Subgroups — C₂×#20265105 #27174359

Order 12928.

820 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265105	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×#13108053	—	2	4	no	—
14	C₂×#13108053	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊Dic₁₀₁	—	2	4	no	—
16	C₂×C₂²⋊Dic₁₀₁	—	2	4	no	—
17	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
19	#15692162	C₂²	1	4	yes	complemented
20	#15692162	C₂²	1	4	yes	complemented
21	#15692162	C₂²	1	4	yes	complemented
22	C₂⁵×C₁₀₁	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₀₁	—	2	4	no	—
26	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
27	C₂²×Dic₁₀₁	—	2	2	no	—
28	C₂²×Dic₁₀₁	—	2	2	no	—
29	C₂²×Dic₁₀₁	—	2	2	no	—
30	C₂²×Dic₁₀₁	—	2	2	no	—
31	C₂²×Dic₁₀₁	—	2	2	no	—
32	C₂²×Dic₁₀₁	—	2	2	no	—
33	C₂²×Dic₁₀₁	—	2	2	no	—
34	C₂²×Dic₁₀₁	—	2	2	no	—
35	C₂²×Dic₁₀₁	—	2	2	no	—
36	C₂²×Dic₁₀₁	—	2	2	no	—
37	C₂²×Dic₁₀₁	—	2	2	no	—
38	C₂²×Dic₁₀₁	D₄	1	4	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	#13108053	—	2	8	no	—
44	C₂²⋊Dic₁₀₁	—	2	8	no	—
45	C₂²⋊Dic₁₀₁	—	2	8	no	—
46	C₂⁴×C₁₀₁	—	2	2	no	—
47	C₂⁴×C₁₀₁	—	2	2	no	—
48	C₂⁴×C₁₀₁	—	2	2	no	—
49	C₂⁴×C₁₀₁	—	2	2	no	—
50	C₂⁴×C₁₀₁	—	2	2	no	—
51	C₂⁴×C₁₀₁	—	2	2	no	—
52	C₂⁴×C₁₀₁	D₄	1	4	yes	—
53	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
54	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₁	—	2	2	no	—
56	C₂²×C₄×C₁₀₁	—	2	2	no	—
57	C₂²×C₄×C₁₀₁	—	2	2	no	—
58	C₂²×C₄×C₁₀₁	—	2	2	no	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	—	2	2	no	—
61	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
64	C₂³×C₁₀₁	—	2	8	no	—
65	C₂³×C₁₀₁	—	2	8	no	—
66	C₂³×C₁₀₁	—	2	8	no	—
67	C₂³×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	—	2	2	no	—
69	C₂³×C₁₀₁	—	2	2	no	—
70	C₂³×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	—	2	2	no	—
73	C₂³×C₁₀₁	—	4	4	no	—
74	C₂³×C₁₀₁	—	4	4	no	—
75	C₂³×C₁₀₁	—	4	4	no	—
76	C₂³×C₁₀₁	—	4	4	no	—
77	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
78	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
79	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
80	C₂³×C₁₀₁	#41	1	2	yes	—
81	C₂³×C₁₀₁	#41	1	2	yes	—
82	C₂³×C₁₀₁	#41	1	2	yes	—
83	C₂³×C₁₀₁	#41	1	2	yes	—
84	C₂³×C₁₀₁	#41	1	2	yes	—
85	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	2	2	no	—
91	C₂×C₄×C₁₀₁	—	2	2	no	—
92	C₂×C₄×C₁₀₁	—	4	4	no	—
93	C₂×C₄×C₁₀₁	—	4	4	no	—
94	C₂×Dic₁₀₁	—	2	8	no	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	2	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	2	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	—	2	2	no	—
106	C₂×Dic₁₀₁	—	4	4	no	—
107	C₂×Dic₁₀₁	—	4	4	no	—
108	C₂×Dic₁₀₁	—	4	4	no	—
109	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	—	4	2	no	—
114	C₂²×C₁₀₁	—	4	2	no	—
115	C₂²×C₁₀₁	—	4	2	no	—
116	C₂²×C₁₀₁	—	4	2	no	—
117	C₂²×C₁₀₁	—	4	16	no	—
118	C₂²×C₁₀₁	#121	1	4	yes	—
119	C₂²×C₁₀₁	#123	1	4	yes	—
120	C₂²×C₁₀₁	#123	1	4	yes	—
121	C₂²×C₁₀₁	#123	1	4	yes	—
122	C₂²×C₁₀₁	#124	1	4	yes	—
123	C₂²×C₁₀₁	#126	1	4	yes	—
124	C₂²×C₁₀₁	#126	1	4	yes	—
125	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
132	C₄×C₁₀₁	—	4	2	no	—
133	C₄×C₁₀₁	—	4	2	no	—
134	Dic₁₀₁	—	2	4	no	—
135	Dic₁₀₁	—	4	2	no	—
136	Dic₁₀₁	—	4	2	no	—
137	Dic₁₀₁	—	4	2	no	—
138	C₂×C₁₀₁	—	4	4	no	—
139	C₂×C₁₀₁	#397	1	8	yes	complemented
140	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
141	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
142	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
143	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
144	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
145	C₂×C₁₀₁	C₂×#126	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×#126	1	1	yes	characteristic
147	C₂×#397	—	101	1	no	maximal, Sylow
148	C₁₀₁	C₂×#397	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄⋊C₄	—	101	1	no	—
150	C₂²×C₂²⋊C₄	—	101	1	no	—
151	C₂²×C₂²⋊C₄	—	101	1	no	—
152	C₂²×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic
153	#397	—	101	8	no	—
154	C₂×#95	—	101	1	no	—
155	C₂×#95	—	101	1	no	—
156	C₂×#95	—	101	1	no	—
157	C₂⁵	C₂×D₁₀₁	1	1	yes	characteristic
