Subgroups — #380×D₁₀₁ #27173501

Order 12928.

1232 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×D₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	2	yes	maximal
4	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265036	C₂	1	2	yes	maximal, complemented
6	#20265035	C₂	1	2	yes	maximal, complemented
7	#20265034	C₂	1	2	yes	maximal, complemented
8	#95×D₁₀₁	C₂	1	2	yes	maximal, complemented
9	#380×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	—	2	8	no	—
13	C₂²×C₄×D₁₀₁	C₄	1	8	yes	complemented
14	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
20	C₂×#13108053	C₂²	1	4	yes	complemented
21	C₄⋊C₄×D₁₀₁	—	2	8	no	—
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×#13108051	C₂²	1	4	yes	—
24	C₂²⋊C₄×D₁₀₁	—	2	8	no	—
25	#15692162	C₂²	1	4	yes	complemented
26	#15692162	C₂²	1	2	yes	complemented
27	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
30	#95×C₁₀₁	C₂²	1	2	yes	complemented
31	C₂³×D₁₀₁	—	2	4	no	—
32	C₂³×D₁₀₁	—	2	4	no	—
33	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
34	C₂³×D₁₀₁	D₄	1	4	yes	—
35	C₂³×D₁₀₁	Q₈	1	4	yes	—
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂×C₄×D₁₀₁	—	2	4	no	—
44	C₂×C₄×D₁₀₁	—	2	4	no	—
45	C₂×C₄×D₁₀₁	—	2	4	no	—
46	C₂×C₄×D₁₀₁	—	2	32	no	—
47	C₂×C₄×D₁₀₁	—	2	8	no	—
48	C₂×C₄×D₁₀₁	—	2	8	no	—
49	C₂×C₄×D₁₀₁	—	2	8	no	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
51	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
53	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
54	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
55	C₄⋊Dic₁₀₁	—	2	4	no	—
56	#13108053	—	2	8	no	—
57	C₂²⋊Dic₁₀₁	—	2	4	no	—
58	#13108051	—	2	8	no	—
59	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	—	2	4	no	—
61	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
62	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₁	—	2	4	no	—
66	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
67	C₂³×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	—	2	2	no	—
69	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
71	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
72	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
73	C₂²×D₁₀₁	—	2	2	no	—
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	—	2	8	no	—
76	C₂²×D₁₀₁	—	2	8	no	—
77	C₂²×D₁₀₁	—	2	16	no	—
78	C₂²×D₁₀₁	—	2	16	no	—
79	C₂²×D₁₀₁	C₄⋊C₄	1	16	yes	complemented
80	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
83	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
84	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
85	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
86	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
87	C₂²×D₁₀₁	#41	1	4	yes	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	8	no	—
90	C₂×C₄×C₁₀₁	—	2	4	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	8	no	—
98	C₂×Dic₁₀₁	—	2	4	no	—
99	C₂×Dic₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	—	2	4	no	—
101	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
102	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
104	C₄×D₁₀₁	—	2	8	no	—
105	C₄×D₁₀₁	—	2	8	no	—
106	C₄×D₁₀₁	—	2	8	no	—
107	C₄×D₁₀₁	—	4	16	no	—
108	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	complemented
