Subgroups — #27173591

Order 12928.

652 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	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692212	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265077	C₂	1	2	yes	maximal
4	#20265072	C₂	1	2	yes	maximal, complemented
5	#20265071	C₂	1	2	yes	maximal, complemented
6	#20265067	C₂	1	2	yes	maximal, complemented
7	C₄×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13108051	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
15	#15692212	—	2	4	no	—
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	2	yes	—
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×#13108051	C₂²	1	2	yes	—
23	C₂×#13108051	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	#117×C₁₀₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
40	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
45	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₁₀₁	—	2	2	no	—
47	C₂×C₄×D₁₀₁	—	2	2	no	—
48	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
49	#13108053	—	2	4	no	—
50	#13108053	—	2	4	no	—
51	#13108053	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₁₀₁	—	2	4	no	—
53	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
54	#13108051	—	2	4	no	—
55	#13108051	—	2	4	no	—
56	#13108051	C₂×C₄	1	4	yes	—
57	C₄×Dic₁₀₁	—	2	2	no	—
58	C₄×Dic₁₀₁	—	2	2	no	—
59	C₄×Dic₁₀₁	—	2	4	no	—
60	C₄×Dic₁₀₁	—	2	4	no	—
61	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
62	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	—	2	2	no	—
64	C₂²×C₄×C₁₀₁	—	2	2	no	—
65	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
66	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₁	—	2	4	no	—
71	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
72	C₄²×C₁₀₁	—	2	2	no	—
73	C₄²×C₁₀₁	D₄	1	4	yes	complemented
74	C₂³×C₁₀₁	—	2	1	no	—
75	C₂³×C₁₀₁	—	2	1	no	—
76	C₂³×C₁₀₁	—	2	2	no	—
77	C₂³×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
80	C₂²×D₁₀₁	—	2	1	no	—
81	C₂²×D₁₀₁	—	2	2	no	—
82	C₂²×D₁₀₁	—	2	1	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₁	—	2	1	no	—
86	C₂×C₄×C₁₀₁	—	2	1	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	8	no	—
90	C₂×C₄×C₁₀₁	—	2	2	no	—
91	C₂×C₄×C₁₀₁	—	2	2	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₁	#41	1	2	yes	—
98	C₂²⋊D₁₀₁	—	2	8	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	1	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	—	2	2	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
108	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
109	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
110	C₂×Dic₁₀₁	#41	1	4	yes	—
111	C₂×Dic₁₀₁	#41	1	4	yes	—
112	C₄×D₁₀₁	—	2	8	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	4	2	no	—
117	C₂²×C₁₀₁	—	4	2	no	—
118	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
121	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
122	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	—	4	2	no	—
127	C₂×D₁₀₁	—	4	2	no	—
128	C₄×C₁₀₁	—	2	2	no	—
129	C₄×C₁₀₁	—	2	4	no	—
130	C₄×C₁₀₁	—	4	2	no	—
131	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
132	Dic₁₀₁	—	2	2	no	—
133	Dic₁₀₁	—	2	2	no	—
134	Dic₁₀₁	—	2	4	no	—
135	Dic₁₀₁	—	4	2	no	—
136	Dic₁₀₁	#117	1	4	yes	complemented
137	C₂×C₁₀₁	—	4	2	no	—
138	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
139	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
141	C₂×C₁₀₁	#529	1	2	yes	—
142	C₂×C₁₀₁	#532	1	2	yes	—
143	D₁₀₁	—	4	2	no	—
144	#2293	—	101	1	no	maximal, Sylow
145	C₁₀₁	#2293	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄×D₄	—	101	1	no	—
147	C₂×#117	—	101	1	no	—
148	C₂×#117	D₁₀₁	1	1	yes	characteristic, complemented
149	#390	—	101	2	no	—
150	#389	—	101	2	no	—
151	#388	—	101	2	no	—
152	#388	—	101	2	no	—
153	C₄×C₄⋊C₄	—	101	1	no	—
154	C₄×C₂²⋊C₄	—	101	1	no	—
155	C₄×C₂²⋊C₄	—	101	1	no	—
156	C₄³	—	101	1	no	—
157	C₂²×D₄	—	101	1	no	—
158	C₂³×C₄	—	101	1	no	—
159	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
