Subgroups — C₄×#15692251 #27173695

Order 12928.

924 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692251	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	4	yes	maximal, complemented
4	#20265080	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265078	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265077	C₂	1	4	yes	maximal, complemented
7	#20265046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
12	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
13	#15692251	C₄	1	8	yes	direct product factor
14	C₄×D₄₀₄	—	2	4	no	—
15	C₄×C₂²⋊D₁₀₁	—	2	16	no	—
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×#13108053	C₂²	1	4	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
19	C₂×#13108051	C₂²	1	2	yes	—
20	C₂×#13108051	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
27	C₄×D₄×C₁₀₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂³×D₁₀₁	C₂³	1	2	yes	—
32	C₂×D₄₀₄	—	2	2	no	—
33	C₂×D₄₀₄	—	2	2	no	—
34	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
35	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
36	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
37	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	16	yes	complemented
38	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂×C₄×D₁₀₁	—	2	8	no	—
42	C₂×C₄×D₁₀₁	—	2	4	no	—
43	C₄⋊Dic₁₀₁	—	2	2	no	—
44	#13108053	—	2	8	no	—
45	C₂²⋊Dic₁₀₁	—	2	8	no	—
46	#13108051	—	2	4	no	—
47	#13108051	—	2	8	no	—
48	C₄×Dic₁₀₁	—	2	2	no	—
49	C₄×Dic₁₀₁	—	2	4	no	—
50	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
51	C₄×Dic₁₀₁	D₄	1	8	yes	complemented
52	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
53	C₂×D₄×C₁₀₁	—	2	2	no	—
54	C₂×D₄×C₁₀₁	—	2	2	no	—
55	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
56	C₂²×C₄×C₁₀₁	—	2	4	no	—
57	C₂²×C₄×C₁₀₁	—	2	8	no	—
58	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₁	—	2	2	no	—
63	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
64	C₄²×C₁₀₁	D₄	1	4	yes	complemented
65	C₂³×C₁₀₁	—	2	2	no	—
66	C₂³×C₁₀₁	—	2	4	no	—
67	C₂³×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	—	2	4	no	—
69	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
70	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
71	C₂²×D₁₀₁	—	2	4	no	—
72	C₂²×D₁₀₁	—	2	2	no	—
73	C₂²×D₁₀₁	—	2	4	no	—
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
76	D₄×C₁₀₁	—	2	8	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	4	no	—
80	C₂×C₄×C₁₀₁	—	4	8	no	—
81	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	#41	1	2	yes	—
87	C₂²⋊D₁₀₁	—	2	32	no	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
93	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
94	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
95	C₂×Dic₁₀₁	#41	1	4	yes	—
96	D₄₀₄	—	2	8	no	—
97	C₄×D₁₀₁	—	4	8	no	—
98	C₂²×C₁₀₁	—	2	8	no	—
99	C₂²×C₁₀₁	—	2	4	no	—
100	C₂²×C₁₀₁	—	4	2	no	—
101	C₂²×C₁₀₁	—	4	2	no	—
102	C₂²×C₁₀₁	—	4	4	no	—
103	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
104	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
105	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
107	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
108	C₂×D₁₀₁	—	2	4	no	—
109	C₂×D₁₀₁	—	2	8	no	—
110	C₂×D₁₀₁	—	4	2	no	—
111	C₂×D₁₀₁	—	4	2	no	—
112	C₂×D₁₀₁	—	4	4	no	—
113	C₄×C₁₀₁	—	2	2	no	—
114	C₄×C₁₀₁	—	4	2	no	—
115	C₄×C₁₀₁	C₄×D₄	1	4	yes	complemented
116	C₄×C₁₀₁	#127	1	4	yes	complemented
117	Dic₁₀₁	—	2	4	no	—
118	Dic₁₀₁	—	4	2	no	—
119	Dic₁₀₁	C₄×D₄	1	8	yes	complemented
120	C₂×C₁₀₁	—	4	4	no	—
121	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
122	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×C₁₀₁	C₂×#127	1	1	yes	characteristic
124	C₂×C₁₀₁	#532	1	1	yes	characteristic
125	C₂×C₁₀₁	#532	1	2	yes	—
126	D₁₀₁	—	4	4	no	—
127	C₄×#127	—	101	1	no	maximal, Sylow
128	C₁₀₁	C₄×#127	1	1	yes	characteristic, complemented, Sylow
129	C₂×#127	—	101	1	no	—
130	C₂×C₄×D₄	—	101	4	no	—
131	C₂×C₄×D₄	—	101	1	no	—
132	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
133	#390	—	101	1	no	—
134	#390	—	101	1	no	—
135	#390	—	101	4	no	—
136	#384	—	101	1	no	—
137	C₄³	—	101	1	no	—
138	C₂²×D₄	—	101	4	no	—
139	C₂²×D₄	—	101	1	no	—
140	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
