Subgroups — #27179928

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#41×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265645	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265286	C₂	1	2	yes	maximal, complemented
5	#20265281	C₂	1	2	yes	maximal, complemented
6	#20265280	C₂	1	2	yes	maximal, complemented
7	#20265274	C₂	1	2	yes	maximal, complemented
8	#20265213	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265174	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692165	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×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	#15692215	—	2	2	no	—
25	#15692214	—	2	1	no	—
26	#15692214	C₂²	1	1	yes	characteristic
27	#15692212	—	2	1	no	—
28	C₂×C₄×Dic₁₀₁	—	2	1	no	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₄²×D₁₀₁	—	2	1	no	—
31	#15692188	C₂²	1	2	yes	—
32	#15692186	C₂²	1	2	yes	—
33	#15692180	C₂²	1	2	yes	—
34	#15692177	C₂²	1	2	yes	—
35	#15692165	C₂²	1	2	yes	—
36	#15692165	C₂²	1	2	yes	—
37	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
39	#117×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	2	no	—
48	C₂²×Dic₁₀₁	D₄	1	2	yes	—
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	2	no	—
59	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
60	#13108053	—	2	2	no	—
61	C₂²⋊Dic₁₀₁	—	2	1	no	—
62	C₂²⋊Dic₁₀₁	—	2	2	no	—
63	#13108051	—	2	2	no	—
64	C₄×Dic₁₀₁	—	2	1	no	—
65	C₄×Dic₁₀₁	—	2	2	no	—
66	C₄×Dic₁₀₁	—	2	2	no	—
67	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
68	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₀₁	—	2	2	no	—
78	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
81	C₄²×C₁₀₁	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₁	#41	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₀₁	—	2	4	no	—
98	C₂×Dic₁₀₁	—	2	1	no	—
99	C₂×Dic₁₀₁	—	2	2	no	—
100	C₂×Dic₁₀₁	—	4	2	no	—
101	C₂×Dic₁₀₁	—	4	4	no	—
102	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
103	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
104	D₄₀₄	—	2	1	no	—
105	D₄₀₄	C₂²×C₄	1	2	yes	—
106	C₄×D₁₀₁	—	2	4	no	—
107	C₄×D₁₀₁	—	4	4	no	—
108	Dic₂₀₂	—	2	1	no	—
109	Dic₂₀₂	C₂²×C₄	1	2	yes	—
110	C₈×C₁₀₁	—	2	2	no	—
111	#11717579	C₂²×C₄	1	2	yes	—
112	#11717579	C₂²×C₄	1	2	yes	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	4	2	no	—
116	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
117	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₁	—	2	2	no	—
119	C₂×D₁₀₁	—	4	2	no	—
120	C₄×C₁₀₁	—	2	2	no	—
121	C₄×C₁₀₁	—	4	2	no	—
122	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
123	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
125	Dic₁₀₁	—	2	2	no	—
126	Dic₁₀₁	—	2	2	no	—
127	Dic₁₀₁	—	2	2	no	—
128	Dic₁₀₁	—	4	2	no	—
129	C₂×C₁₀₁	—	2	2	no	—
130	C₂×C₁₀₁	—	4	2	no	—
131	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₀₁	#575	1	2	yes	—
133	D₁₀₁	—	4	2	no	—
134	#2943	—	101	1	no	maximal, Sylow
135	C₁₀₁	#2943	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	101	1	no	—
137	C₄×#41	—	101	1	no	—
138	C₄×#41	—	101	1	no	—
139	#442	—	101	2	no	—
140	#441	—	101	2	no	—
141	#440	—	101	2	no	—
142	#440	—	101	2	no	—
143	#427	—	101	1	no	—
144	#416	—	101	1	no	—
145	#416	D₁₀₁	1	1	yes	characteristic, complemented
146	C₂×#97	—	101	1	no	—
147	C₂×#41	—	101	1	no	—
148	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
149	#135	—	101	8	no	—
150	C₂×Q₁₆	—	101	1	no	—
151	C₂×SD₁₆	—	101	1	no	—
152	C₂×SD₁₆	—	101	1	no	—
153	C₂×D₈	—	101	1	no	—
154	C₂²×C₈	—	101	1	no	—
155	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
156	C₄×Q₈	—	101	2	no	—
157	C₄×Q₈	—	101	2	no	—
