Subgroups — #27180003

Order 12928.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692275	C₂	1	1	yes	maximal, characteristic
2	#20265660	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265658	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265280	C₂	1	2	yes	maximal, complemented
5	#20265274	C₂	1	2	yes	maximal, complemented
6	#20265269	C₂	1	2	yes	maximal, complemented
7	#20265257	C₂	1	2	yes	maximal, complemented
8	C₂×#15692188	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265179	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265135	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	#15692275	C₄	1	8	yes	complemented
14	C₂×#13108064	C₂²	1	1	yes	characteristic
15	C₂×#13108062	C₂²	1	1	yes	characteristic
16	C₂×#13108061	C₂²	1	1	yes	characteristic
17	C₂×#13108059	C₂²	1	1	yes	characteristic
18	C₂²×#11717579	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₀₁	C₂²	1	2	yes	—
20	#15692220	C₂²	1	2	yes	—
21	D₄×Dic₁₀₁	—	2	2	no	—
22	D₄×Dic₁₀₁	C₂²	1	2	yes	—
23	#15692217	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₀₁	—	2	2	no	—
25	#15692214	—	2	1	no	—
26	#15692214	C₂²	1	1	yes	characteristic
27	#15692213	—	2	1	no	—
28	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
29	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₀₁	—	2	1	no	—
31	#15692188	C₂²	1	2	yes	—
32	#15692188	C₂²	1	2	yes	—
33	#15692180	C₂²	1	2	yes	—
34	#15692177	C₂²	1	2	yes	—
35	#15692165	C₂²	1	2	yes	—
36	C₈×Dic₁₀₁	C₂²	1	2	yes	—
37	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
40	#103×C₁₀₁	C₂²	1	2	yes	—
41	#102×C₁₀₁	C₂²	1	2	yes	—
42	#13108074	—	2	2	no	—
43	#13108074	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
45	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
47	C₂²×Dic₁₀₁	—	2	1	no	—
48	C₂²×Dic₁₀₁	—	2	2	no	—
49	C₂×C₄×D₁₀₁	—	2	1	no	—
50	C₂×C₄×D₁₀₁	—	2	2	no	—
51	#13108064	C₂×C₄	1	4	yes	complemented
52	#13108062	C₂×C₄	1	4	yes	complemented
53	#13108061	C₂×C₄	1	4	yes	complemented
54	#13108059	C₂×C₄	1	4	yes	complemented
55	C₂×#11717579	C₂×C₄	1	4	yes	complemented
56	C₂×#11717579	C₂³	1	1	yes	characteristic
57	C₂×#11717579	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₁	—	2	1	no	—
59	C₄⋊Dic₁₀₁	—	2	2	no	—
60	C₄⋊Dic₁₀₁	D₄	1	1	yes	characteristic
61	C₄⋊Dic₁₀₁	D₄	1	1	yes	characteristic
62	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
63	#13108053	—	2	2	no	—
64	C₂²⋊Dic₁₀₁	—	2	2	no	—
65	C₂²⋊Dic₁₀₁	D₄	1	2	yes	—
66	#13108051	—	2	2	no	—
67	C₄×Dic₁₀₁	—	2	2	no	—
68	C₄×Dic₁₀₁	—	2	2	no	—
69	C₄×Dic₁₀₁	C₂³	1	2	yes	—
70	#41×C₁₀₁	—	2	2	no	—
71	#41×C₁₀₁	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₁₀₁	—	2	1	no	—
74	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	—	2	1	no	—
76	C₂²×C₄×C₁₀₁	—	2	1	no	—
77	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	M₁₆×C₁₀₁	—	2	2	no	—
79	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₀₁	—	2	1	no	—
81	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₀₁	—	2	1	no	—
85	Q₈×C₁₀₁	—	2	1	no	—
86	Q₈×C₁₀₁	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₁	—	2	4	no	—
