Subgroups — #27173528

Order 12928.

568 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265044	C₂	1	2	yes	maximal, complemented
3	#20265042	C₂	1	2	yes	maximal, complemented
4	#20265039	C₂	1	2	yes	maximal
5	#20265034	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13108051	C₂	1	2	yes	maximal
8	#20265023	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
14	C₂×#13108051	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₄²×D₁₀₁	C₄	1	8	yes	complemented
20	#15692162	C₂²	1	4	yes	—
21	#15692162	C₂²	1	4	yes	—
22	#15692162	C₂²	1	4	yes	—
23	#15692162	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
27	#95×C₁₀₁	C₂²	1	2	yes	—
28	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
30	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
33	C₂×C₄×D₁₀₁	—	2	4	no	—
34	C₂×C₄×D₁₀₁	—	2	2	no	—
35	C₂×C₄×D₁₀₁	—	2	4	no	—
36	C₂×C₄×D₁₀₁	—	2	2	no	—
37	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₁₀₁	—	2	4	no	—
41	#13108051	—	2	8	no	—
42	C₄×Dic₁₀₁	—	2	4	no	—
43	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
47	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₁₀₁	—	2	4	no	—
51	C₄²×C₁₀₁	—	2	4	no	—
52	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
53	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
54	C₂²×D₁₀₁	—	2	1	no	—
55	C₂²×D₁₀₁	—	2	2	no	—
56	C₂²×D₁₀₁	—	2	1	no	—
57	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
58	C₂²×D₁₀₁	#41	1	4	yes	—
59	C₂×C₄×C₁₀₁	—	2	2	no	—
60	C₂×C₄×C₁₀₁	—	2	4	no	—
61	C₂×C₄×C₁₀₁	—	2	4	no	—
62	C₂×C₄×C₁₀₁	—	2	4	no	—
63	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₀₁	#41	1	2	yes	—
67	C₂×C₄×C₁₀₁	#41	1	4	yes	—
68	C₂×C₄×C₁₀₁	#41	1	2	yes	—
69	C₂×Dic₁₀₁	—	2	2	no	—
70	C₂×Dic₁₀₁	—	2	1	no	—
71	C₂×Dic₁₀₁	—	2	2	no	—
72	C₂×Dic₁₀₁	—	2	1	no	—
73	C₂×Dic₁₀₁	—	2	4	no	—
74	C₂×Dic₁₀₁	—	2	4	no	—
75	C₂×Dic₁₀₁	—	2	4	no	—
76	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
79	C₄×D₁₀₁	—	2	8	no	—
80	C₄×D₁₀₁	—	2	8	no	—
81	C₄×D₁₀₁	—	2	8	no	—
82	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
84	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
85	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
86	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
87	C₂×D₁₀₁	—	2	2	no	—
88	C₂×D₁₀₁	—	2	2	no	—
89	C₂×D₁₀₁	—	2	4	no	—
90	C₂×D₁₀₁	—	2	4	no	—
91	C₂×D₁₀₁	#117	1	4	yes	—
92	C₄×C₁₀₁	—	2	2	no	—
93	C₄×C₁₀₁	—	4	4	no	—
94	C₄×C₁₀₁	#117	1	4	yes	—
95	C₄×C₁₀₁	#117	1	4	yes	—
96	Dic₁₀₁	—	2	2	no	—
97	Dic₁₀₁	—	2	2	no	—
98	Dic₁₀₁	—	2	2	no	—
99	Dic₁₀₁	—	4	4	no	—
100	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
101	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
102	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
103	C₂×C₁₀₁	#529	1	2	yes	—
104	C₂×C₁₀₁	#529	1	2	yes	—
105	D₁₀₁	—	2	4	no	—
106	#2283	—	101	1	no	maximal, Sylow
107	C₁₀₁	#2283	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₄²	—	101	1	no	—
109	#383	—	101	2	no	—
110	#382	—	101	2	no	—
111	#381	—	101	2	no	—
112	#380	—	101	2	no	—
113	C₄×C₄⋊C₄	—	101	1	no	—
114	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
115	C₄×C₂²⋊C₄	—	101	2	no	—
116	#376	—	101	2	no	—
117	C₂³×C₄	—	101	1	no	—
