Subgroups — #27173837

Order 12928.

696 conjugacy classes of subgroups.

The subgroup lattice has 300 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	2	yes	maximal, complemented
2	#20265070	C₂	1	2	yes	maximal, complemented
3	#20265063	C₂	1	2	yes	maximal, complemented
4	#20265057	C₂	1	2	yes	maximal, complemented
5	#20265057	C₂	1	2	yes	maximal, complemented
6	#20265052	C₂	1	2	yes	maximal, complemented
7	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#121×C₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	D₄×Dic₁₀₁	—	2	8	no	—
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	#15692162	C₂²	1	2	yes	—
23	#15692162	C₂²	1	2	yes	—
24	#15692162	C₂²	1	2	yes	—
25	#15692162	C₂²	1	2	yes	—
26	#15692162	C₂²	1	2	yes	—
27	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	#121×C₁₀₁	C₄	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
33	C₂²×Dic₁₀₁	—	2	2	no	—
34	C₂²×Dic₁₀₁	—	2	4	no	—
35	C₂²×Dic₁₀₁	—	4	1	no	—
36	C₂²×Dic₁₀₁	—	4	1	no	—
37	C₂²×Dic₁₀₁	—	4	2	no	—
38	C₂²×Dic₁₀₁	—	4	2	no	—
39	C₂²×Dic₁₀₁	—	4	1	no	—
40	C₂²×Dic₁₀₁	—	4	1	no	—
41	C₂²×Dic₁₀₁	D₄	1	4	yes	complemented
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₄⋊Dic₁₀₁	—	2	4	no	—
47	C₄⋊Dic₁₀₁	—	4	1	no	—
48	C₄⋊Dic₁₀₁	—	4	1	no	—
49	C₂²⋊Dic₁₀₁	—	2	4	no	—
50	C₂²⋊Dic₁₀₁	—	2	4	no	—
51	C₂²⋊Dic₁₀₁	—	4	2	no	—
52	C₂²⋊Dic₁₀₁	—	4	2	no	—
53	C₄×Dic₁₀₁	—	2	4	no	—
54	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
55	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₀₁	—	2	4	no	—
57	C₂×D₄×C₁₀₁	—	2	4	no	—
58	C₂×D₄×C₁₀₁	—	4	1	no	—
59	C₂×D₄×C₁₀₁	—	4	1	no	—
60	C₂×D₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
61	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
62	C₂²×C₄×C₁₀₁	—	4	1	no	—
63	C₂²×C₄×C₁₀₁	—	4	1	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	1	yes	characteristic
68	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
69	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
70	C₂³×C₁₀₁	—	2	1	no	—
71	C₂³×C₁₀₁	—	2	1	no	—
72	C₂³×C₁₀₁	—	2	1	no	—
73	C₂³×C₁₀₁	—	2	1	no	—
74	C₂³×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	2	no	—
76	C₂³×C₁₀₁	—	2	2	no	—
77	C₂³×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	—	2	2	no	—
80	C₂³×C₁₀₁	C₂²×C₄	1	4	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	16	no	—
86	D₄×C₁₀₁	—	4	4	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	1	no	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	1	no	—
93	C₂×C₄×C₁₀₁	—	4	4	no	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	2	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	8	no	—
110	C₂×Dic₁₀₁	—	4	4	no	—
111	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	2	no	—
117	C₂²×C₁₀₁	—	2	2	no	—
118	C₂²×C₁₀₁	—	2	4	no	—
119	C₂²×C₁₀₁	—	2	4	no	—
120	C₂²×C₁₀₁	—	2	4	no	—
121	C₂²×C₁₀₁	—	4	2	no	—
122	C₂²×C₁₀₁	—	4	2	no	—
123	C₂²×C₁₀₁	—	4	2	no	—
124	C₂²×C₁₀₁	—	4	2	no	—
125	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
126	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
133	C₄×C₁₀₁	—	2	4	no	—
134	C₄×C₁₀₁	—	2	2	no	—
135	C₄×C₁₀₁	—	2	2	no	—
136	Dic₁₀₁	—	2	4	no	—
137	Dic₁₀₁	—	4	2	no	—
138	Dic₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	4	2	no	—
140	C₂×C₁₀₁	—	2	4	no	—
141	C₂×C₁₀₁	—	4	4	no	—
142	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×C₁₀₁	#518	1	1	yes	characteristic
144	C₂×C₁₀₁	#520	1	1	yes	characteristic
145	C₂×C₁₀₁	#534	1	1	yes	characteristic
146	C₂×C₁₀₁	#537	1	1	yes	characteristic
147	C₂×C₁₀₁	#538	1	1	yes	characteristic
148	C₂×C₁₀₁	#539	1	1	yes	characteristic
149	#2325	—	101	1	no	maximal, Sylow
150	C₁₀₁	#2325	1	1	yes	characteristic, complemented, Sylow
151	C₂×#121	D₁₀₁	1	1	yes	characteristic
152	C₂×C₄×D₄	—	101	2	no	—
153	#388	—	101	2	no	—
154	#387	—	101	2	no	—
155	#386	—	101	2	no	—
156	#386	—	101	2	no	—
157	#385	—	101	2	no	—
158	#380	—	101	1	no	—
159	C₂×#95	—	101	1	no	—
160	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
161	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
162	C₂³×C₄	—	101	1	no	—
163	C₂³×C₄	—	101	2	no	—
