Subgroups — C₂×#20265099 #27174352

Order 12928.

908 conjugacy classes of subgroups.

The subgroup lattice has 304 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×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
6	#20265099	C₂	1	8	yes	maximal, direct product factor
7	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₀₁	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×#13108053	—	2	4	no	—
16	C₂×#13108053	C₂²	1	4	yes	complemented
17	C₂×#13108051	—	2	4	no	—
18	C₂×#13108051	—	2	4	no	—
19	C₂×#13108051	—	2	4	no	—
20	C₂×#13108051	C₂²	1	4	yes	—
21	C₂×#13108051	C₂²	1	4	yes	—
22	C₂×#13108051	C₂²	1	4	yes	—
23	#15692162	C₂²	1	4	yes	complemented
24	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	complemented
29	C₂³×D₁₀₁	—	2	2	no	—
30	C₂³×D₁₀₁	—	2	2	no	—
31	C₂³×D₁₀₁	—	2	2	no	—
32	C₂³×D₁₀₁	—	2	2	no	—
33	C₂³×D₁₀₁	—	2	2	no	—
34	C₂³×D₁₀₁	—	2	2	no	—
35	C₂³×D₁₀₁	Q₈	1	4	yes	—
36	C₂³×D₁₀₁	C₂³	1	2	yes	—
37	C₂²×Dic₁₀₁	—	2	2	no	—
38	C₂²×Dic₁₀₁	—	2	2	no	—
39	C₂²×Dic₁₀₁	—	2	2	no	—
40	C₂²×Dic₁₀₁	—	2	2	no	—
41	C₂²×Dic₁₀₁	—	2	2	no	—
42	C₂²×Dic₁₀₁	—	2	2	no	—
43	C₂²×Dic₁₀₁	—	2	2	no	—
44	C₂²×Dic₁₀₁	—	2	2	no	—
45	C₂²×Dic₁₀₁	D₄	1	4	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₄⋊Dic₁₀₁	—	2	8	no	—
50	#13108053	—	2	8	no	—
51	#13108051	—	2	8	no	—
52	#13108051	—	2	8	no	—
53	#13108051	—	2	8	no	—
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	—	2	2	no	—
62	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
63	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
64	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₀₁	—	2	8	no	—
68	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
69	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
71	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
72	C₂³×C₁₀₁	#41	1	2	yes	—
73	C₂³×C₁₀₁	#41	1	2	yes	—
74	C₂³×C₁₀₁	#41	1	2	yes	—
75	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
76	C₂²×D₁₀₁	—	2	8	no	—
77	C₂²×D₁₀₁	—	2	8	no	—
78	C₂²×D₁₀₁	—	2	8	no	—
79	C₂²×D₁₀₁	—	2	2	no	—
80	C₂²×D₁₀₁	—	2	2	no	—
81	C₂²×D₁₀₁	—	2	2	no	—
82	C₂²×D₁₀₁	—	2	2	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	—	2	2	no	—
85	C₂²×D₁₀₁	—	4	4	no	—
86	C₂²×D₁₀₁	—	4	4	no	—
87	C₂²×D₁₀₁	—	4	4	no	—
88	C₂²×D₁₀₁	—	4	4	no	—
89	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
90	C₂×C₄×C₁₀₁	—	2	8	no	—
91	C₂×C₄×C₁₀₁	—	2	8	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	—	4	4	no	—
100	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
102	C₂×Dic₁₀₁	—	2	8	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	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	4	4	no	—
112	C₂×Dic₁₀₁	—	4	4	no	—
113	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
114	C₂²×C₁₀₁	#121	1	4	yes	—
115	C₂²×C₁₀₁	#121	1	4	yes	—
116	C₂²×C₁₀₁	#121	1	4	yes	—
117	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
118	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
119	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
120	C₂²×C₁₀₁	#126	1	4	yes	—
121	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
128	C₂×D₁₀₁	—	2	4	no	—
129	C₂×D₁₀₁	—	2	4	no	—
130	C₂×D₁₀₁	—	2	4	no	—
131	C₂×D₁₀₁	—	4	2	no	—
132	C₂×D₁₀₁	—	4	2	no	—
133	C₂×D₁₀₁	—	4	2	no	—
134	C₂×D₁₀₁	—	4	2	no	—
135	C₂×D₁₀₁	—	4	16	no	—
136	C₄×C₁₀₁	—	2	4	no	—
137	C₄×C₁₀₁	—	2	4	no	—
138	C₄×C₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	2	4	no	—
140	Dic₁₀₁	—	4	2	no	—
141	Dic₁₀₁	—	4	2	no	—
142	C₂×C₁₀₁	#396	1	8	yes	complemented
143	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
144	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
145	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₁₀₁	C₂×#126	1	1	yes	characteristic
150	D₁₀₁	—	4	4	no	—
151	C₂×#396	—	101	1	no	maximal, Sylow
152	C₁₀₁	C₂×#396	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₄⋊C₄	—	101	1	no	—
154	C₂²×C₄⋊C₄	—	101	1	no	—
155	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
156	C₂²×C₂²⋊C₄	—	101	1	no	—
157	C₂²×C₂²⋊C₄	—	101	1	no	—
158	C₂²×C₂²⋊C₄	—	101	1	no	—
159	#396	—	101	8	no	—
160	C₂×#95	—	101	1	no	—
161	C₂⁵	—	101	1	no	—
162	C₂³×C₄	—	101	1	no	—
163	C₂³×C₄	—	101	1	no	—
164	C₂³×C₄	—	101	1	no	—
165	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