118	C₂³×C₄	—	101	1	no	—
119	C₂³×C₄	—	101	1	no	—
120	C₂×C₄⋊C₄	—	101	2	no	—
121	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
122	C₂×C₂²⋊C₄	—	101	4	no	—
123	C₂×C₄²	—	101	1	no	—
124	C₂×C₄²	—	101	1	no	—
125	C₂×C₄²	—	101	1	no	—
126	C₂×C₄²	—	101	8	no	—
127	C₂×C₄²	—	101	2	no	—
128	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
129	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
130	#95	—	101	4	no	—
131	#95	—	101	4	no	—
132	#95	—	101	4	no	—
133	#95	—	101	2	no	—
134	#95	C₂×D₁₀₁	1	2	yes	—
135	C₂⁴	—	101	1	no	—
136	C₂²×C₄	—	101	1	no	—
137	C₂²×C₄	—	101	1	no	—
138	C₂²×C₄	—	101	1	no	—
139	C₂²×C₄	—	101	4	no	—
140	C₂²×C₄	—	101	4	no	—
141	C₂²×C₄	—	101	4	no	—
142	C₂²×C₄	—	101	4	no	—
143	C₂²×C₄	—	202	4	no	—
144	C₂²×C₄	—	202	2	no	—
145	C₂²×C₄	—	202	4	no	—
146	C₂²×C₄	—	202	2	no	—
147	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
148	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	202	4	no	—
153	C₂²⋊C₄	—	202	8	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	101	4	no	—
156	C₄²	—	101	4	no	—
157	C₄²	—	101	4	no	—
158	C₄²	—	202	4	no	—
159	C₄²	C₄×D₁₀₁	1	4	yes	—
160	C₂³	—	101	2	no	—
161	C₂³	—	101	4	no	—
162	C₂³	—	202	1	no	—
163	C₂³	—	202	2	no	—
164	C₂³	—	202	1	no	—
165	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	101	2	no	—
171	C₂×C₄	—	101	2	no	—
172	C₂×C₄	—	101	2	no	—
173	C₂×C₄	—	202	8	no	—
174	C₂×C₄	—	202	4	no	—
175	C₂×C₄	—	202	4	no	—
176	C₂×C₄	—	202	4	no	—
177	C₂×C₄	—	202	2	no	—
178	C₂×C₄	—	202	1	no	—
179	C₂×C₄	—	202	2	no	—
180	C₂×C₄	—	202	1	no	—
181	C₂×C₄	—	202	8	no	—
182	C₂×C₄	—	202	8	no	—
183	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
184	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
185	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
186	C₂×C₄	#13108073	1	2	yes	—
187	C₂×C₄	#13108074	1	4	yes	—
188	C₂×C₄	#13108075	1	2	yes	—
189	C₂²	—	101	4	no	—
190	C₂²	—	202	2	no	—
191	C₂²	—	202	2	no	—
192	C₂²	—	202	4	no	—
193	C₂²	—	202	4	no	—
194	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
195	C₂²	#41×D₁₀₁	1	2	yes	—
196	C₂²	C₂×#13108073	1	1	yes	characteristic
197	C₂²	C₂×#13108074	1	2	yes	—
198	C₂²	C₂×#13108075	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	404	4	no	—
202	C₄	—	202	2	no	—
203	C₄	—	202	2	no	—
204	C₄	—	202	2	no	—
205	C₄	#15692212	1	4	yes	—
206	C₄	#15692213	1	4	yes	—
207	C₂	—	202	4	no	—
208	C₂	#117×D₁₀₁	1	1	yes	characteristic
209	C₂	C₂×#15692212	1	1	yes	characteristic
210	C₂	C₂×#15692213	1	1	yes	characteristic
211	C₂	#20265709	1	2	yes	—
212	C₂	#20265714	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₄×C₄⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2283
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20265044	C₂	semidirect
3. #20265042	C₂	semidirect
6. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
8. #20265023	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₀₁	C₂	semidirect
19. C₄²×D₁₀₁	C₄	semidirect
107. C₁₀₁	#2283	semidirect
114. C₄×C₄⋊C₄	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×D₁₀₁	2
#20265044	2
#20265042	2
#20265039	2
#20265034	2
C₄×C₄⋊Dic₁₀₁	2
C₄×#13108051	2
#20265023	2
C₄×C₄⋊C₄×C₁₀₁	2
#2283	101