Subgroups — C₂×#20266204 #27187834

Order 12928.

616 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₀₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#11717581	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266204	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15692322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊#11717581	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#11717581	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692215	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₀₁⋊C₈	—	2	4	no	—
10	C₂×D₁₀₁⋊C₈	C₂²	1	4	yes	complemented
11	C₂²×#11717581	—	2	4	no	—
12	C₂²×#11717581	C₂²	1	1	yes	characteristic
13	C₂²×#11717581	C₂²	1	1	yes	characteristic
14	C₂²×#11717581	C₂²	1	1	yes	characteristic
15	C₂²×#11717581	C₂²	1	1	yes	characteristic
16	C₂²×#11717581	C₂²	1	4	yes	complemented
17	#15692322	C₂²	1	4	yes	complemented
18	#15692312	C₂²	1	4	yes	complemented
19	C₂²⋊#11717581	C₂²	1	4	yes	complemented
20	C₄×#11717581	C₂²	1	4	yes	complemented
21	C₂²×C₂²⋊D₁₀₁	C₄	1	1	yes	characteristic
22	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
24	#15692215	C₄	1	4	yes	—
25	#15692215	C₂²	1	4	yes	—
26	C₂×#13108053	C₄	1	1	yes	characteristic
27	C₂×#13108051	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
30	D₁₀₁⋊C₈	—	2	8	no	—
31	C₂×#11717581	—	2	2	no	—
32	C₂×#11717581	—	2	2	no	—
33	C₂×#11717581	—	2	2	no	—
34	C₂×#11717581	—	2	8	no	—
35	C₂×#11717581	D₄	1	4	yes	complemented
36	C₂×#11717581	C₂³	1	2	yes	—
37	C₂×#11717581	C₂³	1	2	yes	—
38	C₂×#11717581	C₂³	1	2	yes	—
39	C₂×#11717581	C₂³	1	2	yes	—
40	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₀₁	C₈	1	8	yes	complemented
42	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
43	C₂²×Dic₁₀₁	—	2	2	no	—
44	C₂²×Dic₁₀₁	—	2	2	no	—
45	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₀₁	—	2	2	no	—
51	C₂×C₄×D₁₀₁	—	2	2	no	—
52	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
54	#13108053	C₂×C₄	1	4	yes	—
55	#13108051	C₂×C₄	1	4	yes	—
56	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₀₁	C₂³	1	2	yes	—
58	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
62	C₂³×C₁₀₁	—	2	2	no	—
63	C₂³×C₁₀₁	—	2	2	no	—
64	C₂³×C₁₀₁	C₂×C₈	1	4	yes	complemented
65	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₁₀₁	—	2	2	no	—
68	C₂²×D₁₀₁	—	2	2	no	—
69	C₂²×D₁₀₁	C₂×C₈	1	4	yes	complemented
70	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₀₁	—	2	2	no	—
72	C₂×C₄×C₁₀₁	—	2	2	no	—
73	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂²⋊D₁₀₁	C₂×C₈	1	16	yes	complemented
76	C₂×Dic₁₀₁	—	2	2	no	—
77	C₂×Dic₁₀₁	—	2	4	no	—
78	C₂×Dic₁₀₁	—	2	2	no	—
79	C₂×Dic₁₀₁	—	2	2	no	—
80	C₂×Dic₁₀₁	C₂×C₈	1	4	yes	complemented
81	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
85	C₂×Dic₁₀₁	#41	1	2	yes	—
86	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
