Subgroups — #391×D₁₀₁ #27173598

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265084	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265083	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265082	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265081	C₂	1	4	yes	maximal, complemented
8	#95×D₁₀₁	C₂	1	4	yes	maximal
9	#391×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
11	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
12	C₂×Q₈×D₁₀₁	—	2	4	no	—
13	C₂×Q₈×D₁₀₁	—	2	4	no	—
14	C₂×Q₈×D₁₀₁	C₄	1	8	yes	complemented
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
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
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₄⋊C₄×D₁₀₁	—	2	4	no	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₄²×D₁₀₁	—	2	4	no	—
25	#15692162	C₂²	1	4	yes	—
26	#15692162	C₂²	1	4	yes	—
27	#15692162	C₂²	1	4	yes	—
28	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
31	#95×C₁₀₁	C₂²	1	4	yes	—
32	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₂₀₂	—	2	2	no	—
34	C₂×Dic₂₀₂	—	2	2	no	—
35	C₂×Dic₂₀₂	—	2	4	no	—
36	C₂×Dic₂₀₂	—	2	4	no	—
37	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	complemented
38	C₂×Dic₂₀₂	C₂×C₄	1	8	yes	complemented
39	Q₈×D₁₀₁	—	2	32	no	—
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	1	yes	characteristic
44	C₂×C₄×D₁₀₁	—	2	4	no	—
45	C₂×C₄×D₁₀₁	—	2	4	no	—
46	C₂×C₄×D₁₀₁	—	2	4	no	—
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	—	2	8	no	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	—
51	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
53	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
54	C₂×C₄×D₁₀₁	Q₈	1	8	yes	complemented
55	C₄⋊Dic₁₀₁	—	2	2	no	—
56	#13108053	—	2	4	no	—
57	C₄×Dic₁₀₁	—	2	2	no	—
58	C₄×Dic₁₀₁	—	2	4	no	—
59	C₂×Q₈×C₁₀₁	—	2	2	no	—
60	C₂×Q₈×C₁₀₁	—	2	2	no	—
61	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
62	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₁	—	2	2	no	—
67	C₄²×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
69	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
71	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
72	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
73	C₂²×D₁₀₁	#41	1	4	yes	—
74	Q₈×C₁₀₁	—	2	8	no	—
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	2	2	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	2	no	—
80	C₂×C₄×C₁₀₁	—	2	4	no	—
81	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	complemented
86	C₂×Dic₁₀₁	—	2	2	no	—
87	C₂×Dic₁₀₁	—	2	2	no	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
93	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
95	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
96	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	complemented
97	C₄×D₁₀₁	—	2	16	no	—
98	C₄×D₁₀₁	—	2	16	no	—
99	C₄×D₁₀₁	—	4	8	no	—
100	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	—
101	Dic₂₀₂	—	2	8	no	—
102	Dic₂₀₂	—	2	16	no	—
103	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
104	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₁₀₁	C₂²×Q₈	1	2	yes	—
107	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
108	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×D₁₀₁	C₄×Q₈	1	8	yes	—
110	C₂×D₁₀₁	C₂²⋊Q₈	1	8	yes	—
111	C₂×D₁₀₁	#124	1	4	yes	—
112	C₂×D₁₀₁	C₄⋊Q₈	1	4	yes	—
113	C₄×C₁₀₁	—	2	4	no	—
114	C₄×C₁₀₁	—	2	4	no	—
115	C₄×C₁₀₁	—	4	2	no	—
116	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
117	Dic₁₀₁	—	2	4	no	—
118	Dic₁₀₁	—	2	4	no	—
119	Dic₁₀₁	—	4	2	no	—
120	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
123	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	2	yes	—
124	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
125	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	D₁₀₁	#391	1	8	yes	direct product factor
127	C₂×#391	—	101	1	no	maximal, Sylow
128	C₁₀₁	C₂×#391	1	1	yes	characteristic, complemented, Sylow
129	C₂³×Q₈	—	101	1	no	—
130	C₂²×C₄⋊C₄	—	101	1	no	—
131	C₂²×C₄²	—	101	1	no	—
132	#391	—	101	1	no	—
133	#391	—	101	1	no	—
134	#391	—	101	1	no	—
135	#391	—	101	4	no	—
136	#391	D₁₀₁	1	1	yes	characteristic, direct product factor
137	C₂×#95	—	101	4	no	—
138	C₂²×Q₈	—	101	8	no	—
139	C₂²×Q₈	—	101	2	no	—
140	C₂²×Q₈	—	101	1	no	—
141	C₂²×Q₈	—	202	4	no	—
