Subgroups — #27174001

Order 12928.

604 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692214	C₂	1	2	yes	maximal
2	#20265076	C₂	1	2	yes	maximal, complemented
3	#20265063	C₂	1	4	yes	maximal, complemented
4	#20265040	C₂	1	2	yes	maximal
5	#20265037	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	#15692214	—	2	8	no	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
12	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
17	#15692162	C₂²	1	2	yes	—
18	#15692162	C₂²	1	4	yes	—
19	#15692162	C₂²	1	2	yes	—
20	#15692162	C₂²	1	4	yes	—
21	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₀₁	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
28	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
30	C₄⋊Dic₁₀₁	—	2	4	no	—
31	C₄⋊Dic₁₀₁	—	2	8	no	—
32	C₂²⋊Dic₁₀₁	—	2	4	no	—
33	C₂²⋊Dic₁₀₁	—	2	2	no	—
34	C₂²⋊Dic₁₀₁	D₄	1	4	yes	—
35	C₄×Dic₁₀₁	—	2	4	no	—
36	C₄×Dic₁₀₁	—	2	8	no	—
37	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₁	—	2	4	no	—
39	C₂²×C₄×C₁₀₁	—	2	4	no	—
40	C₂²×C₄×C₁₀₁	—	2	4	no	—
41	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	—
42	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	—
48	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
49	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
50	C₂³×C₁₀₁	—	2	2	no	—
51	C₂³×C₁₀₁	—	2	2	no	—
52	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
54	C₂³×C₁₀₁	#41	1	2	yes	—
55	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₁₀₁	—	2	16	no	—
57	C₂×C₄×C₁₀₁	—	2	2	no	—
58	C₂×C₄×C₁₀₁	—	2	2	no	—
59	C₂×C₄×C₁₀₁	—	2	2	no	—
60	C₂×C₄×C₁₀₁	—	2	2	no	—
61	C₂×C₄×C₁₀₁	—	2	2	no	—
62	C₂×C₄×C₁₀₁	—	2	8	no	—
63	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₀₁	#41	1	4	yes	—
68	C₂×Dic₁₀₁	—	2	2	no	—
69	C₂×Dic₁₀₁	—	2	2	no	—
70	C₂×Dic₁₀₁	—	2	4	no	—
71	C₂×Dic₁₀₁	—	2	2	no	—
72	C₂×Dic₁₀₁	—	2	2	no	—
73	C₂×Dic₁₀₁	—	2	4	no	—
74	C₂×Dic₁₀₁	—	2	4	no	—
75	C₂×Dic₁₀₁	—	2	2	no	—
76	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
77	C₂²×C₁₀₁	—	2	2	no	—
78	C₂²×C₁₀₁	—	2	2	no	—
79	C₂²×C₁₀₁	—	2	4	no	—
80	C₂²×C₁₀₁	—	2	4	no	—
81	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
82	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
85	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
86	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
87	C₄×C₁₀₁	—	2	2	no	—
88	C₄×C₁₀₁	—	2	4	no	—
89	C₄×C₁₀₁	—	2	4	no	—
90	C₄×C₁₀₁	#117	1	4	yes	complemented
91	Dic₁₀₁	—	2	4	no	—
92	Dic₁₀₁	—	4	4	no	—
93	Dic₁₀₁	—	4	2	no	—
94	C₂×C₁₀₁	—	2	4	no	—
95	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
96	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂×C₁₀₁	#519	1	1	yes	characteristic
98	C₂×C₁₀₁	#547	1	2	yes	—
99	C₂×C₁₀₁	#552	1	2	yes	—
100	#2344	—	101	1	no	maximal, Sylow
101	C₁₀₁	#2344	1	1	yes	characteristic, complemented, Sylow
102	C₂×#117	—	101	2	no	—
103	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
104	#389	—	101	2	no	—
105	#387	—	101	4	no	—
106	#381	—	101	2	no	—
107	#380	—	101	1	no	—
108	C₄×C₄⋊C₄	—	101	2	no	—
109	C₄×C₂²⋊C₄	—	101	1	no	—
110	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
111	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
112	#117	—	202	8	no	—
113	C₂×C₄⋊C₄	—	101	4	no	—
114	C₂×C₄⋊C₄	—	101	2	no	—
115	C₂×C₄⋊C₄	Dic₁₀₁	1	8	yes	complemented
116	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	—	101	1	no	—
120	C₂×C₂²⋊C₄	—	101	2	no	—
121	C₂×C₂²⋊C₄	—	101	1	no	—
122	C₂×C₄²	—	101	4	no	—
123	C₂×C₄²	—	101	2	no	—
124	#95	—	101	2	no	—
125	#95	—	101	4	no	—
126	#95	—	101	2	no	—
127	#95	—	101	4	no	—
128	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	101	2	no	—
133	C₂²×C₄	—	101	2	no	—
134	C₂²×C₄	—	101	4	no	—
135	C₂²×C₄	C₂×Dic₁₀₁	1	8	yes	—
136	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
137	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
139	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
140	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
141	C₄⋊C₄	—	202	8	no	—
142	C₄⋊C₄	—	202	4	no	—
143	C₄⋊C₄	C₂×Dic₁₀₁	1	8	yes	—
144	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
145	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
146	C₂²⋊C₄	—	101	4	no	—
147	C₂²⋊C₄	—	202	4	no	—
148	C₂²⋊C₄	—	202	2	no	—
149	C₄²	—	202	8	no	—
150	C₄²	—	202	4	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
154	C₂³	D₄×D₁₀₁	1	2	yes	—
155	C₂³	#13108073	1	2	yes	—
156	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	101	4	no	—
165	C₂×C₄	—	202	4	no	—
166	C₂×C₄	—	202	2	no	—
167	C₂×C₄	—	202	2	no	—
168	C₂×C₄	—	202	4	no	—
169	C₂×C₄	—	202	4	no	—
170	C₂×C₄	—	202	2	no	—
171	C₂×C₄	—	202	2	no	—
172	C₂×C₄	—	202	2	no	—
173	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
174	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
175	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
176	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
177	C₂×C₄	#13108074	1	4	yes	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	D₄×Dic₁₀₁	1	4	yes	—
183	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
184	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
185	C₂²	C₂×#13108073	1	1	yes	characteristic
186	C₂²	C₂×#13108074	1	2	yes	—
187	C₂²	#15692303	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	404	4	no	—
192	C₄	—	404	2	no	—
193	C₄	—	202	4	no	—
194	C₄	#15692214	1	4	yes	complemented
195	C₂	—	2	4	no	—
196	C₂	C₂×#15692214	1	1	yes	characteristic
197	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
198	C₂	#20265655	1	1	yes	characteristic
199	C₂	#20265828	1	2	yes	—
200	C₂	#20265862	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2344
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20265076	C₂	semidirect
3. #20265063	C₂	semidirect
6. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
23. C₂×C₄⋊C₄×C₁₀₁	C₄	semidirect
90. C₄×C₁₀₁	#117	semidirect
101. C₁₀₁	#2344	semidirect
115. C₂×C₄⋊C₄	Dic₁₀₁	semidirect
194. C₄	#15692214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692214	2
#20265076	2
#20265063	2
#20265040	2
#20265037	2
C₄⋊C₄×Dic₁₀₁	2
C₄×C₂²⋊Dic₁₀₁	2
C₂²×C₄⋊C₄×C₁₀₁	2
#2344	101