160	C₄×D₄	—	101	8	no	—
161	#117	—	2	4	no	—
162	#117	—	202	4	no	—
163	C₂×C₄⋊C₄	—	101	1	no	—
164	C₂×C₄⋊C₄	—	101	2	no	—
165	C₂×C₄⋊C₄	—	101	2	no	—
166	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
167	C₂×C₂²⋊C₄	—	101	1	no	—
168	C₂×C₂²⋊C₄	—	101	1	no	—
169	C₂×C₂²⋊C₄	—	101	2	no	—
170	C₂×C₂²⋊C₄	—	101	2	no	—
171	C₂×C₂²⋊C₄	—	101	2	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
173	C₂×C₄²	—	101	1	no	—
174	C₂×C₄²	—	101	2	no	—
175	C₂×C₄²	—	101	2	no	—
176	C₂×C₄²	—	101	1	no	—
177	C₂×C₄²	—	101	1	no	—
178	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
179	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
180	#95	—	101	2	no	—
181	#95	—	101	2	no	—
182	#95	—	101	2	no	—
183	C₂⁴	—	101	1	no	—
184	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
185	C₂×D₄	—	101	4	no	—
186	C₂×D₄	—	202	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	1	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	1	no	—
193	C₂²×C₄	—	101	4	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	202	2	no	—
196	C₂²×C₄	—	202	2	no	—
197	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
198	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	101	4	no	—
204	C₄⋊C₄	—	202	4	no	—
205	C₄⋊C₄	—	202	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	101	4	no	—
208	C₂²⋊C₄	—	101	4	no	—
209	C₂²⋊C₄	—	202	4	no	—
210	C₂²⋊C₄	—	202	4	no	—
211	C₂²⋊C₄	—	202	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	101	4	no	—
214	C₄²	—	202	4	no	—
215	C₄²	—	202	4	no	—
216	C₄²	—	202	2	no	—
217	C₄²	—	202	2	no	—
218	C₄²	C₂²⋊D₁₀₁	1	4	yes	complemented
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	101	2	no	—
224	C₂³	—	202	1	no	—
225	C₂³	—	202	2	no	—
226	C₂³	—	202	2	no	—
227	C₂³	—	202	1	no	—
228	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
229	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
230	D₄	—	202	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	101	2	no	—
240	C₂×C₄	—	101	4	no	—
241	C₂×C₄	—	101	2	no	—
242	C₂×C₄	—	101	2	no	—
243	C₂×C₄	—	101	4	no	—
244	C₂×C₄	—	202	2	no	—
245	C₂×C₄	—	202	2	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	1	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	—	202	1	no	—
252	C₂×C₄	—	202	8	no	—
253	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
254	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
256	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
257	C₂×C₄	#13108074	1	2	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	404	2	no	—
264	C₂²	—	404	2	no	—
265	C₂²	—	202	2	no	—
266	C₂²	—	202	2	no	—
267	C₂²	—	202	2	no	—
268	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
270	C₂²	#41×D₁₀₁	1	2	yes	—
271	C₂²	#41×D₁₀₁	1	2	yes	—
272	C₂²	C₂×#13108074	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	101	4	no	—
277	C₄	—	404	2	no	—
278	C₄	—	202	2	no	—
279	C₄	—	202	2	no	—
280	C₄	—	202	4	no	—
281	C₄	C₄×C₂²⋊D₁₀₁	1	4	yes	—
282	C₂	—	4	2	no	—
283	C₂	—	404	2	no	—
284	C₂	#117×D₁₀₁	1	1	yes	characteristic
285	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
286	C₂	#20265645	1	1	yes	characteristic
287	C₂	#20265705	1	2	yes	—
288	C₂	#20265723	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×#117×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	#2293
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#15692212	C₂	semidirect
4. #20265072	C₂	semidirect
5. #20265071	C₂	semidirect
6. #20265067	C₂	semidirect
7. C₄×#13108053	C₂	semidirect
8. C₄×#13108051	C₂	semidirect
9. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
10. C₄²×Dic₁₀₁	C₂	semidirect
11. C₂×#117×C₁₀₁	C₂	semidirect
14. C₄×C₂²⋊D₁₀₁	C₄	semidirect
73. C₄²×C₁₀₁	D₄	semidirect
136. Dic₁₀₁	#117	semidirect
145. C₁₀₁	#2293	semidirect
148. C₂×#117	D₁₀₁	semidirect
218. C₄²	C₂²⋊D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692212	2
#20265077	2
#20265072	2
#20265071	2
#20265067	2
C₄×#13108053	2
C₄×#13108051	2
C₄×C₂²⋊Dic₁₀₁	2
C₄²×Dic₁₀₁	2
C₂×#117×C₁₀₁	2
#2293	101