142	C₂²×Q₈	—	202	4	no	—
143	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
144	C₂³×C₄	—	101	1	no	—
145	C₂³×C₄	—	101	2	no	—
146	C₂³×C₄	—	101	2	no	—
147	C₂³×C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	2	no	—
149	C₂×C₄⋊C₄	—	101	1	no	—
150	C₂×C₄⋊C₄	—	202	4	no	—
151	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
152	C₂×C₄²	—	101	2	no	—
153	C₂×C₄²	—	101	1	no	—
154	C₂×C₄²	—	202	4	no	—
155	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
156	#95	—	101	4	no	—
157	#95	—	101	4	no	—
158	#95	—	101	4	no	—
159	#95	C₂×D₁₀₁	1	4	yes	—
160	C₂⁴	—	101	1	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	101	8	no	—
164	C₂×Q₈	—	101	4	no	—
165	C₂×Q₈	—	202	4	no	—
166	C₂×Q₈	—	202	4	no	—
167	C₂×Q₈	—	202	32	no	—
168	C₂×Q₈	—	202	2	no	—
169	C₂×Q₈	—	202	2	no	—
170	C₂×Q₈	C₄×D₁₀₁	1	4	yes	complemented
171	C₂²×C₄	—	101	1	no	—
172	C₂²×C₄	—	101	4	no	—
173	C₂²×C₄	—	101	4	no	—
174	C₂²×C₄	—	101	4	no	—
175	C₂²×C₄	—	101	2	no	—
176	C₂²×C₄	—	101	2	no	—
177	C₂²×C₄	—	101	8	no	—
178	C₂²×C₄	—	101	8	no	—
179	C₂²×C₄	—	101	2	no	—
180	C₂²×C₄	—	202	4	no	—
181	C₂²×C₄	—	202	8	no	—
182	C₂²×C₄	—	202	4	no	—
183	C₂²×C₄	—	202	4	no	—
184	C₂²×C₄	—	202	4	no	—
185	C₂²×C₄	—	202	4	no	—
186	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
187	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
188	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
189	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	202	4	no	—
192	C₄⋊C₄	—	202	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	202	4	no	—
195	C₄²	—	202	2	no	—
196	C₂³	—	101	2	no	—
197	C₂³	—	101	2	no	—
198	C₂³	—	101	2	no	—
199	C₂³	—	101	4	no	—
200	C₂³	—	101	4	no	—
201	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
202	Q₈	—	2	8	no	—
203	Q₈	—	202	16	no	—
204	Q₈	—	202	8	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	101	2	no	—
212	C₂×C₄	—	101	2	no	—
213	C₂×C₄	—	101	2	no	—
214	C₂×C₄	—	101	16	no	—
215	C₂×C₄	—	101	4	no	—
216	C₂×C₄	—	101	4	no	—
217	C₂×C₄	—	404	8	no	—
218	C₂×C₄	—	202	16	no	—
219	C₂×C₄	—	202	16	no	—
220	C₂×C₄	—	202	2	no	—
221	C₂×C₄	—	202	4	no	—
222	C₂×C₄	—	202	2	no	—
223	C₂×C₄	—	202	2	no	—
224	C₂×C₄	—	202	2	no	—
225	C₂×C₄	—	202	2	no	—
226	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
227	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
228	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
229	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
230	C₂×C₄	Q₈×D₁₀₁	1	4	yes	complemented
231	C₂²	—	101	4	no	—
232	C₂²	—	101	8	no	—
233	C₂²	—	101	8	no	—
234	C₂²	—	101	4	no	—
235	C₂²	—	101	4	no	—
236	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
237	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
238	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
239	C₂²	C₂×Q₈×D₁₀₁	1	2	yes	—
240	C₂²	#41×D₁₀₁	1	2	yes	—
241	C₄	—	2	4	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	—	101	4	no	—
245	C₄	—	404	2	no	—
246	C₄	—	202	4	no	—
247	C₄	—	202	4	no	—
248	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
249	C₂	—	101	8	no	—
250	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
251	C₂	C₄×Q₈×D₁₀₁	1	2	yes	—
252	C₂	C₂²⋊Q₈×D₁₀₁	1	2	yes	—
253	C₂	#124×D₁₀₁	1	1	yes	characteristic
254	C₂	C₄⋊Q₈×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#391×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	#391	D₁₀₁
hypocenter	C₁₀₁	C₂×#391
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
4. #20265084	C₂	semidirect
5. #20265083	C₂	semidirect
6. #20265082	C₂	semidirect
7. #20265081	C₂	semidirect
9. #391×C₁₀₁	C₂	semidirect
14. C₂×Q₈×D₁₀₁	C₄	semidirect
37. C₂×Dic₂₀₂	C₂×C₄	semidirect
38. C₂×Dic₂₀₂	C₂×C₄	semidirect
54. C₂×C₄×D₁₀₁	Q₈	semidirect
61. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
85. C₂×C₄×C₁₀₁	C₂×Q₈	semidirect
96. C₂×Dic₁₀₁	C₂×Q₈	semidirect
126. D₁₀₁	#391	direct (N × H)
128. C₁₀₁	C₂×#391	semidirect
136. #391	D₁₀₁	direct (N × H)
170. C₂×Q₈	C₄×D₁₀₁	semidirect
230. C₂×C₄	Q₈×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₀₁	2
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₄²×D₁₀₁	2
#20265084	2
#20265083	2
#20265082	2
#20265081	2
#95×D₁₀₁	2
#391×C₁₀₁	2
C₂×#391	101

Subgroups — #391×D101 #27173598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #391×D₁₀₁ #27173598