Subgroups — C₂×#20265056 #27173550

Order 12928.

1576 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13108051	C₂	1	2	yes	maximal, complemented
4	#20265056	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15692162	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
8	C₂²×C₂²⋊D₁₀₁	—	2	4	no	—
9	C₂²×C₂²⋊D₁₀₁	—	2	4	no	—
10	C₂²×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
11	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
12	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
13	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₁₀₁	—	2	4	no	—
15	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
16	C₂×#13108051	—	2	8	no	—
17	C₂×#13108051	C₂²	1	8	yes	complemented
18	#15692162	C₂²	1	8	yes	complemented
19	C₂⁵×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₁	—	2	4	no	—
21	C₂³×C₄×C₁₀₁	—	2	8	no	—
22	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₀₁	C₂²	1	4	yes	complemented
24	C₂³×D₁₀₁	—	2	2	no	—
25	C₂³×D₁₀₁	—	2	2	no	—
26	C₂³×D₁₀₁	—	2	4	no	—
27	C₂³×D₁₀₁	—	2	4	no	—
28	C₂³×D₁₀₁	C₂×C₄	1	4	yes	complemented
29	C₂³×D₁₀₁	C₂³	1	2	yes	—
30	C₂×C₂²⋊D₁₀₁	—	2	32	no	—
31	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
32	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
33	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	16	yes	complemented
34	C₂²×Dic₁₀₁	—	2	2	no	—
35	C₂²×Dic₁₀₁	—	2	2	no	—
36	C₂²×Dic₁₀₁	—	2	4	no	—
37	C₂²×Dic₁₀₁	—	2	4	no	—
38	C₂²×Dic₁₀₁	—	2	4	no	—
39	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
40	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²⋊Dic₁₀₁	—	2	8	no	—
43	#13108051	—	2	16	no	—
44	C₂⁴×C₁₀₁	—	2	4	no	—
45	C₂⁴×C₁₀₁	—	2	4	no	—
46	C₂⁴×C₁₀₁	C₂×C₄	1	4	yes	—
47	C₂⁴×C₁₀₁	D₄	1	4	yes	—
48	C₂⁴×C₁₀₁	D₄	1	4	yes	—
49	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
50	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₁	—	2	32	no	—
52	C₂²×C₄×C₁₀₁	—	2	4	no	—
53	C₂²×C₄×C₁₀₁	—	2	4	no	—
54	C₂²×C₄×C₁₀₁	—	2	8	no	—
55	C₂²×C₄×C₁₀₁	—	2	16	no	—
56	C₂²×C₄×C₁₀₁	D₄	1	8	yes	complemented
57	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
58	C₂³×C₁₀₁	—	2	16	no	—
59	C₂³×C₁₀₁	—	2	8	no	—
60	C₂³×C₁₀₁	—	2	16	no	—
61	C₂³×C₁₀₁	—	2	8	no	—
62	C₂³×C₁₀₁	—	2	4	no	—
63	C₂³×C₁₀₁	—	2	4	no	—
64	C₂³×C₁₀₁	C₂²⋊C₄	1	16	yes	complemented
65	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
66	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
67	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
68	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
69	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
70	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
71	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
72	C₂³×C₁₀₁	#41	1	4	yes	—
73	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
74	C₂²×D₁₀₁	—	2	8	no	—
75	C₂²×D₁₀₁	—	2	16	no	—
76	C₂²×D₁₀₁	—	2	4	no	—
77	C₂²×D₁₀₁	—	2	2	no	—
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	—	2	4	no	—
80	C₂²×D₁₀₁	—	4	4	no	—
81	C₂²×D₁₀₁	—	4	8	no	—
82	C₂²×D₁₀₁	—	4	4	no	—
83	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	complemented
84	C₂×C₄×C₁₀₁	—	2	32	no	—
85	C₂×C₄×C₁₀₁	—	2	16	no	—
86	C₂×C₄×C₁₀₁	—	2	4	no	—
87	C₂×C₄×C₁₀₁	—	2	4	no	—
88	C₂×C₄×C₁₀₁	C₂×D₄	1	8	yes	complemented
89	C₂²⋊D₁₀₁	—	2	32	no	—
90	C₂×Dic₁₀₁	—	2	8	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	4	no	—
