Subgroups — C₂×#20265024 #27173461

Order 12928.

1360 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108051	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
4	#20265024	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
9	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₀₁	—	2	8	no	—
11	C₂²×C₄×D₁₀₁	C₄	1	8	yes	complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
13	C₂×#13108051	C₄	1	16	yes	complemented
14	C₂×#13108051	C₂²	1	8	yes	—
15	C₂×C₄×Dic₁₀₁	—	2	8	no	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	8	yes	complemented
17	#15692162	C₂²	1	4	yes	complemented
18	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
20	#95×C₁₀₁	C₂²	1	4	yes	complemented
21	C₂³×D₁₀₁	—	2	4	no	—
22	C₂³×D₁₀₁	—	2	2	no	—
23	C₂³×D₁₀₁	—	2	2	no	—
24	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
25	C₂³×D₁₀₁	C₂×C₄	1	8	yes	—
26	C₂³×D₁₀₁	C₂³	1	2	yes	—
27	C₂²×Dic₁₀₁	—	2	4	no	—
28	C₂²×Dic₁₀₁	—	2	4	no	—
29	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
30	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
31	C₂²×Dic₁₀₁	D₄	1	8	yes	—
32	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
33	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
34	C₂×C₄×D₁₀₁	—	2	32	no	—
35	C₂×C₄×D₁₀₁	—	2	16	no	—
36	C₂×C₄×D₁₀₁	C₂×C₄	1	16	yes	complemented
37	#13108051	C₂×C₄	1	32	yes	complemented
38	C₄×Dic₁₀₁	—	2	16	no	—
39	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₁	—	2	4	no	—
41	C₂²×C₄×C₁₀₁	—	2	4	no	—
42	C₂²×C₄×C₁₀₁	—	2	4	no	—
43	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
44	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
45	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
47	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
48	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
49	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
50	C₂³×C₁₀₁	#41	1	2	yes	—
51	C₂³×C₁₀₁	#41	1	2	yes	—
52	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
53	C₂²×D₁₀₁	—	2	8	no	—
54	C₂²×D₁₀₁	—	2	8	no	—
55	C₂²×D₁₀₁	—	2	16	no	—
56	C₂²×D₁₀₁	—	2	16	no	—
57	C₂²×D₁₀₁	—	2	4	no	—
58	C₂²×D₁₀₁	—	2	2	no	—
59	C₂²×D₁₀₁	—	2	2	no	—
60	C₂²×D₁₀₁	C₄²	1	16	yes	complemented
61	C₂²×D₁₀₁	C₂²×C₄	1	8	yes	—
62	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₀₁	—	2	8	no	—
64	C₂×C₄×C₁₀₁	—	2	16	no	—
65	C₂×C₄×C₁₀₁	—	2	4	no	—
66	C₂×C₄×C₁₀₁	—	2	4	no	—
67	C₂×C₄×C₁₀₁	—	2	4	no	—
68	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	complemented
70	C₂×Dic₁₀₁	—	2	16	no	—
71	C₂×Dic₁₀₁	—	2	4	no	—
72	C₂×Dic₁₀₁	—	2	4	no	—
73	C₂×Dic₁₀₁	C₂²⋊C₄	1	16	yes	complemented
74	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
75	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	complemented
76	C₂×Dic₁₀₁	C₂×D₄	1	8	yes	—
77	C₄×D₁₀₁	—	2	32	no	—
78	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	—
79	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
80	C₂²×C₁₀₁	#117	1	4	yes	—
81	C₂²×C₁₀₁	C₄×D₄	1	8	yes	—
82	C₂²×C₁₀₁	C₄×D₄	1	8	yes	—
83	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
85	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
86	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
88	C₂×D₁₀₁	—	2	4	no	—
89	C₂×D₁₀₁	—	2	4	no	—
90	C₂×D₁₀₁	—	2	8	no	—
91	C₂×D₁₀₁	—	2	8	no	—
92	C₂×D₁₀₁	—	2	32	no	—
93	C₂×D₁₀₁	C₂×C₄²	1	8	yes	complemented
94	C₄×C₁₀₁	—	2	4	no	—
95	C₄×C₁₀₁	—	2	8	no	—
96	Dic₁₀₁	—	2	8	no	—
97	Dic₁₀₁	C₂×C₂²⋊C₄	1	8	yes	complemented
98	C₂×C₁₀₁	C₄×C₂²⋊C₄	1	8	yes	complemented
99	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
100	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
102	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
103	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
104	D₁₀₁	—	2	8	no	—
105	C₂×C₄×C₂²⋊C₄	—	101	1	no	maximal, Sylow
106	C₁₀₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴×C₄	—	101	1	no	—
108	C₂²×C₂²⋊C₄	—	101	2	no	—
109	C₂²×C₄²	—	101	2	no	—
110	C₄×C₂²⋊C₄	—	101	8	no	—
111	C₂×#95	—	101	1	no	—
112	C₂×#95	D₁₀₁	1	1	yes	characteristic, complemented
113	C₂⁵	—	101	1	no	—
114	C₂³×C₄	—	101	1	no	—
115	C₂³×C₄	—	101	2	no	—
116	C₂³×C₄	—	101	8	no	—
117	C₂³×C₄	—	101	4	no	—