88	D₄×C₁₀₁	—	2	1	no	—
89	D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	8	1	no	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₁	#41	1	2	yes	—
97	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₁₀₁	—	2	4	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	8	2	no	—
102	C₂×Dic₁₀₁	—	8	1	no	—
103	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
104	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
105	D₄₀₄	—	2	1	no	—
106	D₄₀₄	C₂²×C₄	1	2	yes	—
107	C₄×D₁₀₁	—	2	4	no	—
108	C₄×D₁₀₁	—	8	2	no	—
109	Dic₂₀₂	—	2	1	no	—
110	Dic₂₀₂	C₂²×C₄	1	2	yes	—
111	C₈×C₁₀₁	—	2	2	no	—
112	#11717579	C₂²×C₄	1	2	yes	—
113	#11717579	C₂²×C₄	1	2	yes	—
114	C₂²×C₁₀₁	—	2	1	no	—
115	C₂²×C₁₀₁	—	2	1	no	—
116	C₂²×C₁₀₁	—	2	2	no	—
117	C₂²×C₁₀₁	—	4	1	no	—
118	C₂²×C₁₀₁	—	4	1	no	—
119	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
120	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₀₁	—	2	2	no	—
122	C₂×D₁₀₁	—	4	1	no	—
123	C₂×D₁₀₁	—	4	1	no	—
124	C₄×C₁₀₁	—	2	2	no	—
125	C₄×C₁₀₁	—	4	2	no	—
126	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
127	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
129	Dic₁₀₁	—	2	2	no	—
130	Dic₁₀₁	—	2	2	no	—
131	Dic₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	4	2	no	—
133	C₂×C₁₀₁	—	2	2	no	—
134	C₂×C₁₀₁	—	4	2	no	—
135	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₀₁	#576	1	1	yes	characteristic
137	C₂×C₁₀₁	#578	1	1	yes	characteristic
138	D₁₀₁	—	4	2	no	—
139	#2951	—	101	1	no	maximal, Sylow
140	C₁₀₁	#2951	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	101	1	no	—
142	#520	—	101	1	no	—
143	#520	—	101	1	no	—
144	#440	—	101	2	no	—
145	#440	—	101	2	no	—
146	C₄×Q₁₆	—	101	2	no	—
147	C₄×D₈	—	101	2	no	—
148	C₂×#107	—	101	1	no	—
149	#417	—	101	1	no	—
150	#417	D₁₀₁	1	1	yes	characteristic, complemented
151	#405	—	101	1	no	—
152	C₂×#41	—	101	1	no	—
153	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
154	#135	—	101	8	no	—
155	C₂×Q₁₆	—	101	1	no	—
156	C₂×SD₁₆	—	101	1	no	—
157	C₂×SD₁₆	—	101	1	no	—
158	C₂×D₈	—	101	1	no	—
159	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
160	C₂²×C₈	—	101	1	no	—
161	C₄×Q₈	—	101	2	no	—
162	C₄×Q₈	—	101	2	no	—
163	C₄×D₄	—	101	2	no	—
164	C₄×D₄	—	101	2	no	—
165	C₄×D₄	—	202	2	no	—
166	C₄×D₄	—	202	2	no	—
167	#117	—	101	1	no	—
168	#117	—	202	1	no	—
169	#117	—	202	1	no	—
170	C₂×C₄⋊C₄	—	101	1	no	—
171	C₂×C₄⋊C₄	—	202	1	no	—
172	C₂×C₄⋊C₄	—	202	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
174	#107	—	101	2	no	—
175	#107	—	101	2	no	—
176	#103	—	101	2	no	—
177	#103	C₂×D₁₀₁	1	2	yes	—
178	#102	—	101	2	no	—
179	#102	C₂×D₁₀₁	1	2	yes	—
180	#97	—	101	2	no	—
181	C₄×C₈	—	101	2	no	—
182	#41	—	2	2	no	—
183	#41	—	101	4	no	—
184	#41	—	202	2	no	—
185	#41	C₄×D₁₀₁	1	4	yes	complemented
186	C₂×Q₈	—	101	1	no	—
187	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	101	1	no	—
190	C₂×D₄	—	202	1	no	—
191	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	202	2	no	—
195	C₂²×C₄	—	202	2	no	—
196	C₂²×C₄	—	202	1	no	—