87	C₄×D₁₀₁	—	2	4	no	—
88	C₄×D₁₀₁	—	2	2	no	—
89	C₄×D₁₀₁	—	2	2	no	—
90	#11717581	—	2	2	no	—
91	#11717581	—	2	2	no	—
92	#11717581	—	2	2	no	—
93	#11717581	C₂×D₄	1	4	yes	complemented
94	C₂²×C₁₀₁	—	2	2	no	—
95	C₂²×C₁₀₁	—	2	2	no	—
96	C₂²×C₁₀₁	—	2	8	no	—
97	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
98	C₂²×C₁₀₁	C₂²×C₈	1	2	yes	—
99	C₂²×C₁₀₁	C₂²×C₈	1	4	yes	complemented
100	C₂²×C₁₀₁	#131	1	2	yes	—
101	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₀₁	—	2	2	no	—
103	C₂×D₁₀₁	—	2	2	no	—
104	C₂×D₁₀₁	—	2	8	no	—
105	C₂×D₁₀₁	C₂²×C₈	1	4	yes	complemented
106	C₄×C₁₀₁	—	2	2	no	—
107	C₄×C₁₀₁	—	2	2	no	—
108	Dic₁₀₁	—	2	2	no	—
109	Dic₁₀₁	C₄×D₄	1	2	yes	—
110	Dic₁₀₁	C₂²×C₈	1	4	yes	complemented
111	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
112	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
113	C₂×C₁₀₁	—	2	4	no	—
114	C₂×C₁₀₁	C₈×D₄	1	4	yes	complemented
115	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×C₁₀₁	C₂³×C₈	1	1	yes	characteristic
117	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
118	D₁₀₁	—	2	4	no	—
119	C₂×C₈×D₄	—	101	1	no	maximal, Sylow
120	C₁₀₁	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₈	—	101	1	no	—
122	C₂³×C₈	—	101	1	no	—
123	C₂×C₄×D₄	—	101	1	no	—
124	C₈×D₄	—	101	8	no	—
125	C₂×C₄⋊C₈	—	101	1	no	—
126	C₂×C₂²⋊C₈	—	101	1	no	—
127	C₂×C₂²⋊C₈	—	101	1	no	—
128	C₂×C₄×C₈	—	101	1	no	—
129	C₂²×D₄	—	101	1	no	—
130	C₂³×C₄	—	101	1	no	—
131	C₂³×C₄	—	101	1	no	—
132	C₂²×C₈	—	101	1	no	—
133	C₂²×C₈	—	101	1	no	—
134	C₂²×C₈	—	101	1	no	—
135	C₂²×C₈	—	101	1	no	—
136	C₂²×C₈	—	101	4	no	—
137	C₂²×C₈	—	101	4	no	—
138	C₂²×C₈	—	202	4	no	—
139	C₂²×C₈	—	202	4	no	—
140	C₄×D₄	—	101	4	no	—
141	C₄×D₄	—	101	4	no	—
142	C₂×C₄⋊C₄	—	101	1	no	—
143	C₂×C₂²⋊C₄	—	101	1	no	—
144	C₂×C₂²⋊C₄	#87938	1	1	yes	characteristic
145	C₂×C₄²	—	101	1	no	—
146	C₄⋊C₈	—	101	4	no	—
147	C₂²⋊C₈	—	101	4	no	—
148	C₂²⋊C₈	—	101	4	no	—
149	C₄×C₈	—	101	4	no	—
150	C₂⁴	—	101	1	no	—
151	C₂⁴	C₂×#87938	1	1	yes	characteristic
152	C₂×D₄	—	101	8	no	—
153	C₂×D₄	—	101	4	no	—
154	C₂²×C₄	—	101	1	no	—
155	C₂²×C₄	—	101	1	no	—
156	C₂²×C₄	—	101	2	no	—
157	C₂²×C₄	—	101	2	no	—
158	C₂²×C₄	—	101	1	no	—
159	C₂²×C₄	—	101	2	no	—
160	C₂²×C₄	—	101	2	no	—
161	C₂²×C₄	—	202	2	no	—
162	C₂²×C₄	—	202	2	no	—
163	C₂²×C₄	—	202	2	no	—
164	C₂²×C₄	—	202	2	no	—
165	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
166	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
167	C₂×C₈	—	101	2	no	—
168	C₂×C₈	—	101	4	no	—
169	C₂×C₈	—	101	2	no	—
170	C₂×C₈	—	101	2	no	—
171	C₂×C₈	—	101	2	no	—
172	C₂×C₈	—	202	8	no	—
173	C₂×C₈	—	202	8	no	—
174	C₂×C₈	—	202	2	no	—
175	C₂×C₈	—	202	2	no	—
176	C₂×C₈	—	202	2	no	—
177	C₄⋊C₄	—	101	4	no	—
178	C₂²⋊C₄	—	101	4	no	—
179	C₂²⋊C₄	C₂×#87938	1	4	yes	—
180	C₄²	—	101	2	no	—
181	C₄²	—	101	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	101	4	no	—
