Subgroups — C₂×#20265931 #27185266

Order 12928.

1024 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265931	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692254	C₂	1	2	yes	maximal, complemented
3	C₂×#15692250	C₂	1	2	yes	maximal, complemented
4	C₂×#15692250	C₂	1	2	yes	maximal, complemented
5	C₂×#15692247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692243	C₂	1	1	yes	maximal, characteristic
7	C₂×#15692240	C₂	1	2	yes	maximal
8	C₂×#15692235	C₂	1	2	yes	maximal, complemented
9	C₂×#15692233	C₂	1	2	yes	maximal, complemented
10	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	#15692254	C₂²	1	16	yes	complemented
16	#15692250	C₂²	1	16	yes	complemented
17	#15692250	C₂²	1	16	yes	complemented
18	#15692247	C₂²	1	8	yes	complemented
19	#15692243	C₂²	1	8	yes	—
20	#15692240	C₂²	1	16	yes	—
21	#15692235	C₂²	1	16	yes	complemented
22	#15692233	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×#13108053	C₂²	1	2	yes	—
27	C₂×#13108053	C₂²	1	2	yes	—
28	C₂×#13108053	C₂²	1	2	yes	—
29	C₂×#13108051	C₂²	1	2	yes	—
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	2	yes	—
32	C₂×#13108051	C₂²	1	2	yes	—
33	C₂×#13108051	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
35	#125×C₁₀₁	C₂²	1	8	yes	complemented
36	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂³×D₁₀₁	C₂³	1	2	yes	—
42	C₂×Dic₂₀₂	—	4	4	no	—
43	C₂×Dic₂₀₂	—	4	2	no	—
44	C₂×Dic₂₀₂	C₂³	1	8	yes	—
45	C₂×Dic₂₀₂	C₂³	1	4	yes	—
46	C₂×D₄₀₄	—	4	2	no	—
47	C₂×D₄₀₄	C₂³	1	4	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂×C₄×D₁₀₁	—	4	4	no	—
52	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
53	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
54	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
55	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
56	#13108053	C₂³	1	8	yes	—
57	#13108053	C₂³	1	8	yes	—
58	#13108053	C₂³	1	8	yes	—
59	#13108051	C₂³	1	8	yes	—
60	#13108051	C₂³	1	8	yes	—
61	#13108051	C₂³	1	8	yes	—
62	#13108051	C₂³	1	8	yes	—
63	#13108051	C₂³	1	8	yes	—
64	C₄×Dic₁₀₁	C₂³	1	8	yes	—
65	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
71	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
72	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
73	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
74	C₄²×C₁₀₁	C₂³	1	4	yes	—
75	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
76	C₂²×D₁₀₁	—	2	4	no	—
77	C₂²×D₁₀₁	—	2	4	no	—
78	C₂²×D₁₀₁	—	2	4	no	—
79	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
80	C₂×C₄×C₁₀₁	—	2	4	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	—	2	4	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
88	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
89	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
90	C₂×Dic₁₀₁	—	2	4	no	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	—	2	4	no	—
93	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
94	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
95	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
96	D₄₀₄	—	4	4	no	—
97	C₄×D₁₀₁	—	4	8	no	—
98	Dic₂₀₂	—	4	4	no	—
99	Dic₂₀₂	—	4	8	no	—
100	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
101	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
102	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
103	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
104	C₂×D₁₀₁	—	2	4	no	—
105	C₂×D₁₀₁	—	2	4	no	—
106	C₂×D₁₀₁	—	2	4	no	—
107	C₂×D₁₀₁	—	4	8	no	—
108	C₄×C₁₀₁	—	2	4	no	—
109	C₄×C₁₀₁	—	2	2	no	—
110	C₄×C₁₀₁	—	2	2	no	—
111	C₄×C₁₀₁	—	2	4	no	—
112	C₄×C₁₀₁	—	2	2	no	—
113	Dic₁₀₁	—	2	4	no	—
114	Dic₁₀₁	—	2	4	no	—
115	Dic₁₀₁	—	2	4	no	—
116	C₂×C₁₀₁	#563	1	4	yes	complemented
117	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
120	D₁₀₁	—	4	4	no	—
121	C₂×#563	—	101	1	no	maximal, Sylow
122	C₁₀₁	C₂×#563	1	1	yes	characteristic, complemented, Sylow
123	#563	—	101	16	no	—
124	C₂×C₄⋊Q₈	—	101	2	no	—
125	C₂×#126	—	101	2	no	—
126	C₂×#126	—	101	2	no	—
127	C₂×#125	—	101	1	no	—
128	C₂×#125	D₁₀₁	1	1	yes	characteristic, complemented
129	C₂×#124	—	101	1	no	—
130	C₂×#123	—	101	2	no	—
