×C₁₃	C₂²	1	12	yes	—
8	C₂²×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
10	C₂³×#1055	C₂²	1	3	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	Q₈×#1055	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊#1055	C₂²	1	36	yes	—
14	C₂×C₄×#1055	C₂²	1	36	yes	—
15	C₂²×Q₈×C₁₃	C₈	1	16	yes	complemented
16	C₂²×Q₈×C₁₃	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₂×C₄	1	48	yes	—
21	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	36	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	18	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	18	yes	—
25	C₂²×#1055	—	2	12	no	—
26	C₂²×#1055	Q₈	1	8	yes	direct product factor
27	C₂²×#1055	C₂³	1	18	yes	—
28	C₂²×#1055	C₂³	1	6	yes	—
29	C₄⋊#1055	C₂³	1	48	yes	—
30	C₄×#1055	C₂³	1	48	yes	—
31	C₂²×C₈×Q₈	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₁₃	C₂×C₈	1	96	yes	complemented
34	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
35	C₂²×C₄×C₁₃	—	2	12	no	—
36	C₂²×C₄×C₁₃	C₂×C₈	1	24	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	18	yes	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	18	yes	—
40	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₁₃	#41	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
43	C₁₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
44	C₄²×C₁₃	C₂²×C₄	1	36	yes	—
45	C₄²×C₁₃	C₂⁴	1	12	yes	—
46	C₂×#1055	—	2	36	no	—
47	C₂×#1055	C₂×Q₈	1	24	yes	direct product factor
48	C₂×#1055	C₂⁴	1	12	yes	—
49	C₂×#1055	C₂⁴	1	4	yes	—
50	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
51	C₂×C₈×Q₈	—	13	24	no	—
52	C₂²×C₄⋊C₈	—	13	3	no	—
53	C₂²×C₄×C₈	—	13	3	no	—
54	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
55	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
56	C₂³×C₁₃	#131	1	4	yes	—
57	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
58	Q₈×C₁₃	C₂²×C₈	1	64	yes	complemented
59	C₂×C₄×C₁₃	—	2	36	no	—
60	C₂×C₄×C₁₃	C₄×Q₈	1	12	yes	—
61	C₂×C₄×C₁₃	C₂²×C₈	1	72	yes	—
62	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
63	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
64	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
65	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
66	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
67	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
68	#1055	—	2	12	no	—
69	#1055	C₂²×Q₈	1	8	yes	direct product factor
70	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
71	C₂³×C₈	—	13	1	no	—
72	C₂³×C₈	—	13	3	no	—
73	C₂×C₄×Q₈	Dic₁₃	1	12	yes	—
74	C₂×C₄×Q₈	C₂×D₁₃	1	12	yes	—
75	C₂²×C₄⋊C₄	Dic₁₃	1	3	yes	—
76	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
77	C₈×Q₈	—	13	64	no	—
78	C₂×C₄⋊C₈	—	13	36	no	—
79	C₂×C₄×C₈	—	13	36	no	—
80	C₂²×C₁₃	C₈×Q₈	1	16	yes	complemented
81	C₂²×C₁₃	C₂×C₄×Q₈	1	6	yes	—
82	C₂²×C₁₃	C₂³×C₈	1	6	yes	—
83	C₂²×C₁₃	C₂×#131	1	6	yes	—
84	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
85	C₄×C₁₃	—	2	12	no	—
86	C₄×C₁₃	C₂×C₄×Q₈	1	6	yes	—
87	C₄×C₁₃	C₂³×C₈	1	24	yes	—
88	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
89	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
90	C₂²×Q₈	#1055	1	16	yes	direct product factor
91	C₂²×Q₈	C₂×Dic₁₃	1	12	yes	—
92	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
93	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