164	C₂³×C₄	—	101	1	no	—
165	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
166	#121	Dic₁₀₁	1	8	yes	complemented
167	C₄×D₄	—	202	8	no	—
168	C₂×C₄⋊C₄	—	101	1	no	—
169	C₂×C₄⋊C₄	—	101	1	no	—
170	C₂×C₄⋊C₄	—	101	2	no	—
171	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	101	2	no	—
173	C₂×C₂²⋊C₄	—	101	2	no	—
174	C₂×C₂²⋊C₄	—	101	2	no	—
175	C₂×C₂²⋊C₄	—	101	2	no	—
176	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
177	C₂×C₄²	—	101	2	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	2	yes	—
184	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	C₂×Dic₁₀₁	1	8	yes	complemented
190	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	complemented
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	101	4	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	101	2	no	—
197	C₂²×C₄	—	101	2	no	—
198	C₂²×C₄	—	404	1	no	—
199	C₂²×C₄	—	404	1	no	—
200	C₂²×C₄	—	404	2	no	—
201	C₂²×C₄	—	404	2	no	—
202	C₂²×C₄	—	404	1	no	—
203	C₂²×C₄	—	404	1	no	—
204	C₂²×C₄	—	202	4	no	—
205	C₂²×C₄	—	202	2	no	—
206	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	complemented
207	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
210	C₄⋊C₄	—	404	1	no	—
211	C₄⋊C₄	—	404	1	no	—
212	C₄⋊C₄	—	202	4	no	—
213	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	complemented
214	C₂²⋊C₄	—	404	2	no	—
215	C₂²⋊C₄	—	404	2	no	—
216	C₂²⋊C₄	—	202	4	no	—
217	C₂²⋊C₄	—	202	4	no	—
218	C₂²⋊C₄	C₂×Dic₁₀₁	1	8	yes	complemented
219	C₄²	—	202	4	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
231	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
232	C₂³	D₄×D₁₀₁	1	2	yes	—
233	C₂³	#13108073	1	2	yes	—
234	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
235	D₄	—	2	16	no	—
236	D₄	—	4	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	101	4	no	—
245	C₂×C₄	—	404	4	no	—
246	C₂×C₄	—	404	8	no	—
247	C₂×C₄	—	404	4	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	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
260	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
261	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	D₄×Dic₁₀₁	1	4	yes	—
276	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
277	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
278	C₂²	C₂×#13108073	1	1	yes	characteristic
279	C₂²	#15692300	1	1	yes	characteristic
280	C₂²	#15692300	1	1	yes	characteristic
281	C₂²	#15692301	1	1	yes	characteristic
282	C₂²	#15692302	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	404	2	no	—
287	C₄	—	404	2	no	—
288	C₄	—	404	2	no	—
289	C₄	—	202	4	no	—
290	C₂	—	2	4	no	—
291	C₂	—	4	4	no	—
292	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
293	C₂	#20265652	1	1	yes	characteristic
294	C₂	#20265660	1	1	yes	characteristic
295	C₂	#20265735	1	1	yes	characteristic
296	C₂	#20265754	1	1	yes	characteristic
297	C₂	#20265764	1	1	yes	characteristic
298	C₂	#20265772	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	C₂×#121×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#121	D₁₀₁
hypocenter	C₁₀₁	#2325
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₁₀₁	C₂	semidirect
2. #20265070	C₂	semidirect
3. #20265063	C₂	semidirect
4. #20265057	C₂	semidirect
5. #20265057	C₂	semidirect
6. #20265052	C₂	semidirect
7. #20265037	C₂	semidirect
8. C₂×#15692162	C₂	semidirect
30. #121×C₁₀₁	C₄	semidirect
41. C₂²×Dic₁₀₁	D₄	semidirect
60. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
61. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
64. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
68. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
69. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
150. C₁₀₁	#2325	semidirect
166. #121	Dic₁₀₁	semidirect
189. C₂×D₄	C₂×Dic₁₀₁	semidirect
190. C₂×D₄	C₂×Dic₁₀₁	semidirect
206. C₂²×C₄	C₂×Dic₁₀₁	semidirect
213. C₄⋊C₄	C₂×Dic₁₀₁	semidirect
218. 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
#20265070	2
#20265063	2
#20265057	2
#20265052	2
#20265037	2
C₂×#15692162	2
C₂×#121×C₁₀₁	2
#2325	101