Subgroups — #27169359

Order 12928.

568 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692162	C₂	1	2	yes	maximal
2	C₂×#15692162	C₂	1	2	yes	maximal
3	C₂×#15692162	C₂	1	2	yes	maximal
4	C₂²×C₄²×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₁	C₂²	1	4	yes	—
6	#15692162	—	2	4	no	—
7	#15692162	—	2	4	no	—
8	#15692162	C₄	1	8	yes	complemented
9	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₀₁	—	2	4	no	—
13	C₂×C₄²×C₁₀₁	C₄	1	4	yes	complemented
14	C₂²×Dic₁₀₁	—	2	4	no	—
15	C₂²×Dic₁₀₁	—	2	4	no	—
16	C₂²×Dic₁₀₁	—	2	4	no	—
17	C₂²×Dic₁₀₁	—	2	4	no	—
18	C₂²×Dic₁₀₁	—	2	4	no	—
19	C₂²×Dic₁₀₁	—	2	4	no	—
20	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
21	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	—	2	2	no	—
23	C₂²×C₄×C₁₀₁	—	2	2	no	—
24	C₂²×C₄×C₁₀₁	—	2	2	no	—
25	C₂²×C₄×C₁₀₁	—	2	2	no	—
26	C₂²×C₄×C₁₀₁	—	2	2	no	—
27	C₂²×C₄×C₁₀₁	—	2	2	no	—
28	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
34	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
35	C₄²×C₁₀₁	—	2	8	no	—
36	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
37	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
39	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₁₀₁	#41	1	2	yes	—
42	C₂³×C₁₀₁	#41	1	2	yes	—
43	C₂³×C₁₀₁	#41	1	2	yes	—
44	C₂³×C₁₀₁	#41	1	2	yes	—
45	C₂×C₄×C₁₀₁	—	2	2	no	—
46	C₂×C₄×C₁₀₁	—	2	2	no	—
47	C₂×C₄×C₁₀₁	—	2	2	no	—
48	C₂×C₄×C₁₀₁	—	2	2	no	—
49	C₂×C₄×C₁₀₁	—	2	2	no	—
50	C₂×C₄×C₁₀₁	—	2	2	no	—
51	C₂×C₄×C₁₀₁	—	2	8	no	—
52	C₂×C₄×C₁₀₁	—	2	4	no	—
53	C₂×C₄×C₁₀₁	—	2	4	no	—
54	C₂×C₄×C₁₀₁	—	2	8	no	—
55	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
56	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
58	C₂×Dic₁₀₁	—	2	8	no	—
59	C₂×Dic₁₀₁	—	2	8	no	—
60	C₂×Dic₁₀₁	—	2	8	no	—
61	C₂×Dic₁₀₁	—	4	4	no	—
62	C₂×Dic₁₀₁	—	4	4	no	—
63	C₂×Dic₁₀₁	—	4	4	no	—
64	C₂×Dic₁₀₁	—	4	4	no	—
65	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₀₁	#117	1	4	yes	—
69	C₂²×C₁₀₁	#117	1	1	yes	characteristic
70	C₂²×C₁₀₁	#117	1	1	yes	characteristic
71	C₂²×C₁₀₁	#117	1	1	yes	characteristic
72	C₂²×C₁₀₁	#117	1	1	yes	characteristic
73	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
74	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
75	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₀₁	#121	1	2	yes	—
77	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
78	C₂²×C₁₀₁	#123	1	4	yes	—
79	C₂²×C₁₀₁	#124	1	2	yes	—
80	C₂²×C₁₀₁	#125	1	2	yes	—
81	C₂²×C₁₀₁	#126	1	4	yes	—
82	C₄×C₁₀₁	—	2	4	no	—
83	C₄×C₁₀₁	—	2	4	no	—
84	C₄×C₁₀₁	—	2	4	no	—
85	Dic₁₀₁	—	4	8	no	—
86	C₂×C₁₀₁	#376	1	1	yes	characteristic
87	C₂×C₁₀₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₁₀₁	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₁₀₁	#380	1	1	yes	characteristic
90	C₂×C₁₀₁	#381	1	1	yes	characteristic
91	C₂×C₁₀₁	#382	1	1	yes	characteristic
92	C₂×C₁₀₁	#383	1	1	yes	characteristic
93	C₂×C₁₀₁	#387	1	4	yes	—
94	C₂×C₁₀₁	#388	1	4	yes	—
95	#1434	—	101	1	no	maximal, Sylow
96	C₁₀₁	#1434	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₄²	D₁₀₁	1	1	yes	characteristic