185	C₂³	—	101	2	no	—
186	C₂³	—	202	2	no	—
187	C₂³	—	202	2	no	—
188	C₂³	D₁₀₁⋊C₈	1	4	yes	complemented
189	C₂³	C₂²×#87938	1	2	yes	—
190	C₂³	C₂²×#87938	1	1	yes	characteristic
191	D₄	—	101	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	101	2	no	—
195	C₂×C₄	—	101	2	no	—
196	C₂×C₄	—	101	4	no	—
197	C₂×C₄	—	101	2	no	—
198	C₂×C₄	—	101	1	no	—
199	C₂×C₄	—	101	1	no	—
200	C₂×C₄	—	101	2	no	—
201	C₂×C₄	—	202	2	no	—
202	C₂×C₄	—	202	2	no	—
203	C₂×C₄	—	202	4	no	—
204	C₂×C₄	—	202	2	no	—
205	C₂×C₄	—	202	2	no	—
206	C₂×C₄	—	202	4	no	—
207	C₂×C₄	—	202	2	no	—
208	C₂×C₄	C₂²×#87938	1	2	yes	—
209	C₂×C₄	C₂²×#87938	1	2	yes	—
210	C₈	—	101	4	no	—
211	C₈	—	202	2	no	—
212	C₈	—	202	2	no	—
213	C₈	—	202	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	101	4	no	—
218	C₂²	—	202	2	no	—
219	C₂²	—	202	2	no	—
220	C₂²	—	202	8	no	—
221	C₂²	D₄×#87938	1	2	yes	—
222	C₂²	C₂×D₁₀₁⋊C₈	1	2	yes	—
223	C₂²	C₂×D₁₀₁⋊C₈	1	4	yes	complemented
224	C₂²	#15692342	1	2	yes	—
225	C₂²	C₂³×#87938	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	101	1	no	—
229	C₄	—	101	1	no	—
230	C₄	—	101	2	no	—
231	C₄	—	101	4	no	—
232	C₄	—	202	2	no	—
233	C₂	—	2	4	no	—
234	C₂	—	202	4	no	—
235	C₂	#20266204	1	4	yes	direct product factor
236	C₂	C₂×D₄×#87938	1	1	yes	characteristic
237	C₂	C₂²×D₁₀₁⋊C₈	1	1	yes	characteristic
238	C₂	C₂×#15692342	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×#87938
commutator	C₂×C₁₀₁	C₂³×C₈
Frattini	C₂²	C₂³×#87938
Fitting	C₂×C₂²⋊C₄×C₁₀₁	C₄
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂²×C₂²⋊D₁₀₁	C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#87938
hypocenter	C₁₀₁	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₀₁⋊C₈	C₂	semidirect
2. C₂³×#11717581	C₂	semidirect
3. #20266204	C₂	direct (N × H)
4. C₂×#15692322	C₂	semidirect
5. C₂×#15692312	C₂	semidirect
6. C₂×C₂²⋊#11717581	C₂	semidirect
7. C₂×C₄×#11717581	C₂	semidirect
10. C₂×D₁₀₁⋊C₈	C₂²	semidirect
16. C₂²×#11717581	C₂²	semidirect
17. #15692322	C₂²	semidirect
18. #15692312	C₂²	semidirect
19. C₂²⋊#11717581	C₂²	semidirect
20. C₄×#11717581	C₂²	semidirect
35. C₂×#11717581	D₄	semidirect
41. C₂×C₂²⋊D₁₀₁	C₈	semidirect
64. C₂³×C₁₀₁	C₂×C₈	semidirect
69. C₂²×D₁₀₁	C₂×C₈	semidirect
75. C₂²⋊D₁₀₁	C₂×C₈	semidirect
80. C₂×Dic₁₀₁	C₂×C₈	semidirect
93. #11717581	C₂×D₄	semidirect
99. C₂²×C₁₀₁	C₂²×C₈	semidirect
105. C₂×D₁₀₁	C₂²×C₈	semidirect
110. Dic₁₀₁	C₂²×C₈	semidirect
114. C₂×C₁₀₁	C₈×D₄	semidirect
120. C₁₀₁	C₂×C₈×D₄	semidirect
188. C₂³	D₁₀₁⋊C₈	semidirect
223. C₂²	C₂×D₁₀₁⋊C₈	semidirect
235. C₂	#20266204	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₀₁⋊C₈	2
C₂³×#11717581	2
#20266204	2
C₂×#15692322	2
C₂×#15692312	2
C₂×C₂²⋊#11717581	2
C₂×C₄×#11717581	2
C₂×#15692215	2
C₂×C₈×D₄	101

Subgroups — C2×#20266204 #27187834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20266204 #27187834