Subgroups — C₂×#20265070 #27173566

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₀₁	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	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265070	C₂	1	8	yes	maximal, direct product factor
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×C₄⋊Dic₁₀₁	—	2	4	no	—
14	C₂×C₄⋊Dic₁₀₁	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	C₂×C₄×Dic₁₀₁	—	2	4	no	—
20	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	complemented
21	#15692162	C₂²	1	4	yes	complemented
22	#15692162	C₂²	1	4	yes	complemented
23	C₂⁵×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	8	yes	complemented
27	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
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	D₄	1	4	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₄⋊Dic₁₀₁	—	2	8	no	—
45	C₂²⋊Dic₁₀₁	—	2	8	no	—
46	C₂²⋊Dic₁₀₁	—	2	8	no	—
47	C₄×Dic₁₀₁	—	2	8	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	—	2	2	no	—
53	C₂⁴×C₁₀₁	—	2	2	no	—
54	C₂⁴×C₁₀₁	C₂×C₄	1	4	yes	—
55	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
56	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
62	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
63	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	16	yes	complemented
66	C₂³×C₁₀₁	—	2	8	no	—
67	C₂³×C₁₀₁	—	2	8	no	—
68	C₂³×C₁₀₁	—	2	8	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	—	2	2	no	—
74	C₂³×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	4	4	no	—
76	C₂³×C₁₀₁	—	4	4	no	—
77	C₂³×C₁₀₁	—	4	4	no	—
78	C₂³×C₁₀₁	—	4	4	no	—
79	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
80	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
82	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
83	C₂³×C₁₀₁	#41	1	2	yes	—
84	C₂³×C₁₀₁	#41	1	2	yes	—
85	C₂³×C₁₀₁	#41	1	2	yes	—
86	C₂³×C₁₀₁	#41	1	2	yes	—
87	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₁₀₁	—	2	8	no	—
89	C₂×C₄×C₁₀₁	—	2	8	no	—
90	C₂×C₄×C₁₀₁	—	2	2	no	—
91	C₂×C₄×C₁₀₁	—	2	2	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
96	C₂×Dic₁₀₁	—	2	8	no	—
97	C₂×Dic₁₀₁	—	2	8	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	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	4	4	no	—
109	C₂×Dic₁₀₁	—	4	4	no	—
110	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
111	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	—	2	4	no	—
114	C₂²×C₁₀₁	—	2	4	no	—
115	C₂²×C₁₀₁	—	4	2	no	—
116	C₂²×C₁₀₁	—	4	2	no	—
117	C₂²×C₁₀₁	—	4	2	no	—
118	C₂²×C₁₀₁	—	4	2	no	—
119	C₂²×C₁₀₁	—	4	16	no	—
120	C₂²×C₁₀₁	#117	1	4	yes	—
121	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
122	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
123	C₂²×C₁₀₁	#121	1	4	yes	—
124	C₂²×C₁₀₁	#123	1	4	yes	—
125	C₂²×C₁₀₁	#124	1	4	yes	—
126	C₂²×C₁₀₁	#126	1	4	yes	—
127	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
134	C₄×C₁₀₁	—	2	4	no	—
135	C₄×C₁₀₁	—	2	4	no	—
136	Dic₁₀₁	—	2	4	no	—
137	Dic₁₀₁	—	2	4	no	—
138	Dic₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	4	2	no	—
140	C₂×C₁₀₁	—	4	4	no	—
141	C₂×C₁₀₁	#388	1	8	yes	complemented
142	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
143	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
147	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
148	C₂×C₁₀₁	C₂×#126	1	1	yes	characteristic
149	C₂×#388	—	101	1	no	maximal, Sylow
150	C₁₀₁	C₂×#388	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₄⋊C₄	—	101	1	no	—
152	C₂²×C₂²⋊C₄	—	101	1	no	—
153	C₂²×C₂²⋊C₄	—	101	1	no	—
154	C₂²×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic
155	C₂²×C₄²	—	101	1	no	—
156	#388	—	101	8	no	—
157	C₂×#95	—	101	1	no	—
158	C₂×#95	—	101	1	no	—
159	C₂⁵	C₂×D₁₀₁	1	1	yes	characteristic
160	C₂³×C₄	—	101	1	no	—
161	C₂³×C₄	—	101	1	no	—
162	C₂³×C₄	—	101	1	no	—
163	C₂³×C₄	—	101	1	no	—
164	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
165	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	101	4	no	—
167	C₂×C₄⋊C₄	—	202	4	no	—
168	C₂×C₂²⋊C₄	—	101	4	no	—