94	C₂×Dic₁₀₁	—	2	4	no	—
95	C₂×Dic₁₀₁	—	2	4	no	—
96	C₂×Dic₁₀₁	—	4	8	no	—
97	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	complemented
98	C₂²×C₁₀₁	—	2	8	no	—
99	C₂²×C₁₀₁	—	2	4	no	—
100	C₂²×C₁₀₁	—	2	8	no	—
101	C₂²×C₁₀₁	—	2	4	no	—
102	C₂²×C₁₀₁	—	2	32	no	—
103	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	8	yes	complemented
105	C₂²×C₁₀₁	C₄×D₄	1	8	yes	—
106	C₂²×C₁₀₁	#120	1	4	yes	—
107	C₂²×C₁₀₁	#121	1	8	yes	—
108	C₂²×C₁₀₁	#123	1	4	yes	—
109	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
110	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
112	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
113	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
114	C₂×D₁₀₁	—	2	4	no	—
115	C₂×D₁₀₁	—	2	8	no	—
116	C₂×D₁₀₁	—	4	2	no	—
117	C₂×D₁₀₁	—	4	4	no	—
118	C₂×D₁₀₁	—	4	2	no	—
119	C₂×D₁₀₁	—	4	16	no	—
120	C₄×C₁₀₁	—	2	8	no	—
121	Dic₁₀₁	—	2	4	no	—
122	Dic₁₀₁	—	4	4	no	—
123	C₂×C₁₀₁	—	2	8	no	—
124	C₂×C₁₀₁	#386	1	8	yes	complemented
125	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
127	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
128	C₂×C₁₀₁	C₂×#121	1	2	yes	—
129	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
130	D₁₀₁	—	4	4	no	—
131	C₂×#386	—	101	1	no	maximal, Sylow
132	C₁₀₁	C₂×#386	1	1	yes	characteristic, complemented, Sylow
133	C₂³×D₄	—	101	1	no	—
134	C₂⁴×C₄	D₁₀₁	1	1	yes	characteristic, complemented
135	C₂²×C₂²⋊C₄	—	101	1	no	—
136	C₂²×C₂²⋊C₄	—	101	2	no	—
137	#386	—	101	8	no	—
138	C₂×#95	—	101	2	no	—
139	C₂⁵	—	101	1	no	—
140	C₂⁵	C₂×D₁₀₁	1	1	yes	characteristic
141	C₂²×D₄	—	101	8	no	—
142	C₂²×D₄	—	101	4	no	—
143	C₂²×D₄	—	202	4	no	—
144	C₂²×D₄	—	202	4	no	—
145	C₂³×C₄	—	2	4	no	—
146	C₂³×C₄	—	2	8	no	—
147	C₂³×C₄	—	101	1	no	—
148	C₂³×C₄	—	101	2	no	—
149	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
150	C₂³×C₄	C₂×D₁₀₁	1	4	yes	complemented
151	C₂×C₂²⋊C₄	—	101	4	no	—
152	C₂×C₂²⋊C₄	—	101	8	no	—
153	C₂×C₂²⋊C₄	—	202	4	no	—
154	C₂×C₂²⋊C₄	—	202	8	no	—
155	#95	—	101	8	no	—
156	C₂⁴	—	2	4	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	—	101	4	no	—
159	C₂⁴	—	101	2	no	—
160	C₂⁴	—	202	4	no	—
161	C₂⁴	—	202	2	no	—
162	C₂⁴	—	202	2	no	—
163	C₂⁴	—	202	4	no	—
164	C₂⁴	C₄×D₁₀₁	1	4	yes	—
165	C₂⁴	D₄₀₄	1	4	yes	—
166	C₂⁴	C₂²⋊D₁₀₁	1	4	yes	—
167	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
168	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
169	C₂×D₄	—	101	16	no	—
170	C₂×D₄	—	202	32	no	—
171	C₂×D₄	—	202	8	no	—
172	C₂×D₄	—	202	8	no	—
173	C₂²×C₄	—	2	32	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	16	no	—
178	C₂²×C₄	—	101	4	no	—
179	C₂²×C₄	—	101	2	no	—
180	C₂²×C₄	—	101	4	no	—
181	C₂²×C₄	—	202	4	no	—
182	C₂²×C₄	—	202	4	no	—
183	C₂²×C₄	—	202	4	no	—
184	C₂²×C₄	—	202	2	no	—
185	C₂²×C₄	—	202	2	no	—
186	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	complemented
187	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
188	C₂²⋊C₄	—	202	8	no	—
189	C₂²⋊C₄	—	202	16	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	16	no	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	16	no	—
195	C₂³	—	2	8	no	—
196	C₂³	—	101	4	no	—
197	C₂³	—	404	4	no	—
198	C₂³	—	404	8	no	—
199	C₂³	—	404	4	no	—
200	C₂³	—	202	4	no	—
201	C₂³	—	202	2	no	—
202	C₂³	—	202	2	no	—
203	C₂³	—	202	4	no	—
204	C₂³	—	202	8	no	—
205	C₂³	—	202	16	no	—
206	C₂³	#13108051	1	16	yes	complemented