167	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	101	4	no	—
170	C₂×C₄⋊C₄	—	101	4	no	—
171	C₂×C₄⋊C₄	—	202	4	no	—
172	C₂×C₄⋊C₄	—	202	4	no	—
173	C₂×C₄⋊C₄	C₂×D₁₀₁	1	4	yes	complemented
174	C₂×C₂²⋊C₄	—	101	4	no	—
175	C₂×C₂²⋊C₄	—	101	4	no	—
176	C₂×C₂²⋊C₄	—	101	4	no	—
177	C₂×C₂²⋊C₄	—	202	4	no	—
178	C₂×C₂²⋊C₄	—	202	4	no	—
179	C₂×C₂²⋊C₄	—	202	4	no	—
180	#95	—	101	4	no	—
181	C₂⁴	—	101	4	no	—
182	C₂⁴	—	101	2	no	—
183	C₂⁴	—	202	2	no	—
184	C₂⁴	—	202	2	no	—
185	C₂⁴	—	202	2	no	—
186	C₂⁴	—	202	2	no	—
187	C₂⁴	—	202	2	no	—
188	C₂⁴	—	202	2	no	—
189	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	101	4	no	—
198	C₂²×C₄	—	101	2	no	—
199	C₂²×C₄	—	101	2	no	—
200	C₂²×C₄	—	101	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	—	202	2	no	—
208	C₂²×C₄	—	202	2	no	—
209	C₂²×C₄	D₄₀₄	1	4	yes	—
210	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
211	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
212	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
213	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	202	8	no	—
216	C₄⋊C₄	—	202	8	no	—
217	C₂²⋊C₄	—	202	8	no	—
218	C₂²⋊C₄	—	202	8	no	—
219	C₂²⋊C₄	—	202	8	no	—
220	C₂³	—	101	4	no	—
221	C₂³	—	404	4	no	—
222	C₂³	—	404	4	no	—
223	C₂³	—	404	4	no	—
224	C₂³	—	404	4	no	—
225	C₂³	—	202	2	no	—
226	C₂³	—	202	2	no	—
227	C₂³	—	202	2	no	—
228	C₂³	—	202	2	no	—
229	C₂³	—	202	2	no	—
230	C₂³	—	202	2	no	—
231	C₂³	—	202	8	no	—
232	C₂³	—	202	8	no	—
233	C₂³	—	202	8	no	—
234	C₂³	D₄×D₁₀₁	1	2	yes	—
235	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
236	C₂³	C₂×D₄₀₄	1	2	yes	—
237	C₂³	Q₈×D₁₀₁	1	2	yes	—
238	C₂³	#13108073	1	2	yes	—
239	C₂³	#13108074	1	2	yes	—
240	C₂³	#13108075	1	2	yes	—
241	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	101	4	no	—
253	C₂×C₄	—	404	4	no	—
254	C₂×C₄	—	404	4	no	—
255	C₂×C₄	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	2	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	—	202	8	no	—
264	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
265	C₂×C₄	C₂×D₄₀₄	1	4	yes	—
266	C₂²	—	404	16	no	—
267	C₂²	—	404	2	no	—
268	C₂²	—	404	2	no	—
269	C₂²	—	404	2	no	—
270	C₂²	—	404	2	no	—
271	C₂²	—	202	4	no	—
272	C₂²	—	202	4	no	—
273	C₂²	—	202	4	no	—
274	C₂²	#15692227	1	4	yes	—
275	C₂²	#15692229	1	4	yes	—
276	C₂²	#15692230	1	4	yes	—
277	C₂²	#15692231	1	4	yes	—
278	C₂²	#15692233	1	4	yes	—
279	C₂²	#15692235	1	4	yes	—
280	C₂²	#15692250	1	4	yes	—
281	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
282	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
283	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
284	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
285	C₂²	C₂×#13108073	1	1	yes	characteristic
286	C₂²	C₂×#13108074	1	1	yes	characteristic
287	C₂²	C₂×#13108075	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	2	no	—
291	C₄	—	404	2	no	—
292	C₄	—	404	2	no	—
293	C₄	—	202	4	no	—
294	C₂	—	404	4	no	—
295	C₂	#20265099	1	8	yes	direct product factor
296	C₂	C₂×#15692227	1	1	yes	characteristic
297	C₂	C₂×#15692229	1	1	yes	characteristic
298	C₂	C₂×#15692230	1	1	yes	characteristic
299	C₂	C₂×#15692231	1	1	yes	characteristic
300	C₂	C₂×#15692233	1	1	yes	characteristic
301	C₂	C₂×#15692235	1	1	yes	characteristic
302	C₂	C₂×#15692250	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle	C₂⁴×C₁₀₁	C₂³
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#396
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
2. C₂²×#13108053	C₂	semidirect
6. #20265099	C₂	direct (N × H)
7. C₂×#15692162	C₂	semidirect
8. C₂²×C₄⋊C₄×C₁₀₁	C₂	semidirect
14. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
16. C₂×#13108053	C₂²	semidirect
23. #15692162	C₂²	semidirect
28. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
142. C₂×C₁₀₁	#396	semidirect
152. C₁₀₁	C₂×#396	semidirect
155. C₂²×C₄⋊C₄	D₁₀₁	semidirect
173. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
295. C₂	#20265099	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×#13108053	2
C₂²×#13108051	2
#20265099	2
C₂×#15692162	2
C₂²×C₄⋊C₄×C₁₀₁	2
C₂×#396	101

Subgroups — C2×#20265099 #27174352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265099 #27174352