Subgroups — #390×D₁₀₁ #27173585

Order 12928.

1360 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	4	yes	maximal, complemented
4	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265080	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265079	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265078	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265077	C₂	1	4	yes	maximal, complemented
9	#390×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	2	yes	—
11	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
13	C₂×D₄×D₁₀₁	—	2	4	no	—
14	C₂×D₄×D₁₀₁	—	2	4	no	—
15	C₂×D₄×D₁₀₁	C₄	1	8	yes	complemented
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13108053	C₂²	1	2	yes	complemented
21	C₄⋊C₄×D₁₀₁	—	2	4	no	—
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
23	C₂×#13108051	C₂²	1	4	yes	complemented
24	C₂×#13108051	C₂²	1	4	yes	complemented
25	C₂²⋊C₄×D₁₀₁	—	2	16	no	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
27	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
28	C₄²×D₁₀₁	—	2	4	no	—
29	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	complemented
32	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
33	C₂³×D₁₀₁	—	2	4	no	—
34	C₂³×D₁₀₁	—	2	4	no	—
35	C₂³×D₁₀₁	—	2	8	no	—
36	C₂³×D₁₀₁	—	2	8	no	—
37	C₂³×D₁₀₁	C₂×C₄	1	8	yes	—
38	C₂³×D₁₀₁	C₂³	1	2	yes	—
39	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
40	C₂×D₄₀₄	—	2	2	no	—
41	C₂×D₄₀₄	—	2	2	no	—
42	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
43	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
44	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
45	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
46	D₄×D₁₀₁	—	2	32	no	—
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₀₁	—	2	4	no	—
51	C₂×C₄×D₁₀₁	—	2	4	no	—
52	C₂×C₄×D₁₀₁	—	2	8	no	—
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
55	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
56	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
57	C₂×C₄×D₁₀₁	D₄	1	8	yes	complemented
58	C₄⋊Dic₁₀₁	—	2	2	no	—
59	#13108053	—	2	4	no	—
60	C₂²⋊Dic₁₀₁	—	2	8	no	—
61	#13108051	—	2	8	no	—
62	#13108051	—	2	8	no	—
63	C₄×Dic₁₀₁	—	2	2	no	—
64	C₄×Dic₁₀₁	—	2	4	no	—
65	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
66	C₂×D₄×C₁₀₁	—	2	2	no	—
67	C₂×D₄×C₁₀₁	—	2	2	no	—
68	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₁	—	2	2	no	—
73	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
74	C₄²×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	4	no	—
76	C₂³×C₁₀₁	—	2	2	no	—
77	C₂³×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	—	2	4	no	—
79	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
80	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
81	C₂²×D₁₀₁	—	2	4	no	—
82	C₂²×D₁₀₁	—	2	2	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	—	2	4	no	—
85	C₂²×D₁₀₁	—	2	16	no	—
86	C₂²×D₁₀₁	—	2	32	no	—
87	C₂²×D₁₀₁	—	4	16	no	—
88	C₂²×D₁₀₁	—	4	8	no	—
89	C₂²×D₁₀₁	—	4	8	no	—
90	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
91	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
93	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
94	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
95	C₂²×D₁₀₁	#41	1	4	yes	—
96	D₄×C₁₀₁	—	2	8	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	—	2	4	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	complemented
105	C₂²⋊D₁₀₁	—	2	16	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	4	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
111	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
113	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	complemented
114	D₄₀₄	—	2	8	no	—
