Subgroups — #419×D₁₀₁ #27179635

Order 12928.

752 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	C₂²×Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265187	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265186	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265185	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265184	C₂	1	4	yes	maximal, complemented
8	#103×D₁₀₁	C₂	1	4	yes	maximal, complemented
9	#419×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₀₂	—	2	1	no	—
11	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
12	C₂×Q₈×D₁₀₁	—	2	4	no	—
13	C₂×Q₈×D₁₀₁	—	2	1	no	—
14	C₂×Q₈×D₁₀₁	—	2	1	no	—
15	C₂×Q₈×D₁₀₁	C₄	1	8	yes	complemented
16	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	—
17	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	—	2	1	no	—
19	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108058	C₂²	1	1	yes	characteristic
21	C₂×#13108057	C₂²	1	2	yes	—
22	M₁₆×D₁₀₁	—	2	4	no	—
23	C₂×C₈×D₁₀₁	C₂²	1	2	yes	—
24	#15692214	C₂²	1	1	yes	characteristic
25	#15692213	C₂²	1	2	yes	—
26	#15692212	—	2	1	no	—
27	#15692211	—	2	1	no	—
28	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×D₁₀₁	—	2	1	no	—
30	C₄²×D₁₀₁	—	2	1	no	—
31	#15692180	C₂²	1	4	yes	complemented
32	#15692179	C₂²	1	4	yes	complemented
33	#15692178	C₂²	1	4	yes	complemented
34	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
36	#117×C₁₀₁	C₂²	1	1	yes	characteristic
37	#103×C₁₀₁	C₂²	1	4	yes	complemented
38	C₂³×D₁₀₁	D₄	1	1	yes	characteristic
39	C₂×Dic₂₀₂	—	2	1	no	—
40	C₂×Dic₂₀₂	—	2	2	no	—
41	C₂×Dic₂₀₂	—	2	8	no	—
42	C₂×Dic₂₀₂	—	2	2	no	—
43	C₂×Dic₂₀₂	—	2	2	no	—
44	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₂₀₂	C₂³	1	2	yes	—
46	Q₈×D₁₀₁	—	2	8	no	—
47	Q₈×D₁₀₁	—	2	8	no	—
48	Q₈×D₁₀₁	—	2	4	no	—
49	Q₈×D₁₀₁	—	2	4	no	—
50	Q₈×D₁₀₁	C₂×C₄	1	8	yes	—
51	Q₈×D₁₀₁	C₂×C₄	1	8	yes	complemented
52	C₂²×Dic₁₀₁	—	2	1	no	—
53	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₁	—	2	8	no	—
55	C₂×C₄×D₁₀₁	—	2	2	no	—
56	C₂×C₄×D₁₀₁	—	2	2	no	—
57	C₂×C₄×D₁₀₁	—	2	2	no	—
58	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
59	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
60	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
62	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
64	#13108058	—	2	2	no	—
65	#13108057	—	2	4	no	—
66	C₂×#11717579	C₂³	1	2	yes	—
67	C₈×D₁₀₁	—	2	4	no	—
68	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
69	#13108053	—	2	2	no	—
70	C₂²⋊Dic₁₀₁	—	2	1	no	—
71	#13108051	—	2	2	no	—
72	C₄×Dic₁₀₁	—	2	1	no	—
73	C₄×Dic₁₀₁	—	2	2	no	—
74	C₂×Q₈×C₁₀₁	—	2	2	no	—
75	C₂×Q₈×C₁₀₁	—	2	1	no	—
76	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
77	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₁	—	2	1	no	—
79	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₀₁	—	2	2	no	—
81	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
82	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
84	C₄²×C₁₀₁	—	2	1	no	—
85	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₁₀₁	—	2	4	no	—
87	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
88	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₁₀₁	—	2	4	no	—
91	Q₈×C₁₀₁	—	2	2	no	—