109	C₂²×C₁₀₁	—	2	2	no	—
110	C₂²×C₁₀₁	—	2	2	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	complemented
114	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
119	C₂×D₁₀₁	—	2	2	no	—
120	C₂×D₁₀₁	—	2	2	no	—
121	C₂×D₁₀₁	—	2	4	no	—
122	C₂×D₁₀₁	—	2	4	no	—
123	C₂×D₁₀₁	—	2	32	no	—
124	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	complemented
127	C₂×D₁₀₁	#117	1	4	yes	—
128	C₂×D₁₀₁	#121	1	8	yes	—
129	C₂×D₁₀₁	C₂²⋊Q₈	1	8	yes	—
130	C₄×C₁₀₁	—	2	2	no	—
131	C₄×C₁₀₁	—	4	4	no	—
132	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
133	Dic₁₀₁	—	2	2	no	—
134	Dic₁₀₁	—	4	4	no	—
135	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
136	C₂×C₁₀₁	—	2	4	no	—
137	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
140	C₂×C₁₀₁	C₂×#121	1	2	yes	—
141	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	2	yes	—
142	D₁₀₁	—	2	4	no	—
143	D₁₀₁	#380	1	8	yes	direct product factor
144	C₂×#380	—	101	1	no	maximal, Sylow
145	C₁₀₁	C₂×#380	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴×C₄	—	101	1	no	—
147	C₂²×C₄⋊C₄	—	101	2	no	—
148	C₂²×C₂²⋊C₄	—	101	2	no	—
149	#380	—	101	1	no	—
150	#380	—	101	2	no	—
151	#380	—	101	2	no	—
152	#380	—	101	2	no	—
153	#380	D₁₀₁	1	1	yes	characteristic, direct product factor
154	C₂×#95	—	101	2	no	—
155	C₂⁵	—	101	1	no	—
156	C₂³×C₄	—	101	1	no	—
157	C₂³×C₄	—	101	1	no	—
158	C₂³×C₄	—	101	8	no	—
159	C₂³×C₄	—	101	1	no	—
160	C₂³×C₄	—	101	1	no	—
161	C₂³×C₄	—	101	1	no	—
162	C₂³×C₄	—	101	4	no	—
163	C₂³×C₄	—	202	8	no	—
164	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
165	C₂×C₄⋊C₄	—	101	4	no	—
166	C₂×C₄⋊C₄	—	101	2	no	—
167	C₂×C₄⋊C₄	—	202	8	no	—
168	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
169	C₂×C₂²⋊C₄	—	101	2	no	—
170	C₂×C₂²⋊C₄	—	101	4	no	—
171	C₂×C₂²⋊C₄	—	202	8	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
173	#95	—	101	4	no	—
174	#95	—	101	2	no	—
175	#95	C₂×D₁₀₁	1	2	yes	complemented
176	C₂⁴	—	101	1	no	—
177	C₂⁴	—	101	4	no	—
178	C₂⁴	—	101	4	no	—
179	C₂⁴	—	101	4	no	—
180	C₂⁴	—	101	1	no	—
181	C₂⁴	—	202	4	no	—
182	C₂⁴	—	202	4	no	—
183	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	101	1	no	—
186	C₂²×C₄	—	101	4	no	—
187	C₂²×C₄	—	101	8	no	—
188	C₂²×C₄	—	101	4	no	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	4	no	—
191	C₂²×C₄	—	101	4	no	—
192	C₂²×C₄	—	101	4	no	—
193	C₂²×C₄	—	101	4	no	—
194	C₂²×C₄	—	202	4	no	—
195	C₂²×C₄	—	202	32	no	—
196	C₂²×C₄	—	202	8	no	—
197	C₂²×C₄	—	202	8	no	—
198	C₂²×C₄	—	202	8	no	—
199	C₂²×C₄	—	202	4	no	—
200	C₂²×C₄	—	202	4	no	—
201	C₂²×C₄	—	202	4	no	—
202	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
203	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
204	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	202	8	no	—
208	C₄⋊C₄	—	202	4	no	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	202	8	no	—
211	C₂²⋊C₄	—	202	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	101	2	no	—
215	C₂³	—	101	4	no	—
216	C₂³	—	101	2	no	—
217	C₂³	—	101	2	no	—
218	C₂³	—	101	4	no	—
219	C₂³	—	101	2	no	—
220	C₂³	—	101	2	no	—
221	C₂³	—	101	2	no	—
222	C₂³	—	101	16	no	—
223	C₂³	—	202	8	no	—
224	C₂³	—	202	8	no	—
225	C₂³	—	202	16	no	—
226	C₂³	—	202	16	no	—
227	C₂³	—	202	2	no	—
228	C₂³	—	202	2	no	—
