Subgroups — #27173804

Order 12928.

632 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×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265076	C₂	1	2	yes	maximal, complemented
4	#20265070	C₂	1	2	yes	maximal, complemented
5	#20265063	C₂	1	2	yes	maximal, complemented
6	#20265057	C₂	1	1	yes	maximal, characteristic
7	#20265052	C₂	1	2	yes	maximal, complemented
8	#20265052	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265040	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#123×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	D₄×Dic₁₀₁	—	2	4	no	—
16	C₂×C₄⋊Dic₁₀₁	—	2	4	no	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊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	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	#123×C₁₀₁	C₄	1	8	yes	complemented
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
38	C₂²×Dic₁₀₁	—	2	2	no	—
39	C₂²×Dic₁₀₁	—	2	2	no	—
40	C₂²×Dic₁₀₁	—	4	2	no	—
41	C₂²×Dic₁₀₁	—	4	2	no	—
42	C₂²×Dic₁₀₁	—	4	2	no	—
43	C₂²×Dic₁₀₁	D₄	1	4	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	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
50	C₄⋊Dic₁₀₁	—	2	4	no	—
51	C₄⋊Dic₁₀₁	—	2	4	no	—
52	C₄⋊Dic₁₀₁	—	2	2	no	—
53	C₄⋊Dic₁₀₁	—	4	2	no	—
54	C₂²⋊Dic₁₀₁	—	2	2	no	—
55	C₂²⋊Dic₁₀₁	—	2	2	no	—
56	C₂²⋊Dic₁₀₁	—	2	2	no	—
57	C₂²⋊Dic₁₀₁	—	4	2	no	—
58	C₄×Dic₁₀₁	—	2	2	no	—
59	C₄×Dic₁₀₁	—	4	2	no	—
60	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₀₁	—	2	2	no	—
63	C₂×D₄×C₁₀₁	—	2	2	no	—
64	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
65	C₂²×C₄×C₁₀₁	—	2	2	no	—
66	C₂²×C₄×C₁₀₁	—	2	2	no	—
67	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
72	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
73	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
74	C₂³×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	2	no	—
76	C₂³×C₁₀₁	—	2	1	no	—
77	C₂³×C₁₀₁	—	2	1	no	—
78	C₂³×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	—	2	2	no	—
80	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
83	C₂³×C₁₀₁	#41	1	2	yes	—
84	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
85	D₄×C₁₀₁	—	2	8	no	—
86	C₂×C₄×C₁₀₁	—	2	8	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	1	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	2	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	1	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	2	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	—	2	2	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	4	4	no	—
109	C₂×Dic₁₀₁	—	4	4	no	—
110	C₂×Dic₁₀₁	—	4	4	no	—
111	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
113	C₂²×C₁₀₁	—	2	4	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	4	no	—
117	C₂²×C₁₀₁	—	2	2	no	—
118	C₂²×C₁₀₁	—	2	4	no	—
119	C₂²×C₁₀₁	—	4	1	no	—
120	C₂²×C₁₀₁	—	4	1	no	—
121	C₂²×C₁₀₁	—	4	2	no	—
122	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
123	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
127	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
128	C₄×C₁₀₁	—	2	4	no	—
129	C₄×C₁₀₁	—	2	2	no	—
130	C₄×C₁₀₁	—	2	4	no	—
131	Dic₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	2	2	no	—
133	Dic₁₀₁	—	4	1	no	—
134	Dic₁₀₁	—	4	2	no	—
135	Dic₁₀₁	—	4	1	no	—
136	Dic₁₀₁	—	4	2	no	—
137	C₂×C₁₀₁	—	2	4	no	—
138	C₂×C₁₀₁	—	4	2	no	—
139	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₀₁	#520	1	2	yes	—
141	C₂×C₁₀₁	#534	1	1	yes	characteristic
142	C₂×C₁₀₁	#536	1	1	yes	characteristic
143	C₂×C₁₀₁	#539	1	2	yes	—
144	#2322	—	101	1	no	maximal, Sylow
145	C₁₀₁	#2322	1	1	yes	characteristic, complemented, Sylow
146	C₂×#123	D₁₀₁	1	1	yes	characteristic
147	C₂×C₄×D₄	—	101	1	no	—
148	C₂²×C₄⋊C₄	—	101	1	no	—
149	#389	—	101	2	no	—
150	#388	—	101	2	no	—
151	#387	—	101	2	no	—
152	#386	—	101	1	no	—
153	#385	—	101	2	no	—
154	#385	—	101	1	no	—
155	#381	—	101	1	no	—
156	C₄×C₂²⋊C₄	—	101	1	no	—
157	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
158	C₂³×C₄	—	101	1	no	—
159	C₂³×C₄	—	101	1	no	—
160	C₂³×C₄	—	101	1	no	—
161	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