92	Q₈×C₁₀₁	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	—	2	4	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₁₀₁	—	2	4	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
104	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
106	C₄×D₁₀₁	—	2	8	no	—
107	C₄×D₁₀₁	—	2	8	no	—
108	C₄×D₁₀₁	—	4	4	no	—
109	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
110	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
111	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
112	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
113	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
114	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
115	Dic₂₀₂	—	2	4	no	—
116	Dic₂₀₂	—	2	2	no	—
117	Dic₂₀₂	—	2	8	no	—
118	Dic₂₀₂	—	2	8	no	—
119	Dic₂₀₂	C₂²×C₄	1	4	yes	—
120	C₈×C₁₀₁	—	2	2	no	—
121	#11717579	—	2	2	no	—
122	C₂²×C₁₀₁	—	2	2	no	—
123	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	—	2	8	no	—
127	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×D₁₀₁	#137	1	4	yes	—
130	C₄×C₁₀₁	—	2	4	no	—
131	C₄×C₁₀₁	—	4	2	no	—
132	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
133	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
135	Dic₁₀₁	—	2	4	no	—
136	Dic₁₀₁	—	4	2	no	—
137	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
138	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₁₀₁	—	2	2	no	—
140	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₁₀₁	C₂×#137	1	2	yes	—
142	D₁₀₁	—	2	2	no	—
143	D₁₀₁	#419	1	4	yes	direct product factor
144	C₂×#419	—	101	1	no	maximal, Sylow
145	C₁₀₁	C₂×#419	1	1	yes	characteristic, complemented, Sylow
146	C₂³×Q₈	—	101	1	no	—
147	C₂²×M₁₆	—	101	1	no	—
148	C₂×#117	—	101	1	no	—
149	#419	—	101	1	no	—
150	#419	—	101	1	no	—
151	#419	—	101	1	no	—
152	#419	—	101	4	no	—
153	#419	D₁₀₁	1	1	yes	characteristic, direct product factor
154	C₂×#103	—	101	4	no	—
155	C₂²×Q₈	—	101	8	no	—
156	C₂²×Q₈	—	101	2	no	—
157	C₂²×Q₈	—	101	2	no	—
158	C₂²×Q₈	—	101	1	no	—
159	C₂²×Q₈	—	202	1	no	—
160	C₂²×Q₈	—	202	1	no	—
161	C₂²×Q₈	—	202	1	no	—
162	C₂²×Q₈	—	202	4	no	—
163	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
164	C₂³×C₄	—	101	1	no	—
165	C₂³×C₄	—	202	1	no	—
166	C₂×M₁₆	—	101	1	no	—
167	C₂×M₁₆	—	101	2	no	—
168	C₂×M₁₆	—	202	4	no	—
169	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
170	C₂²×C₈	—	101	2	no	—
171	#117	—	101	1	no	—
172	#117	—	101	2	no	—
173	#117	—	202	1	no	—
174	#117	—	202	1	no	—
175	#117	C₂×D₁₀₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	101	2	no	—
177	C₂×C₂²⋊C₄	—	202	1	no	—
178	C₂×C₄²	—	202	1	no	—
179	#103	—	101	4	no	—
180	#103	—	101	4	no	—
181	#103	—	101	4	no	—
182	#103	C₂×D₁₀₁	1	4	yes	complemented
183	C₂⁴	—	101	1	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	101	8	no	—
187	C₂×Q₈	—	101	8	no	—
188	C₂×Q₈	—	101	4	no	—
189	C₂×Q₈	—	101	2	no	—
190	C₂×Q₈	—	202	2	no	—
191	C₂×Q₈	—	202	2	no	—
192	C₂×Q₈	—	202	8	no	—
193	C₂×Q₈	—	202	4	no	—
194	C₂×Q₈	—	202	4	no	—
195	C₂×Q₈	—	202	8	no	—
196	C₂×Q₈	—	202	8	no	—
197	C₂×Q₈	—	202	1	no	—
198	C₂×Q₈	—	202	2	no	—
199	C₂×Q₈	C₄×D₁₀₁	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	101	1	no	—
203	C₂²×C₄	—	101	4	no	—
204	C₂²×C₄	—	101	1	no	—
205	C₂²×C₄	—	101	1	no	—