118	C₂³×C₄	—	202	8	no	—
119	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
120	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
121	C₂×C₂²⋊C₄	—	101	16	no	—
122	C₂×C₂²⋊C₄	—	101	8	no	—
123	C₂×C₄²	—	101	8	no	—
124	C₂×C₄²	—	202	8	no	—
125	#95	—	101	4	no	—
126	#95	C₂×D₁₀₁	1	4	yes	complemented
127	C₂⁴	—	101	8	no	—
128	C₂⁴	—	101	4	no	—
129	C₂⁴	—	101	2	no	—
130	C₂⁴	—	202	4	no	—
131	C₂⁴	—	202	2	no	—
132	C₂⁴	—	202	2	no	—
133	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	101	4	no	—
138	C₂²×C₄	—	101	8	no	—
139	C₂²×C₄	—	101	2	no	—
140	C₂²×C₄	—	101	8	no	—
141	C₂²×C₄	—	101	4	no	—
142	C₂²×C₄	—	101	16	no	—
143	C₂²×C₄	—	202	16	no	—
144	C₂²×C₄	—	202	32	no	—
145	C₂²×C₄	—	202	4	no	—
146	C₂²×C₄	—	202	4	no	—
147	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
148	C₂²×C₄	C₄×D₁₀₁	1	8	yes	complemented
149	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
150	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
151	C₂²⋊C₄	—	101	32	no	—
152	C₄²	—	202	16	no	—
153	C₂³	—	101	8	no	—
154	C₂³	—	101	4	no	—
155	C₂³	—	101	16	no	—
156	C₂³	—	202	4	no	—
157	C₂³	—	202	2	no	—
158	C₂³	—	202	2	no	—
159	C₂³	—	202	8	no	—
160	C₂³	—	202	8	no	—
161	C₂³	—	202	16	no	—
162	C₂³	—	202	16	no	—
163	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
164	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
165	C₂³	D₄×D₁₀₁	1	4	yes	—
166	C₂³	#13108073	1	2	yes	—
167	C₂³	#13108075	1	2	yes	—
168	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	16	no	—
174	C₂×C₄	—	101	4	no	—
175	C₂×C₄	—	101	8	no	—
176	C₂×C₄	—	101	16	no	—
177	C₂×C₄	—	101	8	no	—
178	C₂×C₄	—	202	32	no	—
179	C₂×C₄	—	202	4	no	—
180	C₂×C₄	—	202	4	no	—
181	C₂×C₄	—	202	16	no	—
182	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
183	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	complemented
184	C₂²	—	101	8	no	—
185	C₂²	—	202	32	no	—
186	C₂²	—	202	4	no	—
187	C₂²	—	202	4	no	—
188	C₂²	—	202	8	no	—
189	C₂²	—	202	8	no	—
190	C₂²	C₄²×D₁₀₁	1	4	yes	—
191	C₂²	C₂²⋊C₄×D₁₀₁	1	4	yes	—
192	C₂²	#15692213	1	4	yes	—
193	C₂²	#15692215	1	8	yes	—
194	C₂²	#15692217	1	8	yes	—
195	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
196	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
197	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
198	C₂²	C₂×#13108073	1	1	yes	characteristic
199	C₂²	C₂×#13108075	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	2	8	no	—
202	C₄	—	101	8	no	—
203	C₄	—	202	8	no	—
204	C₂	—	202	8	no	—
205	C₂	#20265024	1	8	yes	direct product factor
206	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
207	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
208	C₂	C₂×#15692213	1	1	yes	characteristic
209	C₂	C₂×#15692215	1	2	yes	—
210	C₂	C₂×#15692217	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×#95×C₁₀₁	C₂
socle	C₂⁴×C₁₀₁	C₂³
prime radical	C₂⁴×D₁₀₁	C₂²
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#95	D₁₀₁
hypocenter	C₁₀₁	C₂×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×Dic₁₀₁	C₂	semidirect
4. #20265024	C₂	direct (N × H)
5. C₂×#15692162	C₂	semidirect
6. C₂×#95×C₁₀₁	C₂	semidirect
11. C₂²×C₄×D₁₀₁	C₄	semidirect
13. C₂×#13108051	C₄	semidirect
16. C₂×C₄×Dic₁₀₁	C₂²	semidirect
17. #15692162	C₂²	semidirect
20. #95×C₁₀₁	C₂²	semidirect
30. C₂²×Dic₁₀₁	C₂×C₄	semidirect
36. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
37. #13108051	C₂×C₄	semidirect
44. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
60. C₂²×D₁₀₁	C₄²	semidirect
69. C₂×C₄×C₁₀₁	C₂²×C₄	semidirect
73. C₂×Dic₁₀₁	C₂²⋊C₄	semidirect
75. C₂×Dic₁₀₁	C₂²×C₄	semidirect
93. C₂×D₁₀₁	C₂×C₄²	semidirect
97. Dic₁₀₁	C₂×C₂²⋊C₄	semidirect
98. C₂×C₁₀₁	C₄×C₂²⋊C₄	semidirect
106. C₁₀₁	C₂×C₄×C₂²⋊C₄	semidirect
112. C₂×#95	D₁₀₁	semidirect
126. #95	C₂×D₁₀₁	semidirect
148. C₂²×C₄	C₄×D₁₀₁	semidirect
183. C₂×C₄	C₂×C₄×D₁₀₁	semidirect
205. C₂	#20265024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂²×#13108051	2
C₂²×C₄×Dic₁₀₁	2
#20265024	2
C₂×#15692162	2
C₂×#95×C₁₀₁	2
C₂×C₄×C₂²⋊C₄	101

Subgroups — C2×#20265024 #27173461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265024 #27173461