162	#123	Dic₁₀₁	1	8	yes	complemented
163	C₄×D₄	—	202	4	no	—
164	C₂×C₄⋊C₄	—	101	2	no	—
165	C₂×C₄⋊C₄	—	101	2	no	—
166	C₂×C₄⋊C₄	—	101	2	no	—
167	C₂×C₄⋊C₄	—	101	1	no	—
168	C₂×C₄⋊C₄	—	202	4	no	—
169	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
170	C₂×C₂²⋊C₄	—	101	2	no	—
171	C₂×C₂²⋊C₄	—	101	1	no	—
172	C₂×C₂²⋊C₄	—	101	1	no	—
173	C₂×C₂²⋊C₄	—	101	1	no	—
174	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
176	C₂×C₄²	—	101	2	no	—
177	C₂×C₄²	—	101	1	no	—
178	#95	—	101	2	no	—
179	#95	—	101	2	no	—
180	#95	—	101	2	no	—
181	#95	—	101	2	no	—
182	#95	—	101	2	no	—
183	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	101	4	no	—
191	C₂²×C₄	—	101	1	no	—
192	C₂²×C₄	—	101	1	no	—
193	C₂²×C₄	—	101	1	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	1	no	—
196	C₂²×C₄	—	101	2	no	—
197	C₂²×C₄	—	404	2	no	—
198	C₂²×C₄	—	404	2	no	—
199	C₂²×C₄	—	404	2	no	—
200	C₂²×C₄	—	202	2	no	—
201	C₂²×C₄	—	202	2	no	—
202	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	complemented
203	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
204	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
206	C₄⋊C₄	—	404	2	no	—
207	C₄⋊C₄	—	202	4	no	—
208	C₄⋊C₄	—	202	4	no	—
209	C₄⋊C₄	—	202	2	no	—
210	C₄⋊C₄	C₂×Dic₁₀₁	1	8	yes	complemented
211	C₂²⋊C₄	—	404	2	no	—
212	C₂²⋊C₄	—	202	2	no	—
213	C₂²⋊C₄	—	202	2	no	—
214	C₂²⋊C₄	—	202	2	no	—
215	C₂²⋊C₄	C₂×Dic₁₀₁	1	4	yes	complemented
216	C₂²⋊C₄	C₂×Dic₁₀₁	1	8	yes	complemented
217	C₄²	—	404	2	no	—
218	C₄²	—	202	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
226	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
227	C₂³	D₄×D₁₀₁	1	2	yes	—
228	C₂³	#13108073	1	2	yes	—
229	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
230	D₄	—	2	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	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	101	2	no	—
239	C₂×C₄	—	101	2	no	—
240	C₂×C₄	—	404	4	no	—
241	C₂×C₄	—	404	4	no	—
242	C₂×C₄	—	404	4	no	—
243	C₂×C₄	—	202	1	no	—
244	C₂×C₄	—	202	1	no	—
245	C₂×C₄	—	202	2	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	—	202	2	no	—
252	C₂×C₄	—	202	2	no	—
253	C₂×C₄	—	202	2	no	—
254	C₂×C₄	—	202	2	no	—
255	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
256	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
257	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	4	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	D₄×Dic₁₀₁	1	4	yes	—
268	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
269	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
270	C₂²	C₂×#13108073	1	1	yes	characteristic
271	C₂²	#15692301	1	2	yes	—
272	C₂²	#15692302	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	404	2	no	—
277	C₄	—	404	1	no	—
278	C₄	—	404	1	no	—
279	C₄	—	404	2	no	—
280	C₄	—	202	2	no	—
281	C₄	—	202	2	no	—
282	C₂	—	2	4	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
285	C₂	#20265660	1	2	yes	—
286	C₂	#20265738	1	1	yes	characteristic
287	C₂	#20265749	1	1	yes	characteristic
288	C₂	#20265771	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×#123×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄×C₁₀₁	C₂²
hypercenter	C₂×#123	D₁₀₁
hypocenter	C₁₀₁	#2322
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
3. #20265076	C₂	semidirect
4. #20265070	C₂	semidirect
5. #20265063	C₂	semidirect
7. #20265052	C₂	semidirect
8. #20265052	C₂	semidirect
9. #20265040	C₂	semidirect
10. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
34. #123×C₁₀₁	C₄	semidirect
67. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
71. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
72. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
73. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
145. C₁₀₁	#2322	semidirect
162. #123	Dic₁₀₁	semidirect
202. C₂²×C₄	C₂×Dic₁₀₁	semidirect
210. C₄⋊C₄	C₂×Dic₁₀₁	semidirect
215. C₂²⋊C₄	C₂×Dic₁₀₁	semidirect
216. C₂²⋊C₄	C₂×Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₀₁	2
C₂²×C₄⋊Dic₁₀₁	2
#20265076	2
#20265070	2
#20265063	2
#20265057	2
#20265052	2
#20265040	2
C₂²⋊C₄×Dic₁₀₁	2
C₂×#123×C₁₀₁	2
#2322	101