169	C₂×C₂²⋊C₄	—	101	4	no	—
170	C₂×C₂²⋊C₄	—	202	4	no	—
171	C₂×C₂²⋊C₄	—	202	4	no	—
172	C₂×C₂²⋊C₄	Dic₁₀₁	1	8	yes	complemented
173	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	—
174	C₂×C₄²	—	101	4	no	—
175	C₂×C₄²	—	202	4	no	—
176	#95	—	101	4	no	—
177	#95	—	101	4	no	—
178	C₂⁴	—	2	2	no	—
179	C₂⁴	—	2	2	no	—
180	C₂⁴	—	2	2	no	—
181	C₂⁴	—	2	2	no	—
182	C₂⁴	—	2	2	no	—
183	C₂⁴	—	2	2	no	—
184	C₂⁴	C₂×Dic₁₀₁	1	4	yes	—
185	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
186	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	101	4	no	—
192	C₂²×C₄	—	101	4	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	101	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	—	202	2	no	—
207	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
208	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
209	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
211	C₄⋊C₄	—	202	8	no	—
212	C₂²⋊C₄	—	202	8	no	—
213	C₂²⋊C₄	—	202	8	no	—
214	C₂²⋊C₄	C₂×Dic₁₀₁	1	16	yes	complemented
215	C₄²	—	202	8	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	8	no	—
223	C₂³	—	2	8	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	4	no	—
229	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
230	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
231	C₂³	D₄×D₁₀₁	1	2	yes	—
232	C₂³	D₄×D₁₀₁	1	2	yes	—
233	C₂³	#13108073	1	2	yes	—
234	C₂³	#13108073	1	2	yes	—
235	C₂³	#13108074	1	2	yes	—
236	C₂³	#13108074	1	2	yes	—
237	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	101	4	no	—
245	C₂×C₄	—	101	4	no	—
246	C₂×C₄	—	404	4	no	—
247	C₂×C₄	—	404	4	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	2	no	—
258	C₂×C₄	—	202	8	no	—
259	C₂×C₄	—	202	8	no	—
260	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
261	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	16	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	#15692214	1	4	yes	—
271	C₂²	D₄×Dic₁₀₁	1	4	yes	—
272	C₂²	D₄×Dic₁₀₁	1	4	yes	—
273	C₂²	#15692228	1	4	yes	—
274	C₂²	#15692237	1	4	yes	—
275	C₂²	#15692241	1	4	yes	—
276	C₂²	#15692248	1	4	yes	—
277	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
278	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
279	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
280	C₂²	C₂×#13108073	1	1	yes	characteristic
281	C₂²	C₂×#13108073	1	1	yes	characteristic
282	C₂²	C₂×#13108074	1	1	yes	characteristic
283	C₂²	C₂×#13108074	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	4	no	—
286	C₄	—	404	2	no	—
287	C₄	—	404	2	no	—
288	C₄	—	202	4	no	—
289	C₄	—	202	4	no	—
290	C₂	—	4	4	no	—
291	C₂	#20265070	1	8	yes	direct product factor
292	C₂	C₂×#15692214	1	1	yes	characteristic
293	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
294	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
295	C₂	C₂×#15692228	1	1	yes	characteristic
296	C₂	C₂×#15692237	1	1	yes	characteristic
297	C₂	C₂×#15692241	1	1	yes	characteristic
298	C₂	C₂×#15692248	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂²×C₂²⋊C₄×C₁₀₁	C₂
socle	C₂⁴×C₁₀₁	C₂³
prime radical	C₂⁵×C₁₀₁	C₂²
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#388
LCS	G ▷ C₂²×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×C₄⋊Dic₁₀₁	C₂	semidirect
4. C₂²×C₄×Dic₁₀₁	C₂	semidirect
5. #20265070	C₂	direct (N × H)
6. C₂×#15692162	C₂	semidirect
7. C₂×#15692162	C₂	semidirect
14. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
20. C₂×C₄×Dic₁₀₁	C₂²	semidirect
21. #15692162	C₂²	semidirect
22. #15692162	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₁₀₁	C₄	semidirect
65. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
141. C₂×C₁₀₁	#388	semidirect
150. C₁₀₁	C₂×#388	semidirect
172. C₂×C₂²⋊C₄	Dic₁₀₁	semidirect
214. C₂²⋊C₄	C₂×Dic₁₀₁	semidirect
291. C₂	#20265070	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×C₄⋊Dic₁₀₁	2
C₂²×C₂²⋊Dic₁₀₁	2
C₂²×C₄×Dic₁₀₁	2
#20265070	2
C₂×#15692162	2
C₂²×C₂²⋊C₄×C₁₀₁	2
C₂×#388	101

Subgroups — C2×#20265070 #27173566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265070 #27173566