Subgroups — #27173835

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×#15692229	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	2	yes	maximal, complemented
3	#20265069	C₂	1	2	yes	maximal, complemented
4	#20265062	C₂	1	2	yes	maximal, complemented
5	#20265056	C₂	1	2	yes	maximal, complemented
6	#20265055	C₂	1	2	yes	maximal, complemented
7	#20265051	C₂	1	2	yes	maximal, complemented
8	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
12	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
13	#15692229	C₄	1	8	yes	complemented
14	C₄×C₂²⋊D₁₀₁	—	2	8	no	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×#13108053	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×#13108051	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×#13108051	C₂²	1	2	yes	—
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	#15692162	C₂²	1	2	yes	—
26	#15692162	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	#95×C₁₀₁	C₂²	1	2	yes	—
32	#95×C₁₀₁	C₂²	1	2	yes	—
33	C₂³×D₁₀₁	C₂³	1	2	yes	—
34	C₂×D₄₀₄	—	4	1	no	—
35	C₂×D₄₀₄	—	4	1	no	—
36	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
37	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
38	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
39	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂×C₄×D₁₀₁	—	2	4	no	—
44	C₂×C₄×D₁₀₁	—	4	2	no	—
45	C₂×C₄×D₁₀₁	—	4	2	no	—
46	C₄⋊Dic₁₀₁	—	4	1	no	—
47	C₄⋊Dic₁₀₁	—	4	1	no	—
48	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
49	#13108053	—	2	4	no	—
50	C₂²⋊Dic₁₀₁	—	2	4	no	—
51	#13108051	—	2	4	no	—
52	#13108051	—	4	2	no	—
53	#13108051	—	4	2	no	—
54	#13108051	C₂×C₄	1	8	yes	complemented
55	C₄×Dic₁₀₁	—	2	4	no	—
56	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	—	2	2	no	—
58	C₂²×C₄×C₁₀₁	—	4	1	no	—
59	C₂²×C₄×C₁₀₁	—	4	1	no	—
60	C₂²×C₄×C₁₀₁	—	4	1	no	—
61	C₂²×C₄×C₁₀₁	—	4	1	no	—
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	D₄	1	4	yes	complemented
66	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
76	C₂³×C₁₀₁	#41	1	2	yes	—
77	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	—	2	2	no	—
80	C₂²×D₁₀₁	—	2	2	no	—
81	C₂²×D₁₀₁	—	2	2	no	—
82	C₂²×D₁₀₁	—	2	2	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	C₂²×C₄	1	4	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	1	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	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	4	4	no	—
95	C₂×C₄×C₁₀₁	—	4	4	no	—
96	C₂×C₄×C₁₀₁	—	4	4	no	—
97	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
100	C₂²⋊D₁₀₁	—	2	16	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	—	2	2	no	—
109	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
110	D₄₀₄	—	4	4	no	—
111	C₄×D₁₀₁	—	4	8	no	—
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	C₄×D₄	1	4	yes	—
119	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
120	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
122	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂×D₁₀₁	—	2	4	no	—
127	C₂×D₁₀₁	—	2	4	no	—
128	C₂×D₁₀₁	—	2	4	no	—
129	C₂×D₁₀₁	—	4	2	no	—
130	C₂×D₁₀₁	—	4	2	no	—
131	C₂×D₁₀₁	—	4	2	no	—
132	C₂×D₁₀₁	—	4	2	no	—
133	C₄×C₁₀₁	—	2	2	no	—
134	C₄×C₁₀₁	—	2	2	no	—
135	C₄×C₁₀₁	—	2	4	no	—
136	C₄×C₁₀₁	—	4	2	no	—
137	Dic₁₀₁	—	2	4	no	—
138	Dic₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	4	2	no	—
140	C₂×C₁₀₁	—	2	4	no	—
141	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×C₁₀₁	#518	1	1	yes	characteristic
143	C₂×C₁₀₁	#520	1	1	yes	characteristic
144	C₂×C₁₀₁	#534	1	1	yes	characteristic
145	C₂×C₁₀₁	#537	1	1	yes	characteristic
146	C₂×C₁₀₁	#538	1	1	yes	characteristic
147	C₂×C₁₀₁	#539	1	1	yes	characteristic
148	D₁₀₁	—	4	4	no	—
149	#2325	—	101	1	no	maximal, Sylow
150	C₁₀₁	#2325	1	1	yes	characteristic, complemented, Sylow
151	C₂×#121	—	101	1	no	—
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	D₁₀₁	1	1	yes	characteristic, complemented
160	C₂²×D₄	—	101	2	no	—
161	C₂²×D₄	—	101	1	no	—
