Subgroups — #27169362

Order 12928.

656 conjugacy classes of subgroups.

The subgroup lattice has 164 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	—	2	8	no	—
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	8	no	—
14	C₂²×Dic₁₀₁	—	2	4	no	—
15	C₂²×Dic₁₀₁	—	2	4	no	—
16	C₂²×Dic₁₀₁	—	2	8	no	—
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	4	yes	—
23	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
26	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
28	C₂²×C₄×C₁₀₁	Q₈	1	4	yes	—
29	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
30	C₄²×C₁₀₁	—	2	8	no	—
31	C₂³×C₁₀₁	C₂²×C₄	1	2	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×D₄	1	2	yes	—
36	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
37	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
38	C₂³×C₁₀₁	#41	1	4	yes	—
39	C₂×C₄×C₁₀₁	—	2	4	no	—
40	C₂×C₄×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	8	no	—
42	C₂×C₄×C₁₀₁	—	2	8	no	—
43	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
44	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	8	yes	—
45	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×C₁₀₁	C₄⋊C₄	1	8	yes	—
47	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₁₀₁	C₄⋊C₄	1	8	yes	complemented
49	C₂×Dic₁₀₁	—	2	8	no	—
50	C₂×Dic₁₀₁	—	2	16	no	—
51	C₂×Dic₁₀₁	—	4	8	no	—
52	C₂×Dic₁₀₁	—	4	4	no	—
53	C₂×Dic₁₀₁	—	4	4	no	—
54	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
60	C₂²×C₁₀₁	#117	1	2	yes	—
61	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
62	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
63	C₂²×C₁₀₁	#121	1	4	yes	—
64	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
65	C₂²×C₁₀₁	#124	1	2	yes	—
66	C₂²×C₁₀₁	#125	1	2	yes	—
67	C₂²×C₁₀₁	#127	1	1	yes	characteristic
68	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
69	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₁₀₁	—	2	8	no	—
71	C₄×C₁₀₁	#95	1	8	yes	complemented
72	Dic₁₀₁	—	4	8	no	—
73	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
74	C₂×C₁₀₁	C₄×C₄⋊C₄	1	2	yes	—
75	C₂×C₁₀₁	#380	1	2	yes	—
76	C₂×C₁₀₁	#382	1	1	yes	characteristic
77	C₂×C₁₀₁	#384	1	1	yes	characteristic
78	C₂×C₁₀₁	#389	1	4	yes	—
79	C₂×C₁₀₁	#390	1	2	yes	—
80	C₂×C₁₀₁	#391	1	2	yes	—
81	C₄⋊#95	—	101	1	no	maximal, Sylow
82	C₁₀₁	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄⋊C₄	—	101	2	no	—
84	C₂²×C₄²	D₁₀₁	1	1	yes	characteristic
85	C₂×#95	—	101	4	no	—
86	C₂³×C₄	—	101	4	no	—
87	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
88	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	101	8	no	—
90	C₂×C₄⋊C₄	—	202	4	no	—
91	C₂×C₄⋊C₄	—	202	4	no	—
92	C₂×C₄²	—	2	4	no	—
93	C₂×C₄²	Dic₁₀₁	1	4	yes	complemented
94	#95	—	202	8	no	—
95	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
96	C₂²×C₄	—	2	4	no	—
97	C₂²×C₄	—	2	4	no	—
98	C₂²×C₄	—	101	8	no	—
99	C₂²×C₄	—	202	4	no	—
100	C₂²×C₄	—	202	8	no	—
101	C₂²×C₄	—	202	4	no	—
102	C₂²×C₄	—	202	8	no	—
103	C₂²×C₄	Dic₂₀₂	1	4	yes	—
104	C₂²×C₄	Dic₂₀₂	1	2	yes	—
105	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
106	C₂²×C₄	D₄₀₄	1	4	yes	—
107	C₂²×C₄	D₄₀₄	1	2	yes	—
108	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
109	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
111	C₄⋊C₄	—	202	16	no	—
112	C₄²	—	2	8	no	—
113	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
114	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
115	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
116	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
117	C₂³	C₂×D₄₀₄	1	2	yes	—
118	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
119	C₂³	C₂×Dic₂₀₂	1	2	yes	—
120	C₂³	#13108074	1	4	yes	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	—	404	8	no	—
126	C₂×C₄	—	404	4	no	—
127	C₂×C₄	—	404	4	no	—
128	C₂×C₄	—	202	8	no	—
129	C₂×C₄	—	202	16	no	—
130	C₂×C₄	C₄×Dic₁₀₁	1	4	yes	—
131	C₂×C₄	#13108051	1	8	yes	—
132	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
133	C₂×C₄	#13108053	1	8	yes	—
134	C₂×C₄	C₄⋊Dic₁₀₁	1	4	yes	—
135	C₂×C₄	C₄⋊Dic₁₀₁	1	8	yes	complemented
136	C₂²	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
137	C₂²	C₂×#13108051	1	2	yes	—
138	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
139	C₂²	C₂×#13108053	1	2	yes	—
140	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
141	C₂²	C₂×C₄⋊Dic₁₀₁	1	2	yes	—
142	C₂²	#15692214	1	2	yes	—
143	C₂²	C₄×D₄₀₄	1	4	yes	—
144	C₂²	C₄×Dic₂₀₂	1	4	yes	—
145	C₂²	#15692229	1	4	yes	—
146	C₂²	#15692234	1	4	yes	—
147	C₂²	#15692243	1	2	yes	—
148	C₂²	#15692247	1	2	yes	—
149	C₂²	#15692252	1	1	yes	characteristic
150	C₂²	#15692255	1	2	yes	—
151	C₂²	#15692255	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	404	8	no	—
154	C₄	#15692162	1	8	yes	complemented
155	C₂	C₂×#15692162	1	1	yes	characteristic
156	C₂	C₄×C₄⋊Dic₁₀₁	1	2	yes	—
157	C₂	#20265037	1	2	yes	—
158	C₂	#20265043	1	1	yes	characteristic
159	C₂	#20265047	1	1	yes	characteristic
160	C₂	#20265075	1	4	yes	—
161	C₂	#20265079	1	2	yes	—
162	C₂	#20265083	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊Dic₁₀₁	C₄	semidirect
12. C₂×C₄²×C₁₀₁	C₄	semidirect
48. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
71. C₄×C₁₀₁	#95	semidirect
82. C₁₀₁	C₄⋊#95	semidirect
93. C₂×C₄²	Dic₁₀₁	semidirect
135. C₂×C₄	C₄⋊Dic₁₀₁	semidirect
154. C₄	#15692162	semidirect

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