197	C₂²×C₄	—	202	1	no	—
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
199	Q₁₆	—	101	4	no	—
200	SD₁₆	—	101	4	no	—
201	SD₁₆	—	101	4	no	—
202	D₈	—	101	4	no	—
203	M₁₆	—	2	2	no	—
204	C₂×C₈	—	101	4	no	—
205	C₂×C₈	—	101	1	no	—
206	C₂×C₈	—	101	1	no	—
207	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	101	1	no	—
210	C₄⋊C₄	—	101	1	no	—
211	C₄⋊C₄	—	101	2	no	—
212	C₄⋊C₄	—	202	2	no	—
213	C₄⋊C₄	—	202	2	no	—
214	C₄⋊C₄	—	202	1	no	—
215	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
216	C₂²⋊C₄	—	101	2	no	—
217	C₂²⋊C₄	—	202	2	no	—
218	C₂²⋊C₄	—	202	2	no	—
219	C₄²	—	101	2	no	—
220	C₄²	—	202	2	no	—
221	C₄²	—	202	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	202	1	no	—
224	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	101	2	no	—
227	Q₈	—	202	1	no	—
228	Q₈	C₂×C₄×D₁₀₁	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	101	2	no	—
232	D₄	—	202	4	no	—
233	D₄	—	202	1	no	—
234	D₄	C₂×C₄×D₁₀₁	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	808	2	no	—
240	C₂×C₄	—	808	1	no	—
241	C₂×C₄	—	808	2	no	—
242	C₂×C₄	—	101	2	no	—
243	C₂×C₄	—	101	2	no	—
244	C₂×C₄	—	202	2	no	—
245	C₂×C₄	—	202	1	no	—
246	C₂×C₄	—	202	4	no	—
247	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
248	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
249	C₂×C₄	#13108073	1	2	yes	—
250	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	101	2	no	—
253	C₈	—	101	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	404	1	no	—
260	C₂²	—	404	1	no	—
261	C₂²	—	202	2	no	—
262	C₂²	#15692215	1	2	yes	—
263	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	404	2	no	—
267	C₄	—	202	2	no	—
268	C₄	—	202	2	no	—
269	C₄	—	202	2	no	—
270	C₄	#15692215	1	2	yes	—
271	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
272	C₄	C₂×#13108073	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	404	2	no	—
276	C₂	C₂×#15692215	1	1	yes	characteristic
277	C₂	#20265981	1	1	yes	characteristic
278	C₂	#20265993	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#417×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂×#15692275	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#417	D₁₀₁
hypocenter	C₁₀₁	#2951
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #20265660	C₂	semidirect
3. #20265658	C₂	semidirect
4. #20265280	C₂	semidirect
5. #20265274	C₂	semidirect
6. #20265269	C₂	semidirect
7. #20265257	C₂	semidirect
8. C₂×#15692188	C₂	semidirect
9. #20265179	C₂	semidirect
10. #20265135	C₂	semidirect
11. #417×C₁₀₁	C₂	semidirect
13. #15692275	C₄	semidirect
43. #13108074	C₂×C₄	semidirect
51. #13108064	C₂×C₄	semidirect
52. #13108062	C₂×C₄	semidirect
53. #13108061	C₂×C₄	semidirect
54. #13108059	C₂×C₄	semidirect
55. C₂×#11717579	C₂×C₄	semidirect
71. #41×C₁₀₁	C₂×C₄	semidirect
140. C₁₀₁	#2951	semidirect
150. #417	D₁₀₁	semidirect
185. #41	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692275	2
#20265660	2
#20265658	2
#20265280	2
#20265274	2
#20265269	2
#20265257	2
C₂×#15692188	2
#20265179	2
#20265135	2
#417×C₁₀₁	2
#2951	101