229	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
230	C₂³	D₄×D₁₀₁	1	2	yes	—
231	C₂³	Q₈×D₁₀₁	1	2	yes	—
232	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	101	2	no	—
239	C₂×C₄	—	101	4	no	—
240	C₂×C₄	—	101	16	no	—
241	C₂×C₄	—	101	2	no	—
242	C₂×C₄	—	404	16	no	—
243	C₂×C₄	—	202	8	no	—
244	C₂×C₄	—	202	8	no	—
245	C₂×C₄	—	202	8	no	—
246	C₂×C₄	—	202	4	no	—
247	C₂×C₄	—	202	4	no	—
248	C₂×C₄	—	202	4	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	8	no	—
251	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
252	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
253	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	101	4	no	—
259	C₂²	—	101	4	no	—
260	C₂²	—	101	4	no	—
261	C₂²	—	101	8	no	—
262	C₂²	—	101	8	no	—
263	C₂²	—	101	4	no	—
264	C₂²	—	202	32	no	—
265	C₂²	—	202	2	no	—
266	C₂²	—	202	2	no	—
267	C₂²	—	202	4	no	—
268	C₂²	—	202	4	no	—
269	C₂²	C₄⋊C₄×D₁₀₁	1	4	yes	complemented
270	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
271	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
272	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
273	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
274	C₂²	#41×D₁₀₁	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	101	4	no	—
278	C₄	—	404	4	no	—
279	C₄	—	202	2	no	—
280	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	complemented
281	C₂	—	2	4	no	—
282	C₂	—	101	8	no	—
283	C₂	—	202	4	no	—
284	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
285	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
286	C₂	#117×D₁₀₁	1	1	yes	characteristic
287	C₂	#121×D₁₀₁	1	2	yes	—
288	C₂	C₂²⋊Q₈×D₁₀₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#380×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	#380	D₁₀₁
hypocenter	C₁₀₁	C₂×#380
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₄×D₁₀₁	C₂	semidirect
4. #20265037	C₂	semidirect
5. #20265036	C₂	semidirect
6. #20265035	C₂	semidirect
7. #20265034	C₂	semidirect
8. #95×D₁₀₁	C₂	semidirect
9. #380×C₁₀₁	C₂	semidirect
13. C₂²×C₄×D₁₀₁	C₄	semidirect
19. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
20. C₂×#13108053	C₂²	semidirect
25. #15692162	C₂²	semidirect
26. #15692162	C₂²	semidirect
28. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
30. #95×C₁₀₁	C₂²	semidirect
39. C₂²×Dic₁₀₁	C₂×C₄	semidirect
52. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
53. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
61. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
79. C₂²×D₁₀₁	C₄⋊C₄	semidirect
108. C₄×D₁₀₁	C₂²⋊C₄	semidirect
113. C₂²×C₁₀₁	C₂×C₄⋊C₄	semidirect
126. C₂×D₁₀₁	C₂×C₄⋊C₄	semidirect
132. C₄×C₁₀₁	C₂×C₂²⋊C₄	semidirect
135. Dic₁₀₁	C₂×C₂²⋊C₄	semidirect
143. D₁₀₁	#380	direct (N × H)
145. C₁₀₁	C₂×#380	semidirect
153. #380	D₁₀₁	direct (N × H)
168. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
175. #95	C₂×D₁₀₁	semidirect
202. C₂²×C₄	C₄×D₁₀₁	semidirect
269. C₂²	C₄⋊C₄×D₁₀₁	semidirect
280. C₄	C₂²⋊C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₂²⋊C₄×D₁₀₁	2
#20265037	2
#20265036	2
#20265035	2
#20265034	2
#95×D₁₀₁	2
#380×C₁₀₁	2
C₂×#380	101

Subgroups — #380×D101 #27173501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #380×D₁₀₁ #27173501