Subgroups — #27173491

Order 12928.

780 conjugacy classes of subgroups.

The subgroup lattice has 284 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×#15692215	C₂	1	2	yes	maximal
3	C₄×#13108053	C₂	1	2	yes	maximal, complemented
4	#20265028	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
6	C₄×#13108051	C₂	1	2	yes	maximal, complemented
7	#20265024	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265022	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
10	#95×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₄×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
17	#15692215	C₄	1	16	yes	complemented
18	C₂×#13108053	C₂²	1	2	yes	—
19	C₂×#13108053	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	#15692162	C₂²	1	1	yes	characteristic
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	1	yes	characteristic
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
35	#95×C₁₀₁	C₂²	1	1	yes	characteristic
36	#95×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	—
39	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
40	C₂²×Dic₁₀₁	—	4	2	no	—
41	C₂²×Dic₁₀₁	—	4	2	no	—
42	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₀₁	—	4	2	no	—
49	C₂×C₄×D₁₀₁	—	4	2	no	—
50	C₂×C₄×D₁₀₁	—	4	2	no	—
51	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
52	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
53	#13108053	C₂×C₄	1	8	yes	complemented
54	#13108053	C₂×C₄	1	4	yes	complemented
55	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
56	#13108051	C₂×C₄	1	8	yes	complemented
57	#13108051	C₂×C₄	1	4	yes	complemented
58	C₄×Dic₁₀₁	—	4	2	no	—
59	C₄×Dic₁₀₁	—	4	2	no	—
60	C₄×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
61	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
62	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	—	4	2	no	—
64	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
70	C₄²×C₁₀₁	—	4	2	no	—
71	C₂³×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	—	2	1	no	—
73	C₂³×C₁₀₁	—	2	1	no	—
74	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
75	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
76	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
77	C₂²×D₁₀₁	—	2	2	no	—
78	C₂²×D₁₀₁	—	2	1	no	—
79	C₂²×D₁₀₁	—	2	1	no	—
80	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
81	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	4	4	no	—
89	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
93	C₂²⋊D₁₀₁	C₄²	1	16	yes	complemented
94	C₂×Dic₁₀₁	—	2	1	no	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	2	no	—
100	C₂×Dic₁₀₁	—	2	1	no	—
101	C₂×Dic₁₀₁	—	4	8	no	—
102	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
104	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
105	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
106	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
107	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
108	C₄×D₁₀₁	—	4	8	no	—
109	C₄×D₁₀₁	—	4	4	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	2	no	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	—
115	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
116	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	2	no	—
122	C₂×D₁₀₁	—	2	4	no	—
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	C₂×C₄²	1	4	yes	—
125	C₄×C₁₀₁	—	2	2	no	—
126	C₄×C₁₀₁	—	2	4	no	—
127	C₄×C₁₀₁	—	2	2	no	—
128	C₄×C₁₀₁	—	2	4	no	—
129	Dic₁₀₁	—	2	4	no	—
130	Dic₁₀₁	—	2	2	no	—
131	Dic₁₀₁	—	2	4	no	—
132	Dic₁₀₁	—	2	2	no	—
133	Dic₁₀₁	C₂×C₄²	1	4	yes	—
134	C₂×C₁₀₁	—	2	4	no	—
135	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
136	C₂×C₁₀₁	#518	1	2	yes	—
137	C₂×C₁₀₁	#518	1	1	yes	characteristic
138	C₂×C₁₀₁	#520	1	2	yes	—
139	C₂×C₁₀₁	#520	1	1	yes	characteristic
140	D₁₀₁	—	2	4	no	—
141	#2275	—	101	1	no	maximal, Sylow
142	C₁₀₁	#2275	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄×D₄	—	101	1	no	—
144	C₂×C₄×D₄	—	101	2	no	—
145	C₄×C₄⋊C₄	—	101	2	no	—
146	C₄×C₄⋊C₄	—	101	1	no	—
147	C₄×C₂²⋊C₄	—	101	2	no	—
148	C₄×C₂²⋊C₄	—	101	2	no	—
149	C₄×C₂²⋊C₄	—	101	1	no	—
150	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
151	#376	—	101	1	no	—
152	C₂×#95	—	101	1	no	—
153	C₂×#95	—	101	1	no	—
154	C₂²×D₄	—	101	1	no	—
155	C₂³×C₄	—	101	2	no	—
156	C₂³×C₄	—	101	2	no	—
157	C₂³×C₄	—	101	1	no	—
158	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
159	C₄×D₄	—	101	8	no	—
160	C₄×D₄	—	101	16	no	—
161	C₂×C₄⋊C₄	—	101	1	no	—