162	C₂³×C₄	—	101	2	no	—
163	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
164	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
165	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
166	#121	—	101	8	no	—
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	1	no	—
171	C₂×C₄⋊C₄	—	101	2	no	—
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	—	101	2	no	—
177	C₂×C₄²	—	101	2	no	—
178	#95	—	101	2	no	—
179	#95	—	101	2	no	—
180	#95	—	101	2	no	—
181	#95	C₂×D₁₀₁	1	2	yes	—
182	#95	C₂×D₁₀₁	1	2	yes	—
183	C₂⁴	—	101	2	no	—
184	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
185	C₂×D₄	—	101	8	no	—
186	C₂×D₄	—	101	4	no	—
187	C₂×D₄	—	404	1	no	—
188	C₂×D₄	—	404	1	no	—
189	C₂×D₄	—	202	4	no	—
190	C₂×D₄	—	202	4	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	101	2	no	—
199	C₂²×C₄	—	101	2	no	—
200	C₂²×C₄	—	101	2	no	—
201	C₂²×C₄	—	404	2	no	—
202	C₂²×C₄	—	404	2	no	—
203	C₂²×C₄	—	202	4	no	—
204	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
205	C₂²×C₄	D₄₀₄	1	4	yes	complemented
206	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
210	C₄⋊C₄	—	101	4	no	—
211	C₄⋊C₄	—	404	1	no	—
212	C₄⋊C₄	—	404	1	no	—
213	C₄⋊C₄	—	202	4	no	—
214	C₂²⋊C₄	—	101	8	no	—
215	C₂²⋊C₄	—	404	2	no	—
216	C₂²⋊C₄	—	404	2	no	—
217	C₂²⋊C₄	—	202	4	no	—
218	C₂²⋊C₄	—	202	4	no	—
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	—	101	4	no	—
225	C₂³	—	202	2	no	—
226	C₂³	—	202	2	no	—
227	C₂³	—	202	2	no	—
228	C₂³	—	202	2	no	—
229	C₂³	—	202	2	no	—
230	C₂³	—	202	2	no	—
231	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
232	C₂³	C₂×D₄₀₄	1	2	yes	—
233	C₂³	#13108074	1	2	yes	—
234	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
235	D₄	—	404	4	no	—
236	D₄	—	202	16	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	101	4	no	—
250	C₂×C₄	—	404	8	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	C₂×C₄×D₁₀₁	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
261	C₂×C₄	C₂×D₄₀₄	1	4	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	—	404	2	no	—
269	C₂²	—	404	2	no	—
270	C₂²	—	404	2	no	—
271	C₂²	—	404	2	no	—
272	C₂²	—	202	4	no	—
273	C₂²	—	202	4	no	—
274	C₂²	—	202	4	no	—
275	C₂²	C₄×D₄₀₄	1	4	yes	—
276	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
277	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
278	C₂²	C₂×#13108074	1	1	yes	characteristic
279	C₂²	#15692300	1	1	yes	characteristic
280	C₂²	#15692300	1	1	yes	characteristic
281	C₂²	#15692300	1	1	yes	characteristic
282	C₂²	#15692303	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	404	2	no	—
288	C₄	—	404	2	no	—
289	C₄	—	202	4	no	—
290	C₂	—	2	4	no	—
291	C₂	—	404	4	no	—
292	C₂	C₂×C₄×D₄₀₄	1	1	yes	characteristic
293	C₂	#20265650	1	1	yes	characteristic
294	C₂	#20265658	1	1	yes	characteristic
295	C₂	#20265739	1	1	yes	characteristic
296	C₂	#20265758	1	1	yes	characteristic
297	C₂	#20265765	1	1	yes	characteristic
298	C₂	#20265774	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×#95×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692229	C₂
hypercenter	C₂×#95	D₁₀₁
hypocenter	C₁₀₁	#2325
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
3. #20265069	C₂	semidirect
4. #20265062	C₂	semidirect
5. #20265056	C₂	semidirect
6. #20265055	C₂	semidirect
7. #20265051	C₂	semidirect
8. #20265037	C₂	semidirect
9. C₂×#95×C₁₀₁	C₂	semidirect
13. #15692229	C₄	semidirect
36. C₂×D₄₀₄	C₂×C₄	semidirect
39. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
48. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
54. #13108051	C₂×C₄	semidirect
64. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
65. C₂²×C₄×C₁₀₁	D₄	semidirect
150. C₁₀₁	#2325	semidirect
159. C₂×#95	D₁₀₁	semidirect
204. C₂²×C₄	C₄×D₁₀₁	semidirect
205. C₂²×C₄	D₄₀₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692229	2
C₂×C₄×C₂²⋊D₁₀₁	2
#20265069	2
#20265062	2
#20265056	2
#20265055	2
#20265051	2
#20265037	2
C₂×#95×C₁₀₁	2
#2325	101