207	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
208	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
209	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
210	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
211	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
212	C₂³	C₂×D₄₀₄	1	4	yes	—
213	C₂³	C₂×D₄₀₄	1	2	yes	—
214	C₂³	#13108074	1	4	yes	—
215	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
216	D₄	—	202	32	no	—
217	C₂×C₄	—	2	32	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	101	4	no	—
222	C₂×C₄	—	404	8	no	—
223	C₂×C₄	—	202	4	no	—
224	C₂×C₄	—	202	4	no	—
225	C₂×C₄	—	202	4	no	—
226	C₂×C₄	—	202	2	no	—
227	C₂×C₄	—	202	2	no	—
228	C₂×C₄	—	202	8	no	—
229	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	8	yes	complemented
230	C₂²	—	2	32	no	—
231	C₂²	—	2	8	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	404	16	no	—
236	C₂²	—	404	2	no	—
237	C₂²	—	404	4	no	—
238	C₂²	—	404	2	no	—
239	C₂²	—	202	4	no	—
240	C₂²	—	202	8	no	—
241	C₂²	C₂×#13108051	1	4	yes	—
242	C₂²	C₂×#13108051	1	8	yes	complemented
243	C₂²	C₄×C₂²⋊D₁₀₁	1	8	yes	—
244	C₂²	#15692224	1	4	yes	—
245	C₂²	#15692229	1	8	yes	—
246	C₂²	#15692238	1	4	yes	—
247	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
248	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
249	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
250	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
251	C₂²	C₂×#13108074	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	404	4	no	—
254	C₄	—	202	4	no	—
255	C₂	—	2	8	no	—
256	C₂	—	404	4	no	—
257	C₂	#20265056	1	8	yes	direct product factor
258	C₂	C₂²×#13108051	1	1	yes	characteristic
259	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	2	yes	—
260	C₂	C₂×#15692224	1	1	yes	characteristic
261	C₂	C₂×#15692229	1	2	yes	—
262	C₂	C₂×#15692238	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂⁴×C₄×C₁₀₁	C₂
socle	C₂⁴×C₁₀₁	C₂³
prime radical	C₂³×C₂²⋊D₁₀₁	C₂
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂⁴×C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#386
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊Dic₁₀₁	C₂	semidirect
3. C₂²×#13108051	C₂	semidirect
4. #20265056	C₂	direct (N × H)
5. C₂×#15692162	C₂	semidirect
6. C₂⁴×C₄×C₁₀₁	C₂	semidirect
10. C₂²×C₂²⋊D₁₀₁	C₄	semidirect
15. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
17. C₂×#13108051	C₂²	semidirect
18. #15692162	C₂²	semidirect
23. C₂³×C₄×C₁₀₁	C₂²	semidirect
28. C₂³×D₁₀₁	C₂×C₄	semidirect
33. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
39. C₂²×Dic₁₀₁	C₂×C₄	semidirect
56. C₂²×C₄×C₁₀₁	D₄	semidirect
64. C₂³×C₁₀₁	C₂²⋊C₄	semidirect
83. C₂²×D₁₀₁	C₂²×C₄	semidirect
88. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
97. C₂×Dic₁₀₁	C₂²×C₄	semidirect
104. C₂²×C₁₀₁	C₂×C₂²⋊C₄	semidirect
124. C₂×C₁₀₁	#386	semidirect
132. C₁₀₁	C₂×#386	semidirect
134. C₂⁴×C₄	D₁₀₁	semidirect
150. C₂³×C₄	C₂×D₁₀₁	semidirect
186. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
206. C₂³	#13108051	semidirect
229. C₂×C₄	C₂×C₂²⋊D₁₀₁	semidirect
242. C₂²	C₂×#13108051	semidirect
257. C₂	#20265056	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×C₂²⋊D₁₀₁	2
C₂²×C₂²⋊Dic₁₀₁	2
C₂²×#13108051	2
#20265056	2
C₂×#15692162	2
C₂⁴×C₄×C₁₀₁	2
C₂×#386	101

Subgroups — C2×#20265056 #27173550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265056 #27173550