158	C₄×D₄	—	101	2	no	—
159	C₄×D₄	—	101	2	no	—
160	C₄×D₄	—	202	2	no	—
161	C₄×D₄	—	202	2	no	—
162	#117	—	101	1	no	—
163	#117	—	202	1	no	—
164	#117	—	202	1	no	—
165	#117	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂×C₄²	—	101	1	no	—
167	C₂×C₄²	—	202	1	no	—
168	C₂×C₄²	—	202	1	no	—
169	#107	—	101	2	no	—
170	#106	—	101	2	no	—
171	#103	—	101	2	no	—
172	#103	C₂×D₁₀₁	1	2	yes	—
173	#102	—	101	2	no	—
174	#102	C₂×D₁₀₁	1	2	yes	—
175	#97	—	101	2	no	—
176	#97	—	101	2	no	—
177	#41	—	2	2	no	—
178	#41	—	101	4	no	—
179	#41	—	202	2	no	—
180	#41	C₄×D₁₀₁	1	4	yes	complemented
181	C₂×Q₈	—	101	1	no	—
182	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	101	1	no	—
185	C₂×D₄	—	202	1	no	—
186	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	101	2	no	—
189	C₂²×C₄	—	202	2	no	—
190	C₂²×C₄	—	202	2	no	—
191	C₂²×C₄	—	202	1	no	—
192	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
193	Q₁₆	—	101	4	no	—
194	SD₁₆	—	101	4	no	—
195	SD₁₆	—	101	4	no	—
196	D₈	—	101	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	101	4	no	—
199	C₂×C₈	—	101	1	no	—
200	C₂×C₈	—	101	1	no	—
201	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
202	C₄⋊C₄	—	101	2	no	—
203	C₄⋊C₄	—	202	2	no	—
204	C₄⋊C₄	—	202	2	no	—
205	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	202	2	no	—
208	C₂²⋊C₄	—	202	2	no	—
209	C₂²⋊C₄	—	202	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	101	1	no	—
212	C₄²	—	101	1	no	—
213	C₄²	—	101	2	no	—
214	C₄²	—	202	2	no	—
215	C₄²	—	202	2	no	—
216	C₄²	—	202	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	202	1	no	—
219	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	101	2	no	—
222	Q₈	—	202	1	no	—
223	Q₈	C₂×C₄×D₁₀₁	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	101	2	no	—
227	D₄	—	202	4	no	—
228	D₄	—	202	1	no	—
229	D₄	C₂×C₄×D₁₀₁	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	101	2	no	—
234	C₂×C₄	—	101	2	no	—
235	C₂×C₄	—	404	4	no	—
236	C₂×C₄	—	404	4	no	—
237	C₂×C₄	—	404	2	no	—
238	C₂×C₄	—	202	2	no	—
239	C₂×C₄	—	202	1	no	—
240	C₂×C₄	—	202	4	no	—
241	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
242	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
243	C₂×C₄	#13108073	1	2	yes	—
244	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	101	2	no	—
247	C₈	—	101	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	404	2	no	—
252	C₂²	—	202	2	no	—
253	C₂²	#15692215	1	2	yes	—
254	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	404	2	no	—
258	C₄	—	202	2	no	—
259	C₄	—	202	2	no	—
260	C₄	—	202	2	no	—
261	C₄	#15692215	1	2	yes	—
262	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
263	C₄	C₂×#13108073	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	404	2	no	—
267	C₂	C₂×#15692215	1	1	yes	characteristic
268	C₂	#20265974	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#416×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂×#15692275	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#416	D₁₀₁
hypocenter	C₁₀₁	#2943
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₁₀₁	C₂	semidirect
3. #20265645	C₂	semidirect
4. #20265286	C₂	semidirect
5. #20265281	C₂	semidirect
6. #20265280	C₂	semidirect
7. #20265274	C₂	semidirect
8. #20265213	C₂	semidirect
9. #20265174	C₂	semidirect
10. C₂×#15692165	C₂	semidirect
11. #416×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
135. C₁₀₁	#2943	semidirect
145. #416	D₁₀₁	semidirect
180. #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
#41×Dic₁₀₁	2
#20265645	2
#20265286	2
#20265281	2
#20265280	2
#20265274	2
#20265213	2
#20265174	2
C₂×#15692165	2
#416×C₁₀₁	2
#2943	101