206	C₂²×C₄	—	101	4	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	202	2	no	—
210	C₂²×C₄	—	202	2	no	—
211	C₂²×C₄	—	202	2	no	—
212	C₂²×C₄	—	202	8	no	—
213	C₂²×C₄	—	202	1	no	—
214	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	202	4	no	—
217	M₁₆	—	202	2	no	—
218	C₂×C₈	—	101	2	no	—
219	C₂×C₈	—	202	4	no	—
220	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
221	C₄⋊C₄	—	101	2	no	—
222	C₄⋊C₄	—	202	2	no	—
223	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	202	2	no	—
226	C₂²⋊C₄	—	202	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	202	2	no	—
229	C₄²	—	202	1	no	—
230	C₂³	—	101	1	no	—
231	C₂³	—	101	1	no	—
232	C₂³	—	101	4	no	—
233	C₂³	—	202	4	no	—
234	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	4	no	—
237	Q₈	—	101	4	no	—
238	Q₈	—	202	8	no	—
239	Q₈	—	202	8	no	—
240	Q₈	—	202	2	no	—
241	Q₈	—	202	4	no	—
242	Q₈	C₂×C₄×D₁₀₁	1	4	yes	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	101	1	no	—
247	C₂×C₄	—	101	1	no	—
248	C₂×C₄	—	101	4	no	—
249	C₂×C₄	—	101	2	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	101	2	no	—
252	C₂×C₄	—	101	2	no	—
253	C₂×C₄	—	101	4	no	—
254	C₂×C₄	—	101	4	no	—
255	C₂×C₄	—	404	4	no	—
256	C₂×C₄	—	202	8	no	—
257	C₂×C₄	—	202	8	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	4	no	—
261	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
262	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
263	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	202	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	101	2	no	—
269	C₂²	—	101	4	no	—
270	C₂²	—	101	2	no	—
271	C₂²	—	202	8	no	—
272	C₂²	—	202	2	no	—
273	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
274	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	4	2	no	—
277	C₄	—	101	2	no	—
278	C₄	—	101	2	no	—
279	C₄	—	404	2	no	—
280	C₄	—	202	4	no	—
281	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
282	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
283	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	101	4	no	—
286	C₂	—	202	2	no	—
287	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
288	C₂	#137×D₁₀₁	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#419×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂³×D₁₀₁	D₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#419	D₁₀₁
hypocenter	C₁₀₁	C₂×#419
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
4. #20265187	C₂	semidirect
5. #20265186	C₂	semidirect
6. #20265185	C₂	semidirect
7. #20265184	C₂	semidirect
8. #103×D₁₀₁	C₂	semidirect
9. #419×C₁₀₁	C₂	semidirect
15. C₂×Q₈×D₁₀₁	C₄	semidirect
31. #15692180	C₂²	semidirect
32. #15692179	C₂²	semidirect
33. #15692178	C₂²	semidirect
37. #103×C₁₀₁	C₂²	semidirect
44. C₂×Dic₂₀₂	C₂×C₄	semidirect
51. Q₈×D₁₀₁	C₂×C₄	semidirect
76. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
143. D₁₀₁	#419	direct (N × H)
145. C₁₀₁	C₂×#419	semidirect
153. #419	D₁₀₁	direct (N × H)
182. #103	C₂×D₁₀₁	semidirect
199. C₂×Q₈	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×Q₈×D₁₀₁	2
C₂×M₁₆×D₁₀₁	2
#117×D₁₀₁	2
#20265187	2
#20265186	2
#20265185	2
#20265184	2
#103×D₁₀₁	2
#419×C₁₀₁	2
C₂×#419	101

Subgroups — #419×D101 #27179635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #419×D₁₀₁ #27179635