Subgroups — C₄×#15692162 #27169352

Order 12928.

752 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₀₁	C₂	1	2	yes	maximal
2	C₂×#15692162	C₂	1	4	yes	maximal
3	C₂²×C₄²×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₀₁	C₂²	1	4	yes	—
5	C₂×C₄×Dic₁₀₁	—	2	4	no	—
6	C₂×C₄×Dic₁₀₁	—	2	4	no	—
7	C₂×C₄×Dic₁₀₁	C₄	1	8	yes	complemented
8	#15692162	C₄	1	16	yes	direct product factor
9	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₁	—	2	4	no	—
12	C₂×C₄²×C₁₀₁	C₄	1	4	yes	complemented
13	C₂²×Dic₁₀₁	—	2	4	no	—
14	C₂²×Dic₁₀₁	—	2	4	no	—
15	C₂²×Dic₁₀₁	—	2	8	no	—
16	C₂²×Dic₁₀₁	C₂×C₄	1	16	yes	—
17	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
18	C₄×Dic₁₀₁	—	2	16	no	—
19	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₁	—	2	4	no	—
21	C₂²×C₄×C₁₀₁	—	2	4	no	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
28	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
29	C₄²×C₁₀₁	—	2	8	no	—
30	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
31	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
32	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
33	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
34	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
35	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
36	C₂³×C₁₀₁	#41	1	4	yes	—
37	C₂×C₄×C₁₀₁	—	2	4	no	—
38	C₂×C₄×C₁₀₁	—	2	4	no	—
39	C₂×C₄×C₁₀₁	—	2	16	no	—
40	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
41	C₂×C₄×C₁₀₁	C₄²	1	8	yes	complemented
42	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	8	yes	—
43	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
44	C₂×C₄×C₁₀₁	C₄⋊C₄	1	8	yes	—
45	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₁₀₁	—	2	8	no	—
47	C₂×Dic₁₀₁	—	2	8	no	—
48	C₂×Dic₁₀₁	—	2	16	no	—
49	C₂×Dic₁₀₁	—	2	16	no	—
50	C₂×Dic₁₀₁	C₄²	1	16	yes	complemented
51	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	—
52	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
53	C₂²×C₁₀₁	C₂×C₄²	1	2	yes	—
54	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
55	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₀₁	#117	1	4	yes	—
59	C₂²×C₁₀₁	#117	1	2	yes	—
60	C₂²×C₁₀₁	C₄×D₄	1	8	yes	—
61	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
62	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
63	C₄×C₁₀₁	—	2	8	no	—
64	C₄×C₁₀₁	#95	1	8	yes	complemented
65	Dic₁₀₁	—	2	16	no	—
66	C₂×C₁₀₁	C₄³	1	1	yes	characteristic
67	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
68	C₂×C₁₀₁	#376	1	1	yes	characteristic
69	C₂×C₁₀₁	C₄×C₂²⋊C₄	1	2	yes	—
70	C₂×C₁₀₁	C₄×C₂²⋊C₄	1	4	yes	—
71	C₂×C₁₀₁	C₄×C₄⋊C₄	1	2	yes	—
72	C₂×C₁₀₁	C₄×C₄⋊C₄	1	4	yes	—
73	C₄×#95	—	101	1	no	maximal, Sylow
74	C₁₀₁	C₄×#95	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄²	—	101	2	no	—
76	C₂²×C₄²	D₁₀₁	1	1	yes	characteristic
77	C₂×#95	—	101	4	no	—
78	C₂³×C₄	—	101	4	no	—
79	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
80	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	—	101	8	no	—
83	C₂×C₄²	—	202	4	no	—
84	C₂×C₄²	—	202	4	no	—
85	C₂×C₄²	Dic₁₀₁	1	4	yes	complemented
86	#95	—	101	16	no	—
87	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	101	16	no	—
91	C₂²×C₄	—	101	8	no	—
92	C₂²×C₄	—	202	4	no	—
93	C₂²×C₄	—	202	4	no	—
94	C₂²×C₄	—	202	8	no	—
95	C₂²×C₄	Dic₂₀₂	1	2	yes	—
96	C₂²×C₄	C₄×D₁₀₁	1	8	yes	—
97	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
98	C₂²×C₄	D₄₀₄	1	2	yes	—
99	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
100	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
102	C₄²	—	2	8	no	—
103	C₄²	—	202	16	no	—
104	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
105	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
106	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
108	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
109	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
110	C₂³	#13108074	1	4	yes	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	16	no	—
114	C₂×C₄	—	101	16	no	—
115	C₂×C₄	—	202	8	no	—
116	C₂×C₄	—	202	8	no	—
117	C₂×C₄	—	202	16	no	—
118	C₂×C₄	—	202	16	no	—
119	C₂×C₄	C₄×Dic₁₀₁	1	4	yes	—
120	C₂×C₄	C₄×Dic₁₀₁	1	8	yes	complemented
121	C₂×C₄	#13108051	1	8	yes	—
122	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
123	C₂×C₄	#13108053	1	8	yes	—
124	C₂×C₄	C₄⋊Dic₁₀₁	1	4	yes	—
125	C₂²	C₄²×D₁₀₁	1	4	yes	—
126	C₂²	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
127	C₂²	C₂×C₄×Dic₁₀₁	1	2	yes	—
128	C₂²	C₂×#13108051	1	2	yes	—
129	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
130	C₂²	C₂×#13108053	1	2	yes	—
131	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
132	C₂²	#15692212	1	4	yes	—
133	C₂²	#15692214	1	2	yes	—
134	C₂²	C₄×C₂²⋊D₁₀₁	1	8	yes	—
135	C₂²	C₄×D₄₀₄	1	4	yes	—
136	C₂²	C₄×Dic₂₀₂	1	4	yes	—
137	C₄	—	2	8	no	—
138	C₄	—	202	16	no	—
139	C₄	#15692162	1	8	yes	direct product factor
140	C₂	C₄²×Dic₁₀₁	1	1	yes	characteristic
141	C₂	C₂×#15692162	1	1	yes	characteristic
142	C₂	#20265023	1	1	yes	characteristic
143	C₂	C₄×C₂²⋊Dic₁₀₁	1	2	yes	—
144	C₂	C₄×#13108051	1	4	yes	—
145	C₂	C₄×C₄⋊Dic₁₀₁	1	2	yes	—
146	C₂	C₄×#13108053	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₀₁
commutator	C₂×C₁₀₁	C₄³
Frattini	C₂⁴	C₂²×D₁₀₁
Fitting	C₂²×C₄²×C₁₀₁	C₂
socle, prime radical, cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂²×C₄²	D₁₀₁
hypocenter	C₁₀₁	C₄×#95
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×C₄×Dic₁₀₁	C₄	semidirect
8. #15692162	C₄	direct (N × H)
12. C₂×C₄²×C₁₀₁	C₄	semidirect
41. C₂×C₄×C₁₀₁	C₄²	semidirect
50. C₂×Dic₁₀₁	C₄²	semidirect
64. C₄×C₁₀₁	#95	semidirect
74. C₁₀₁	C₄×#95	semidirect
85. C₂×C₄²	Dic₁₀₁	semidirect
120. C₂×C₄	C₄×Dic₁₀₁	semidirect
139. C₄	#15692162	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×Dic₁₀₁	2
C₂×#15692162	2
C₂²×C₄²×C₁₀₁	2
C₄×#95	101

Subgroups — C4×#15692162 #27169352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692162 #27169352