98	C₂×#95	—	101	2	no	—
99	C₂×#95	—	101	2	no	—
100	C₂×#95	—	101	2	no	—
101	C₂³×C₄	—	101	4	no	—
102	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
105	C₂×C₄²	—	2	4	no	—
106	C₂×C₄²	Dic₁₀₁	1	4	yes	complemented
107	#95	—	101	8	no	—
108	#95	—	202	4	no	—
109	#95	—	202	4	no	—
110	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	101	8	no	—
118	C₂²×C₄	—	202	4	no	—
119	C₂²×C₄	—	202	4	no	—
120	C₂²×C₄	—	202	4	no	—
121	C₂²×C₄	—	202	4	no	—
122	C₂²×C₄	—	202	4	no	—
123	C₂²×C₄	—	202	4	no	—
124	C₂²×C₄	Dic₂₀₂	1	2	yes	—
125	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
126	C₂²×C₄	D₄₀₄	1	2	yes	—
127	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
128	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
129	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
130	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
131	C₄²	—	2	8	no	—
132	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
133	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
134	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
135	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
136	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
137	C₂³	#13108074	1	2	yes	—
138	C₂³	#13108074	1	2	yes	—
139	C₂³	#13108074	1	2	yes	—
140	C₂³	#13108074	1	2	yes	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	404	4	no	—
152	C₂×C₄	—	404	4	no	—
153	C₂×C₄	—	404	4	no	—
154	C₂×C₄	—	404	4	no	—
155	C₂×C₄	—	202	8	no	—
156	C₂×C₄	—	202	8	no	—
157	C₂×C₄	—	202	8	no	—
158	C₂×C₄	C₄×Dic₁₀₁	1	4	yes	—
159	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
160	C₂×C₄	C₄⋊Dic₁₀₁	1	4	yes	—
161	C₂²	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
162	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
163	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
164	C₂²	#15692212	1	4	yes	—
165	C₂²	#15692214	1	1	yes	characteristic
166	C₂²	#15692214	1	1	yes	characteristic
167	C₂²	#15692214	1	1	yes	characteristic
168	C₂²	#15692214	1	1	yes	characteristic
169	C₂²	C₄×C₂²⋊D₁₀₁	1	4	yes	—
170	C₂²	C₄×D₄₀₄	1	2	yes	—
171	C₂²	C₄×Dic₂₀₂	1	2	yes	—
172	C₂²	#15692229	1	2	yes	—
173	C₂²	#15692234	1	2	yes	—
174	C₂²	#15692238	1	4	yes	—
175	C₂²	#15692243	1	2	yes	—
176	C₂²	#15692247	1	2	yes	—
177	C₂²	#15692249	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	404	8	no	—
182	C₂	#20265023	1	1	yes	characteristic
183	C₂	C₄×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
184	C₂	C₄×C₄⋊Dic₁₀₁	1	1	yes	characteristic
185	C₂	#20265037	1	1	yes	characteristic
186	C₂	#20265040	1	1	yes	characteristic
187	C₂	#20265043	1	1	yes	characteristic
188	C₂	#20265045	1	1	yes	characteristic
189	C₂	#20265061	1	4	yes	—
190	C₂	#20265068	1	4	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	#1434
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
8. #15692162	C₄	semidirect
13. C₂×C₄²×C₁₀₁	C₄	semidirect
96. C₁₀₁	#1434	semidirect
106. C₂×C₄²	Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692162	2
C₂²×C₄²×C₁₀₁	2
#1434	101