158	C₂³×C₄	—	101	1	no	—
159	C₂³×C₄	—	101	1	no	—
160	C₂³×C₄	—	101	1	no	—
161	C₂³×C₄	—	101	1	no	—
162	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
163	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	101	4	no	—
165	C₂×C₄⋊C₄	—	202	4	no	—
166	C₂×C₂²⋊C₄	—	2	4	no	—
167	C₂×C₂²⋊C₄	—	101	4	no	—
168	C₂×C₂²⋊C₄	—	101	4	no	—
169	C₂×C₂²⋊C₄	—	202	4	no	—
170	C₂×C₂²⋊C₄	—	202	4	no	—
171	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	—
172	#95	—	101	4	no	—
173	#95	—	101	4	no	—
174	#95	—	101	4	no	—
175	C₂⁴	—	2	2	no	—
176	C₂⁴	—	2	2	no	—
177	C₂⁴	—	2	2	no	—
178	C₂⁴	—	2	2	no	—
179	C₂⁴	—	2	2	no	—
180	C₂⁴	—	2	2	no	—
181	C₂⁴	C₂²⋊D₁₀₁	1	4	yes	—
182	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
183	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	101	4	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	2	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	202	2	no	—
196	C₂²×C₄	—	202	2	no	—
197	C₂²×C₄	—	202	2	no	—
198	C₂²×C₄	—	202	2	no	—
199	C₂²×C₄	—	202	2	no	—
200	C₂²×C₄	—	202	2	no	—
201	C₂²×C₄	—	202	2	no	—
202	C₂²×C₄	—	202	2	no	—
203	C₂²×C₄	—	202	2	no	—
204	C₂²×C₄	—	202	2	no	—
205	C₂²×C₄	—	202	2	no	—
206	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
208	C₄⋊C₄	—	202	8	no	—
209	C₂²⋊C₄	—	2	8	no	—
210	C₂²⋊C₄	—	202	8	no	—
211	C₂²⋊C₄	—	202	8	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	8	no	—
219	C₂³	—	2	8	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	D₄×D₁₀₁	1	2	yes	—
226	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
227	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
228	C₂³	#13108073	1	2	yes	—
229	C₂³	#13108073	1	2	yes	—
230	C₂³	#13108073	1	2	yes	—
231	C₂³	#13108074	1	2	yes	—
232	C₂³	#13108074	1	2	yes	—
233	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	101	4	no	—
243	C₂×C₄	—	404	4	no	—
244	C₂×C₄	—	404	4	no	—
245	C₂×C₄	—	404	4	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	—	202	2	no	—
252	C₂×C₄	—	202	2	no	—
253	C₂×C₄	—	202	2	no	—
254	C₂×C₄	—	202	2	no	—
255	C₂×C₄	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	8	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	4	no	—
261	C₂²	—	4	16	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	#15692227	1	4	yes	—
267	C₂²	#15692236	1	4	yes	—
268	C₂²	#15692237	1	4	yes	—
269	C₂²	#15692238	1	4	yes	—
270	C₂²	#15692241	1	4	yes	—
271	C₂²	#15692248	1	4	yes	—
272	C₂²	#15692248	1	4	yes	—
273	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
274	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
275	C₂²	C₂×#13108073	1	1	yes	characteristic
276	C₂²	C₂×#13108073	1	1	yes	characteristic
277	C₂²	C₂×#13108073	1	1	yes	characteristic
278	C₂²	C₂×#13108074	1	1	yes	characteristic
279	C₂²	C₂×#13108074	1	1	yes	characteristic
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	404	2	no	—
283	C₄	—	404	2	no	—
284	C₄	—	404	2	no	—
285	C₄	—	202	4	no	—
286	C₂	—	4	4	no	—
287	C₂	#20265105	1	8	yes	direct product factor
288	C₂	C₂×#15692227	1	1	yes	characteristic
289	C₂	C₂×#15692236	1	1	yes	characteristic
290	C₂	C₂×#15692237	1	1	yes	characteristic
291	C₂	C₂×#15692238	1	1	yes	characteristic
292	C₂	C₂×#15692241	1	1	yes	characteristic
293	C₂	C₂×#15692248	1	1	yes	characteristic
294	C₂	C₂×#15692248	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂²×C₂²⋊C₄×C₁₀₁	C₂
socle	C₂⁴×C₁₀₁	C₂³
prime radical	C₂⁵×C₁₀₁	C₂²
hypercenter	C₂²×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#397
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108053	C₂	semidirect
4. #20265105	C₂	direct (N × H)
5. C₂×#15692162	C₂	semidirect
6. C₂×#15692162	C₂	semidirect
7. C₂×#15692162	C₂	semidirect
14. C₂×#13108053	C₂²	semidirect
19. #15692162	C₂²	semidirect
20. #15692162	C₂²	semidirect
21. #15692162	C₂²	semidirect
139. C₂×C₁₀₁	#397	semidirect
148. C₁₀₁	C₂×#397	semidirect
287. C₂	#20265105	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×#13108053	2
C₂²×C₂²⋊Dic₁₀₁	2
#20265105	2
C₂×#15692162	2
C₂²×C₂²⋊C₄×C₁₀₁	2
C₂×#397	101

Subgroups — C2×#20265105 #27174359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265105 #27174359