Subgroups — #27173625

Order 12928.

524 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692212	C₂	1	1	yes	maximal, characteristic
2	#20265069	C₂	1	2	yes	maximal
3	#20265067	C₂	1	2	yes	maximal
4	#20265064	C₂	1	2	yes	maximal, complemented
5	#20265058	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13108051	C₂	1	1	yes	maximal, characteristic
8	C₄×#13108051	C₂	1	1	yes	maximal, characteristic
9	#20265023	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	#15692212	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×#13108053	C₂²	1	1	yes	characteristic
16	C₂×#13108053	C₂²	1	1	yes	characteristic
17	C₂×#13108051	C₂²	1	2	yes	—
18	C₂×#13108051	C₂²	1	2	yes	—
19	C₂×#13108051	C₂²	1	1	yes	characteristic
20	C₂×#13108051	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	#15692162	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂×C₄×D₁₀₁	—	2	2	no	—
43	C₂×C₄×D₁₀₁	—	2	2	no	—
44	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₁₀₁	—	2	4	no	—
46	#13108053	C₂×C₄	1	4	yes	—
47	#13108053	C₂×C₄	1	4	yes	—
48	#13108051	—	2	4	no	—
49	#13108051	—	2	4	no	—
50	#13108051	C₂×C₄	1	4	yes	—
51	#13108051	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₀₁	—	2	4	no	—
53	C₄×Dic₁₀₁	—	2	2	no	—
54	C₄×Dic₁₀₁	—	2	2	no	—
55	C₄×Dic₁₀₁	—	2	4	no	—
56	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₁	—	2	4	no	—
63	C₄²×C₁₀₁	—	2	2	no	—
64	C₄²×C₁₀₁	—	2	2	no	—
65	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
66	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
67	C₂²×D₁₀₁	—	2	2	no	—
68	C₂²×D₁₀₁	—	2	2	no	—
69	C₂²×D₁₀₁	—	2	1	no	—
70	C₂²×D₁₀₁	—	2	1	no	—
71	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₀₁	—	2	2	no	—
73	C₂×C₄×C₁₀₁	—	2	2	no	—
74	C₂×C₄×C₁₀₁	—	2	2	no	—
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	2	2	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₁	#41	1	2	yes	—
83	C₂×C₄×C₁₀₁	#41	1	4	yes	—
84	C₂×C₄×C₁₀₁	#41	1	2	yes	—
85	C₂×Dic₁₀₁	—	2	2	no	—
86	C₂×Dic₁₀₁	—	2	2	no	—
87	C₂×Dic₁₀₁	—	2	2	no	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	—	2	1	no	—
90	C₂×Dic₁₀₁	—	2	1	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀₁	#41	1	4	yes	—
98	C₄×D₁₀₁	—	2	8	no	—
99	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
100	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
101	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
103	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
104	C₂×D₁₀₁	—	2	2	no	—
105	C₂×D₁₀₁	—	2	2	no	—
106	C₂×D₁₀₁	—	2	2	no	—
107	C₂×D₁₀₁	—	4	2	no	—
108	C₂×D₁₀₁	—	4	2	no	—
109	C₄×C₁₀₁	—	2	2	no	—
110	C₄×C₁₀₁	—	2	2	no	—
111	C₄×C₁₀₁	—	2	4	no	—
112	C₄×C₁₀₁	—	4	2	no	—
113	C₄×C₁₀₁	#117	1	4	yes	—
114	Dic₁₀₁	—	2	2	no	—
115	Dic₁₀₁	—	2	2	no	—
116	Dic₁₀₁	—	2	2	no	—
117	Dic₁₀₁	—	2	4	no	—
118	Dic₁₀₁	—	4	2	no	—
119	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
120	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
121	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
122	C₂×C₁₀₁	#529	1	2	yes	—
123	C₂×C₁₀₁	#529	1	2	yes	—
124	D₁₀₁	—	4	2	no	—
125	#2298	—	101	1	no	maximal, Sylow
126	C₁₀₁	#2298	1	1	yes	characteristic, complemented, Sylow
127	C₂×#117	—	101	1	no	—
128	#388	—	101	2	no	—
129	#388	—	101	2	no	—
130	#387	—	101	2	no	—
131	#387	—	101	2	no	—
132	C₄×C₄⋊C₄	—	101	1	no	—
133	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
134	C₄×C₂²⋊C₄	—	101	1	no	—
135	C₄×C₂²⋊C₄	—	101	1	no	—