94	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	C₂²×C₈	—	13	8	no	—
96	C₂²×C₈	—	13	6	no	—
97	C₂²×C₈	—	13	18	no	—
98	C₂²×C₈	—	26	12	no	—
99	C₄×Q₈	C₂×Dic₁₃	1	48	yes	—
100	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
101	C₂×C₄⋊C₄	C₂×Dic₁₃	1	36	yes	—
102	C₂×C₄²	C₂×Dic₁₃	1	18	yes	—
103	C₂×C₄²	C₂²×D₁₃	1	18	yes	—
104	C₄⋊C₈	—	13	48	no	—
105	C₄×C₈	—	13	48	no	—
106	C₂×C₁₃	C₂×C₈×Q₈	1	12	yes	complemented
107	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
108	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
109	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
110	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
111	C₂×Q₈	C₂×#1055	1	96	yes	direct product factor
112	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
113	C₂²×C₄	—	2	12	no	—
114	C₂²×C₄	C₂×#1055	1	24	yes	—
115	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
116	C₂²×C₄	#6131	1	4	yes	—
117	C₂²×C₄	C₂²×Dic₁₃	1	18	yes	—
118	C₂²×C₄	C₂²×Dic₁₃	1	18	yes	—
119	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
120	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
121	C₂×C₈	—	13	24	no	—
122	C₂×C₈	—	13	4	no	—
123	C₂×C₈	—	13	12	no	—
124	C₂×C₈	—	26	36	no	—
125	C₄⋊C₄	C₂²×Dic₁₃	1	48	yes	—
126	C₄²	C₂²×Dic₁₃	1	36	yes	—
127	C₄²	C₂³×D₁₃	1	12	yes	—
128	C₁₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₂²×#1055	1	4	yes	—
130	C₂³	Q₈×Dic₁₃	1	4	yes	—
131	C₂³	#88202	1	4	yes	—
132	C₂³	C₂³×Dic₁₃	1	3	yes	—
133	Q₈	C₂²×#1055	1	64	yes	direct product factor
134	C₂×C₄	—	2	36	no	—
135	C₂×C₄	C₂²×#1055	1	72	yes	—
136	C₂×C₄	Q₈×Dic₁₃	1	12	yes	—
137	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
138	C₂×C₄	C₂×#6131	1	6	yes	—
139	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
140	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
141	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
142	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
143	C₈	—	13	8	no	—
144	C₈	—	26	12	no	—
145	C₂²	Q₈×#1055	1	16	yes	direct product factor
146	C₂²	C₂³×#1055	1	6	yes	—
147	C₂²	C₂×Q₈×Dic₁₃	1	6	yes	—
148	C₂²	C₂×#88202	1	6	yes	—
149	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂³×#1055	1	24	yes	—
152	C₄	C₂×Q₈×Dic₁₃	1	6	yes	—
153	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
154	C₄	C₂²×#6131	1	1	yes	characteristic
155	C₂	C₂×Q₈×#1055	1	12	yes	direct product factor
156	C₂	C₂²×Q₈×Dic₁₃	1	1	yes	characteristic
157	C₂	C₂⁴×#1055	1	1	yes	characteristic
158	C₂	C₂²×#88202	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
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×#1055	C₂	direct (N × H)
12. Q₈×#1055	C₂²	direct (N × H)
15. C₂²×Q₈×C₁₃	C₈	semidirect
26. C₂²×#1055	Q₈	direct (N × H)
33. C₂×Q₈×C₁₃	C₂×C₈	semidirect
47. C₂×#1055	C₂×Q₈	direct (N × H)
58. Q₈×C₁₃	C₂²×C₈	semidirect
69. #1055	C₂²×Q₈	direct (N × H)
80. C₂²×C₁₃	C₈×Q₈	semidirect
90. C₂²×Q₈	#1055	direct (N × H)
106. C₂×C₁₃	C₂×C₈×Q₈	semidirect
111. C₂×Q₈	C₂×#1055	direct (N × H)
128. C₁₃	C₂²×C₈×Q₈	semidirect
133. Q₈	C₂²×#1055	direct (N × H)
145. C₂²	Q₈×#1055	direct (N × H)
155. C₂	C₂×Q₈×#1055	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×#1055	2
C₂²×C₄⋊#1055	2
C₂²×C₄×#1055	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×C₈×Q₈	13

Subgroups — C22×Q8×#1055 #16727271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×#1055 #16727271