115	C₄×D₁₀₁	—	2	16	no	—
116	C₄×D₁₀₁	—	4	8	no	—
117	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	complemented
118	C₂²×C₁₀₁	—	2	4	no	—
119	C₂²×C₁₀₁	—	2	8	no	—
120	C₂²×C₁₀₁	—	4	4	no	—
121	C₂²×C₁₀₁	—	4	2	no	—
122	C₂²×C₁₀₁	—	4	2	no	—
123	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
127	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
128	C₂×D₁₀₁	—	2	4	no	—
129	C₂×D₁₀₁	—	2	8	no	—
130	C₂×D₁₀₁	—	4	4	no	—
131	C₂×D₁₀₁	—	4	2	no	—
132	C₂×D₁₀₁	—	4	2	no	—
133	C₂×D₁₀₁	—	4	32	no	—
134	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×D₁₀₁	C₄×D₄	1	8	yes	—
136	C₂×D₁₀₁	#121	1	8	yes	—
137	C₂×D₁₀₁	#124	1	4	yes	—
138	C₂×D₁₀₁	#127	1	4	yes	—
139	C₄×C₁₀₁	—	2	4	no	—
140	C₄×C₁₀₁	—	4	2	no	—
141	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
142	Dic₁₀₁	—	2	4	no	—
143	Dic₁₀₁	—	4	2	no	—
144	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
145	C₂×C₁₀₁	—	4	4	no	—
146	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
148	C₂×C₁₀₁	C₂×#121	1	2	yes	—
149	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
150	C₂×C₁₀₁	C₂×#127	1	1	yes	characteristic
151	D₁₀₁	—	4	4	no	—
152	D₁₀₁	#390	1	8	yes	direct product factor
153	C₂×#390	—	101	1	no	maximal, Sylow
154	C₁₀₁	C₂×#390	1	1	yes	characteristic, complemented, Sylow
155	C₂³×D₄	—	101	1	no	—
156	C₂²×C₄⋊C₄	—	101	1	no	—
157	C₂²×C₂²⋊C₄	—	101	4	no	—
158	C₂²×C₄²	—	101	1	no	—
159	#390	—	101	1	no	—
160	#390	—	101	1	no	—
161	#390	—	101	4	no	—
162	#390	—	101	1	no	—
163	#390	D₁₀₁	1	1	yes	characteristic, direct product factor
164	C₂⁵	—	101	2	no	—
165	C₂²×D₄	—	101	2	no	—
166	C₂²×D₄	—	101	8	no	—
167	C₂²×D₄	—	101	1	no	—
168	C₂²×D₄	—	202	4	no	—
169	C₂²×D₄	—	202	4	no	—
170	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
171	C₂³×C₄	—	101	1	no	—
172	C₂³×C₄	—	101	2	no	—
173	C₂³×C₄	—	101	2	no	—
174	C₂×C₄⋊C₄	—	101	2	no	—
175	C₂×C₄⋊C₄	—	101	1	no	—
176	C₂×C₄⋊C₄	—	202	4	no	—
177	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
178	C₂×C₂²⋊C₄	—	101	4	no	—
179	C₂×C₂²⋊C₄	—	101	4	no	—
180	C₂×C₂²⋊C₄	—	101	4	no	—
181	C₂×C₂²⋊C₄	—	202	16	no	—
182	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	complemented
183	C₂×C₄²	—	101	2	no	—
184	C₂×C₄²	—	101	1	no	—
185	C₂×C₄²	—	202	4	no	—
186	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
187	C₂⁴	—	101	1	no	—
188	C₂⁴	—	101	8	no	—
189	C₂⁴	—	101	2	no	—
190	C₂⁴	—	202	8	no	—
191	C₂⁴	—	202	4	no	—
192	C₂⁴	—	202	4	no	—
193	C₂⁴	—	202	8	no	—
194	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	101	8	no	—
198	C₂×D₄	—	101	4	no	—
199	C₂×D₄	—	202	32	no	—
200	C₂×D₄	—	202	4	no	—
201	C₂×D₄	—	202	4	no	—
202	C₂×D₄	—	202	2	no	—
203	C₂×D₄	—	202	2	no	—
204	C₂×D₄	C₄×D₁₀₁	1	4	yes	complemented
205	C₂²×C₄	—	101	1	no	—
206	C₂²×C₄	—	101	4	no	—
207	C₂²×C₄	—	101	4	no	—
208	C₂²×C₄	—	101	4	no	—
209	C₂²×C₄	—	101	2	no	—
210	C₂²×C₄	—	101	8	no	—
211	C₂²×C₄	—	101	2	no	—
212	C₂²×C₄	—	202	8	no	—
213	C₂²×C₄	—	202	4	no	—
214	C₂²×C₄	—	202	4	no	—
215	C₂²×C₄	—	202	4	no	—
216	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
217	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
218	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	202	4	no	—
221	C₄⋊C₄	—	202	2	no	—
222	C₂²⋊C₄	—	2	8	no	—
223	C₂²⋊C₄	—	202	8	no	—
224	C₂²⋊C₄	—	202	8	no	—
225	C₂²⋊C₄	—	202	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	202	4	no	—
228	C₄²	—	202	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	101	2	no	—
234	C₂³	—	101	2	no	—
235	C₂³	—	101	2	no	—
236	C₂³	—	101	4	no	—
237	C₂³	—	101	4	no	—
238	C₂³	—	101	4	no	—
239	C₂³	—	404	16	no	—
240	C₂³	—	404	8	no	—
241	C₂³	—	404	8	no	—
242	C₂³	—	202	16	no	—
243	C₂³	—	202	32	no	—
244	C₂³	—	202	4	no	—
245	C₂³	—	202	2	no	—