162	C₂×C₄⋊C₄	—	101	2	no	—
163	C₂×C₂²⋊C₄	—	101	1	no	—
164	C₂×C₂²⋊C₄	—	101	1	no	—
165	C₂×C₂²⋊C₄	—	101	2	no	—
166	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
167	C₂×C₄²	—	101	2	no	—
168	C₂×C₄²	—	101	1	no	—
169	C₂×C₄²	—	101	2	no	—
170	C₂×C₄²	—	101	2	no	—
171	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
172	#95	—	101	1	no	—
173	#95	—	101	2	no	—
174	#95	—	101	2	no	—
175	#95	—	101	1	no	—
176	#95	—	101	2	no	—
177	#95	C₂×D₁₀₁	1	1	yes	characteristic
178	#95	C₂×D₁₀₁	1	1	yes	characteristic
179	C₂⁴	—	101	1	no	—
180	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
181	C₂×D₄	—	101	4	no	—
182	C₂×D₄	—	101	8	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	101	1	no	—
185	C₂²×C₄	—	101	8	no	—
186	C₂²×C₄	—	101	2	no	—
187	C₂²×C₄	—	101	1	no	—
188	C₂²×C₄	—	101	1	no	—
189	C₂²×C₄	—	101	2	no	—
190	C₂²×C₄	—	101	8	no	—
191	C₂²×C₄	—	101	4	no	—
192	C₂²×C₄	—	404	2	no	—
193	C₂²×C₄	—	404	2	no	—
194	C₂²×C₄	—	404	2	no	—
195	C₂²×C₄	—	404	2	no	—
196	C₂²×C₄	—	404	2	no	—
197	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
202	C₄⋊C₄	—	101	4	no	—
203	C₄⋊C₄	—	101	8	no	—
204	C₂²⋊C₄	—	101	4	no	—
205	C₂²⋊C₄	—	101	4	no	—
206	C₂²⋊C₄	—	101	8	no	—
207	C₂²⋊C₄	C₄×D₁₀₁	1	8	yes	complemented
208	C₄²	—	4	2	no	—
209	C₄²	—	101	4	no	—
210	C₄²	—	101	8	no	—
211	C₄²	—	404	2	no	—
212	C₄²	—	404	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	101	4	no	—
217	C₂³	—	101	2	no	—
218	C₂³	—	202	2	no	—
219	C₂³	—	202	1	no	—
220	C₂³	—	202	1	no	—
221	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
222	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
223	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
224	D₄	—	101	16	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	101	4	no	—
233	C₂×C₄	—	101	2	no	—
234	C₂×C₄	—	101	2	no	—
235	C₂×C₄	—	101	4	no	—
236	C₂×C₄	—	101	4	no	—
237	C₂×C₄	—	101	2	no	—
238	C₂×C₄	—	404	8	no	—
239	C₂×C₄	—	404	4	no	—
240	C₂×C₄	—	404	8	no	—
241	C₂×C₄	—	202	2	no	—
242	C₂×C₄	—	202	2	no	—
243	C₂×C₄	—	202	2	no	—
244	C₂×C₄	—	202	1	no	—
245	C₂×C₄	—	202	2	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	1	no	—
248	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
249	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
251	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	101	4	no	—
257	C₂²	—	202	4	no	—
258	C₂²	—	202	2	no	—
259	C₂²	—	202	4	no	—
260	C₂²	—	202	2	no	—
261	C₂²	C₄²×D₁₀₁	1	4	yes	—
262	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
263	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
264	C₂²	#15692300	1	2	yes	—
265	C₂²	#15692301	1	1	yes	characteristic
266	C₂²	#15692302	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	101	4	no	—
272	C₄	—	202	4	no	—
273	C₄	—	202	4	no	—
274	C₄	—	202	2	no	—
275	C₄	—	202	2	no	—
276	C₂	—	2	4	no	—
277	C₂	—	202	4	no	—
278	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
279	C₂	#20265649	1	2	yes	—
280	C₂	#20265650	1	1	yes	characteristic
281	C₂	#20265656	1	2	yes	—
282	C₂	#20265657	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂×C₁₀₁	C₂²×C₄²
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2275
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₄×#13108053	C₂	semidirect
4. #20265028	C₂	semidirect
5. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
6. C₄×#13108051	C₂	semidirect
7. #20265024	C₂	semidirect
8. #20265022	C₂	semidirect
9. C₂×#15692162	C₂	semidirect
10. #95×D₁₀₁	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
16. C₄×C₂²⋊D₁₀₁	C₄	semidirect
17. #15692215	C₄	semidirect
42. C₂²×Dic₁₀₁	C₂×C₄	semidirect
51. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
52. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
53. #13108053	C₂×C₄	semidirect
54. #13108053	C₂×C₄	semidirect
55. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
56. #13108051	C₂×C₄	semidirect
57. #13108051	C₂×C₄	semidirect
60. C₄×Dic₁₀₁	C₂×C₄	semidirect
61. C₄×Dic₁₀₁	C₂×C₄	semidirect
64. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
69. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
93. C₂²⋊D₁₀₁	C₄²	semidirect
142. C₁₀₁	#2275	semidirect
150. C₄×C₂²⋊C₄	D₁₀₁	semidirect
197. C₂²×C₄	C₄×D₁₀₁	semidirect
207. C₂²⋊C₄	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×C₄×C₂²⋊D₁₀₁	2
C₂×#15692215	2
C₄×#13108053	2
#20265028	2
C₂²⋊C₄×Dic₁₀₁	2
C₄×#13108051	2
#20265024	2
#20265022	2
C₂×#15692162	2
#95×D₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
#2275	101