141	C₂³×C₄	—	101	2	no	—
142	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
143	#127	—	101	8	no	—
144	C₄×D₄	—	2	4	no	—
145	C₄×D₄	—	202	16	no	—
146	C₄×D₄	—	202	4	no	—
147	C₂×C₄⋊C₄	—	101	4	no	—
148	C₂×C₄⋊C₄	—	101	1	no	—
149	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	101	4	no	—
151	C₂×C₂²⋊C₄	—	101	4	no	—
152	C₂×C₂²⋊C₄	—	101	2	no	—
153	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
154	C₂×C₄²	—	101	1	no	—
155	C₂×C₄²	—	101	2	no	—
156	C₂×C₄²	—	101	2	no	—
157	C₂×C₄²	—	101	1	no	—
158	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
159	C₂⁴	—	101	2	no	—
160	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	101	16	no	—
164	C₂×D₄	—	101	4	no	—
165	C₂×D₄	—	202	8	no	—
166	C₂×D₄	—	202	8	no	—
167	C₂×D₄	—	202	2	no	—
168	C₂×D₄	—	202	2	no	—
169	C₂×D₄	C₄×D₁₀₁	1	4	yes	complemented
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	101	2	no	—
173	C₂²×C₄	—	101	2	no	—
174	C₂²×C₄	—	101	2	no	—
175	C₂²×C₄	—	202	8	no	—
176	C₂²×C₄	—	202	4	no	—
177	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
178	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	202	8	no	—
183	C₄⋊C₄	—	202	2	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	202	8	no	—
186	C₂²⋊C₄	—	202	8	no	—
187	C₂²⋊C₄	—	202	4	no	—
188	C₄²	—	101	4	no	—
189	C₄²	—	101	8	no	—
190	C₄²	—	202	4	no	—
191	C₄²	—	202	2	no	—
192	C₄²	C₂²⋊D₁₀₁	1	4	yes	complemented
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	101	4	no	—
198	C₂³	—	202	4	no	—
199	C₂³	—	202	2	no	—
200	C₂³	—	202	4	no	—
201	C₂³	—	202	2	no	—
202	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
203	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
204	D₄	—	2	8	no	—
205	D₄	—	202	32	no	—
206	D₄	—	202	8	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	101	4	no	—
212	C₂×C₄	—	101	4	no	—
213	C₂×C₄	—	101	4	no	—
214	C₂×C₄	—	101	4	no	—
215	C₂×C₄	—	404	8	no	—
216	C₂×C₄	—	202	4	no	—
217	C₂×C₄	—	202	2	no	—
218	C₂×C₄	—	202	2	no	—
219	C₂×C₄	—	202	2	no	—
220	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
221	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
222	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
225	C₂×C₄	#13108074	1	2	yes	—
226	C₂²	—	2	8	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	—	404	2	no	—
232	C₂²	—	404	2	no	—
233	C₂²	—	404	4	no	—
234	C₂²	—	202	8	no	—
235	C₂²	—	202	4	no	—
236	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
237	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
238	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
239	C₂²	#41×D₁₀₁	1	2	yes	—
240	C₂²	C₂×#13108074	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	101	8	no	—
244	C₄	—	404	2	no	—
245	C₄	—	202	4	no	—
246	C₄	C₄×C₂²⋊D₁₀₁	1	4	yes	complemented
247	C₄	#15692251	1	4	yes	direct product factor
248	C₂	—	4	4	no	—
249	C₂	—	404	4	no	—
250	C₂	C₄×D₄×D₁₀₁	1	2	yes	—
251	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
252	C₂	C₂×#15692251	1	1	yes	characteristic
253	C₂	#20265723	1	1	yes	characteristic
254	C₂	#20265724	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692251	C₂
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	C₄×#127
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
4. #20265080	C₂	semidirect
5. #20265078	C₂	semidirect
6. #20265077	C₂	semidirect
7. #20265046	C₂	semidirect
8. C₄²×Dic₁₀₁	C₂	semidirect
9. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
13. #15692251	C₄	direct (N × H)
34. C₂×D₄₀₄	C₂×C₄	semidirect
37. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
50. C₄×Dic₁₀₁	C₂×C₄	semidirect
51. C₄×Dic₁₀₁	D₄	semidirect
55. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
64. C₄²×C₁₀₁	D₄	semidirect
115. C₄×C₁₀₁	C₄×D₄	semidirect
116. C₄×C₁₀₁	#127	semidirect
119. Dic₁₀₁	C₄×D₄	semidirect
128. C₁₀₁	C₄×#127	semidirect
132. C₂×C₄×D₄	D₁₀₁	semidirect
169. C₂×D₄	C₄×D₁₀₁	semidirect
192. C₄²	C₂²⋊D₁₀₁	semidirect
246. C₄	C₄×C₂²⋊D₁₀₁	semidirect
247. C₄	#15692251	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15692251	2
C₂×C₄×D₄₀₄	2
C₂×C₄×C₂²⋊D₁₀₁	2
#20265080	2
#20265078	2
#20265077	2
#20265046	2
C₄²×Dic₁₀₁	2
C₂×C₄×D₄×C₁₀₁	2
C₄×#127	101

Subgroups — C4×#15692251 #27173695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692251 #27173695