131	C₂×C₂²⋊Q₈	—	101	2	no	—
132	C₂×C₂²⋊Q₈	—	101	2	no	—
133	C₂²×Q₈	—	101	2	no	—
134	C₂²×Q₈	—	101	1	no	—
135	C₂²×D₄	—	101	1	no	—
136	C₂³×C₄	—	101	2	no	—
137	C₄⋊Q₈	—	101	16	no	—
138	#126	—	101	16	no	—
139	#126	—	101	16	no	—
140	#125	—	101	8	no	—
141	#125	C₂×D₁₀₁	1	8	yes	complemented
142	#124	—	101	8	no	—
143	#123	—	101	16	no	—
144	C₂²⋊Q₈	—	101	16	no	—
145	C₂²⋊Q₈	—	101	16	no	—
146	C₂×C₄⋊C₄	—	101	1	no	—
147	C₂×C₄⋊C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	2	no	—
149	C₂×C₄⋊C₄	—	101	2	no	—
150	C₂×C₄⋊C₄	—	101	2	no	—
151	C₂×C₄⋊C₄	—	101	1	no	—
152	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
155	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	101	2	no	—
157	C₂×C₂²⋊C₄	—	101	2	no	—
158	C₂×C₂²⋊C₄	—	101	2	no	—
159	C₂×C₂²⋊C₄	—	101	2	no	—
160	C₂×C₂²⋊C₄	—	101	2	no	—
161	C₂×C₄²	—	101	2	no	—
162	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
163	C₂⁴	—	101	2	no	—
164	C₂×Q₈	—	101	8	no	—
165	C₂×Q₈	—	101	4	no	—
166	C₂×Q₈	—	404	4	no	—
167	C₂×Q₈	—	404	2	no	—
168	C₂×D₄	—	101	4	no	—
169	C₂×D₄	—	404	2	no	—
170	C₂²×C₄	—	101	2	no	—
171	C₂²×C₄	—	101	2	no	—
172	C₂²×C₄	—	101	2	no	—
173	C₂²×C₄	—	101	8	no	—
174	C₂²×C₄	—	404	4	no	—
175	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
176	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
180	C₄⋊C₄	—	101	8	no	—
181	C₄⋊C₄	—	101	8	no	—
182	C₄⋊C₄	—	101	8	no	—
183	C₄⋊C₄	—	101	4	no	—
184	C₄⋊C₄	—	101	8	no	—
185	C₄⋊C₄	—	101	4	no	—
186	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
187	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
188	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
189	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
190	C₂²⋊C₄	—	101	8	no	—
191	C₂²⋊C₄	—	101	8	no	—
192	C₂²⋊C₄	—	101	8	no	—
193	C₂²⋊C₄	—	101	8	no	—
194	C₂²⋊C₄	—	101	8	no	—
195	C₄²	—	101	8	no	—
196	C₄²	C₂²×D₁₀₁	1	4	yes	—
197	C₂³	—	101	4	no	—
198	C₂³	—	202	4	no	—
199	C₂³	—	202	4	no	—
200	C₂³	—	202	4	no	—
201	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
202	Q₈	—	404	8	no	—
203	Q₈	—	404	4	no	—
204	D₄	—	404	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	101	4	no	—
211	C₂×C₄	—	101	4	no	—
212	C₂×C₄	—	101	4	no	—
213	C₂×C₄	—	404	8	no	—
214	C₂×C₄	—	202	4	no	—
215	C₂×C₄	—	202	4	no	—
216	C₂×C₄	—	202	4	no	—
217	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
218	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
219	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
220	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
221	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
222	C₂²	—	404	8	no	—
223	C₂²	—	202	4	no	—
224	C₂²	—	202	4	no	—
225	C₂²	—	202	4	no	—
226	C₂²	#15692301	1	2	yes	—
227	C₂²	#15692302	1	2	yes	—
228	C₂²	#15692303	1	2	yes	—
229	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	202	4	no	—
236	C₄	—	202	4	no	—
237	C₄	—	202	4	no	—
238	C₂	—	404	4	no	—
239	C₂	#20265931	1	4	yes	direct product factor
240	C₂	C₂×#15692301	1	1	yes	characteristic
241	C₂	C₂×#15692302	1	1	yes	characteristic
242	C₂	C₂×#15692303	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#125×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	C₂×#563
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #20265931	C₂	direct (N × H)
2. C₂×#15692254	C₂	semidirect
3. C₂×#15692250	C₂	semidirect
4. C₂×#15692250	C₂	semidirect
5. C₂×#15692247	C₂	semidirect
8. C₂×#15692235	C₂	semidirect
9. C₂×#15692233	C₂	semidirect
10. C₂×#125×C₁₀₁	C₂	semidirect
15. #15692254	C₂²	semidirect
16. #15692250	C₂²	semidirect
17. #15692250	C₂²	semidirect
18. #15692247	C₂²	semidirect
21. #15692235	C₂²	semidirect
22. #15692233	C₂²	semidirect
35. #125×C₁₀₁	C₂²	semidirect
116. C₂×C₁₀₁	#563	semidirect
122. C₁₀₁	C₂×#563	semidirect
128. C₂×#125	D₁₀₁	semidirect
141. #125	C₂×D₁₀₁	semidirect
239. C₂	#20265931	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265931	2
C₂×#15692254	2
C₂×#15692250	2
C₂×#15692247	2
C₂×#15692243	2
C₂×#15692240	2
C₂×#15692235	2
C₂×#15692233	2
C₂×#125×C₁₀₁	2
C₂×#563	101

Subgroups — C2×#20265931 #27185266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265931 #27185266