246	C₂³	—	202	2	no	—
247	C₂³	—	202	4	no	—
248	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
249	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
250	D₄	—	2	8	no	—
251	D₄	—	202	16	no	—
252	D₄	—	202	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	101	2	no	—
258	C₂×C₄	—	101	2	no	—
259	C₂×C₄	—	101	2	no	—
260	C₂×C₄	—	101	16	no	—
261	C₂×C₄	—	101	4	no	—
262	C₂×C₄	—	404	8	no	—
263	C₂×C₄	—	202	16	no	—
264	C₂×C₄	—	202	4	no	—
265	C₂×C₄	—	202	2	no	—
266	C₂×C₄	—	202	2	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
269	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
270	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
271	C₂×C₄	D₄×D₁₀₁	1	4	yes	complemented
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	101	4	no	—
278	C₂²	—	101	8	no	—
279	C₂²	—	101	8	no	—
280	C₂²	—	101	4	no	—
281	C₂²	—	101	4	no	—
282	C₂²	—	404	32	no	—
283	C₂²	—	404	4	no	—
284	C₂²	—	404	2	no	—
285	C₂²	—	404	2	no	—
286	C₂²	—	202	4	no	—
287	C₂²	—	202	8	no	—
288	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
289	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
290	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
291	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
292	C₂²	#41×D₁₀₁	1	2	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	4	2	no	—
295	C₄	—	101	4	no	—
296	C₄	—	404	2	no	—
297	C₄	—	202	4	no	—
298	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	complemented
299	C₂	—	4	4	no	—
300	C₂	—	101	8	no	—
301	C₂	—	404	4	no	—
302	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
303	C₂	C₄×D₄×D₁₀₁	1	2	yes	—
304	C₂	#121×D₁₀₁	1	2	yes	—
305	C₂	#124×D₁₀₁	1	1	yes	characteristic
306	C₂	#127×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#390×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄×D₁₀₁	C₂
hypercenter	#390	D₁₀₁
hypocenter	C₁₀₁	C₂×#390
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #390

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₄×D₁₀₁	C₂	semidirect
3. C₂×C₂²⋊C₄×D₁₀₁	C₂	semidirect
4. C₂×C₄²×D₁₀₁	C₂	semidirect
5. #20265080	C₂	semidirect
6. #20265079	C₂	semidirect
7. #20265078	C₂	semidirect
8. #20265077	C₂	semidirect
9. #390×C₁₀₁	C₂	semidirect
15. C₂×D₄×D₁₀₁	C₄	semidirect
19. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
20. C₂×#13108053	C₂²	semidirect
22. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
23. C₂×#13108051	C₂²	semidirect
24. C₂×#13108051	C₂²	semidirect
26. C₂×C₄×Dic₁₀₁	C₂²	semidirect
27. C₂×C₄×Dic₁₀₁	C₂²	semidirect
30. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
31. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
32. C₂×C₄²×C₁₀₁	C₂²	semidirect
42. C₂×D₄₀₄	C₂×C₄	semidirect
45. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
57. C₂×C₄×D₁₀₁	D₄	semidirect
68. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
104. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
113. C₂×Dic₁₀₁	C₂×D₄	semidirect
117. C₄×D₁₀₁	C₂²⋊C₄	semidirect
141. C₄×C₁₀₁	C₂×C₂²⋊C₄	semidirect
144. Dic₁₀₁	C₂×C₂²⋊C₄	semidirect
152. D₁₀₁	#390	direct (N × H)
154. C₁₀₁	C₂×#390	semidirect
163. #390	D₁₀₁	direct (N × H)
177. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
182. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
186. C₂×C₄²	C₂×D₁₀₁	semidirect
204. C₂×D₄	C₄×D₁₀₁	semidirect
271. C₂×C₄	D₄×D₁₀₁	semidirect
298. 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
C₂²×D₄×D₁₀₁	2
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₂²⋊C₄×D₁₀₁	2
C₂×C₄²×D₁₀₁	2
#20265080	2
#20265079	2
#20265078	2
#20265077	2
#390×C₁₀₁	2
C₂×#390	101

Subgroups — #390×D101 #27173585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #390×D₁₀₁ #27173585