136	#376	—	101	1	no	—
137	C₄³	—	101	1	no	—
138	C₂³×C₄	—	101	1	no	—
139	#117	—	101	8	no	—
140	C₂×C₄⋊C₄	—	101	1	no	—
141	C₂×C₄⋊C₄	—	101	1	no	—
142	C₂×C₄⋊C₄	—	101	2	no	—
143	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
144	C₂×C₂²⋊C₄	—	101	1	no	—
145	C₂×C₂²⋊C₄	—	101	1	no	—
146	C₂×C₂²⋊C₄	—	101	2	no	—
147	C₂×C₂²⋊C₄	—	101	2	no	—
148	C₂×C₄²	—	101	1	no	—
149	C₂×C₄²	—	101	2	no	—
150	C₂×C₄²	—	101	2	no	—
151	C₂×C₄²	—	101	1	no	—
152	C₂×C₄²	—	101	1	no	—
153	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
154	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
156	#95	—	101	2	no	—
157	#95	—	101	2	no	—
158	#95	—	101	2	no	—
159	#95	—	101	2	no	—
160	#95	—	101	2	no	—
161	#95	C₂×D₁₀₁	1	2	yes	—
162	C₂⁴	—	101	1	no	—
163	C₂²×C₄	—	101	1	no	—
164	C₂²×C₄	—	101	1	no	—
165	C₂²×C₄	—	101	1	no	—
166	C₂²×C₄	—	101	2	no	—
167	C₂²×C₄	—	101	4	no	—
168	C₂²×C₄	—	101	2	no	—
169	C₂²×C₄	—	202	2	no	—
170	C₂²×C₄	—	202	2	no	—
171	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
172	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
175	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	101	4	no	—
178	C₄⋊C₄	—	101	4	no	—
179	C₄⋊C₄	—	202	4	no	—
180	C₂²⋊C₄	—	101	4	no	—
181	C₂²⋊C₄	—	101	4	no	—
182	C₂²⋊C₄	—	202	4	no	—
183	C₂²⋊C₄	—	202	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	101	4	no	—
187	C₄²	—	202	4	no	—
188	C₄²	—	202	4	no	—
189	C₄²	—	202	2	no	—
190	C₄²	—	202	2	no	—
191	C₄²	C₄×D₁₀₁	1	4	yes	—
192	C₂³	—	101	2	no	—
193	C₂³	—	202	2	no	—
194	C₂³	—	202	1	no	—
195	C₂³	—	202	2	no	—
196	C₂³	—	202	1	no	—
197	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	101	2	no	—
206	C₂×C₄	—	101	2	no	—
207	C₂×C₄	—	101	2	no	—
208	C₂×C₄	—	101	4	no	—
209	C₂×C₄	—	202	2	no	—
210	C₂×C₄	—	202	2	no	—
211	C₂×C₄	—	202	2	no	—
212	C₂×C₄	—	202	2	no	—
213	C₂×C₄	—	202	1	no	—
214	C₂×C₄	—	202	2	no	—
215	C₂×C₄	—	202	2	no	—
216	C₂×C₄	—	202	2	no	—
217	C₂×C₄	—	202	1	no	—
218	C₂×C₄	—	202	8	no	—
219	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
222	C₂×C₄	#13108073	1	2	yes	—
223	C₂×C₄	#13108074	1	4	yes	—
224	C₂×C₄	#13108075	1	2	yes	—
225	C₂²	—	404	2	no	—
226	C₂²	—	404	2	no	—
227	C₂²	—	202	2	no	—
228	C₂²	—	202	2	no	—
229	C₂²	—	202	2	no	—
230	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
231	C₂²	#41×D₁₀₁	1	2	yes	—
232	C₂²	C₂×#13108073	1	1	yes	characteristic
233	C₂²	C₂×#13108074	1	2	yes	—
234	C₂²	C₂×#13108075	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	4	no	—
238	C₄	—	4	2	no	—
239	C₄	—	404	2	no	—
240	C₄	—	202	2	no	—
241	C₄	—	202	2	no	—
242	C₄	—	202	2	no	—
243	C₄	—	202	4	no	—
244	C₄	#15692213	1	4	yes	—
245	C₂	—	404	2	no	—
246	C₂	C₂×#15692213	1	1	yes	characteristic
247	C₂	#20265645	1	1	yes	characteristic
248	C₂	C₄×#13108074	1	1	yes	characteristic
249	C₂	#20265707	1	2	yes	—
250	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	#2298
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
4. #20265064	C₂	semidirect
5. #20265058	C₂	semidirect
6. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
9. #20265023	C₂	semidirect
10. C₄²×Dic₁₀₁	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₀₁	C₂	semidirect
13. #15692212	C₄	semidirect
126. C₁₀₁	#2298	semidirect
133. 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×#15692212	2
#20265069	2
#20265067	2
#20265064	2
#20265058	2
C₄×C₄⋊Dic₁₀₁	2
C₄×#13108051	2
#20265023	2
C₄²×Dic₁₀₁	2
C₄×C₄⋊C₄×C₁₀₁	2
#2298	101