Subgroups — #27173513

Order 12928.

1088 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692212	C₂	1	2	yes	maximal, complemented
3	#20265039	C₂	1	2	yes	maximal, complemented
4	#20265036	C₂	1	2	yes	maximal, complemented
5	#20265035	C₂	1	2	yes	maximal, complemented
6	#20265032	C₂	1	2	yes	maximal
7	C₄×#13108051	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	—	2	8	no	—
13	C₂²×C₄×D₁₀₁	C₄	1	8	yes	complemented
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	#15692212	—	2	8	no	—
19	C₂×#13108053	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	4	yes	—
22	C₂×#13108051	C₂²	1	4	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
24	#15692162	C₂²	1	2	yes	complemented
25	#15692162	C₂²	1	4	yes	complemented
26	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	complemented
29	#95×C₁₀₁	C₂²	1	2	yes	complemented
30	C₂³×D₁₀₁	—	2	4	no	—
31	C₂³×D₁₀₁	—	2	4	no	—
32	C₂³×D₁₀₁	—	2	4	no	—
33	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
34	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
35	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₀₁	—	2	4	no	—
37	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
38	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₁₀₁	—	2	4	no	—
42	C₂×C₄×D₁₀₁	—	2	4	no	—
43	C₂×C₄×D₁₀₁	—	2	4	no	—
44	C₂×C₄×D₁₀₁	—	2	32	no	—
45	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
46	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
50	#13108053	—	2	8	no	—
51	C₂²⋊Dic₁₀₁	—	2	4	no	—
52	#13108051	—	2	8	no	—
53	#13108051	—	2	8	no	—
54	C₄×Dic₁₀₁	—	2	4	no	—
55	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₁	—	2	4	no	—
57	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
62	C₄²×C₁₀₁	—	2	4	no	—
63	C₂³×C₁₀₁	—	2	2	no	—
64	C₂³×C₁₀₁	—	2	2	no	—
65	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₀₁	#41	1	4	yes	—
67	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
68	C₂²×D₁₀₁	—	2	32	no	—
69	C₂²×D₁₀₁	—	2	8	no	—
70	C₂²×D₁₀₁	—	2	2	no	—
71	C₂²×D₁₀₁	—	2	2	no	—
72	C₂²×D₁₀₁	—	2	2	no	—
73	C₂²×D₁₀₁	—	2	2	no	—
74	C₂²×D₁₀₁	—	2	8	no	—
75	C₂²×D₁₀₁	—	2	8	no	—
76	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
79	C₂²×D₁₀₁	#41	1	4	yes	—
80	C₂×C₄×C₁₀₁	—	2	2	no	—
81	C₂×C₄×C₁₀₁	—	2	8	no	—
82	C₂×C₄×C₁₀₁	—	2	4	no	—
83	C₂×C₄×C₁₀₁	—	2	4	no	—
84	C₂×C₄×C₁₀₁	—	2	4	no	—
85	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	—	2	8	no	—
90	C₂×Dic₁₀₁	—	2	4	no	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	—	2	4	no	—
93	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
96	C₄×D₁₀₁	—	2	8	no	—
97	C₄×D₁₀₁	—	2	8	no	—
98	C₄×D₁₀₁	—	2	8	no	—
99	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	complemented
100	C₂²×C₁₀₁	—	2	2	no	—
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	8	no	—
103	C₂²×C₁₀₁	#117	1	4	yes	complemented
104	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
106	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
107	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
108	C₂×D₁₀₁	—	2	32	no	—
109	C₂×D₁₀₁	—	2	2	no	—
110	C₂×D₁₀₁	—	2	2	no	—
111	C₂×D₁₀₁	—	2	2	no	—
112	C₂×D₁₀₁	—	2	2	no	—
113	C₂×D₁₀₁	—	2	8	no	—
114	C₂×D₁₀₁	—	2	8	no	—
115	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×D₁₀₁	#117	1	4	yes	complemented
117	C₄×C₁₀₁	—	2	2	no	—
118	C₄×C₁₀₁	—	4	4	no	—
119	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
120	Dic₁₀₁	—	2	2	no	—
121	Dic₁₀₁	—	4	4	no	—
122	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₁₀₁	—	2	4	no	—
124	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
126	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
127	C₂×C₁₀₁	#525	1	4	yes	—
128	D₁₀₁	—	2	4	no	—
129	D₁₀₁	—	2	4	no	—
130	#2280	—	101	1	no	maximal, Sylow
131	C₁₀₁	#2280	1	1	yes	characteristic, complemented, Sylow
132	C₂⁴×C₄	—	101	1	no	—
133	C₂×#117	—	101	2	no	—
134	#381	—	101	2	no	—
135	#380	—	101	2	no	—
136	#380	—	101	2	no	—
137	#379	—	101	2	no	—
138	C₄×C₂²⋊C₄	—	101	1	no	—
139	C₄×C₂²⋊C₄	—	101	2	no	—
140	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
141	C₂⁵	—	101	1	no	—
142	C₂³×C₄	—	101	1	no	—
143	C₂³×C₄	—	101	1	no	—
144	C₂³×C₄	—	101	8	no	—
145	C₂³×C₄	—	101	1	no	—
146	C₂³×C₄	—	101	1	no	—
147	C₂³×C₄	—	101	1	no	—
148	C₂³×C₄	—	202	8	no	—
149	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
150	#117	—	202	8	no	—
151	C₂×C₄⋊C₄	—	101	4	no	—
152	C₂×C₂²⋊C₄	—	101	2	no	—
153	C₂×C₂²⋊C₄	—	101	4	no	—
154	C₂×C₂²⋊C₄	—	101	4	no	—
155	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
156	C₂×C₄²	—	101	2	no	—
157	C₂×C₄²	C₂×D₁₀₁	1	2	yes	complemented
158	#95	—	101	4	no	—
159	#95	—	101	2	no	—
160	#95	C₂×D₁₀₁	1	2	yes	complemented
161	C₂⁴	—	101	1	no	—
162	C₂⁴	—	101	1	no	—
163	C₂⁴	—	101	4	no	—
164	C₂⁴	—	202	4	no	—
165	C₂⁴	—	202	4	no	—
166	C₂⁴	—	202	4	no	—
167	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	101	1	no	—
170	C₂²×C₄	—	101	4	no	—
171	C₂²×C₄	—	101	8	no	—
172	C₂²×C₄	—	101	4	no	—
173	C₂²×C₄	—	101	1	no	—
174	C₂²×C₄	—	101	4	no	—
175	C₂²×C₄	—	101	4	no	—
176	C₂²×C₄	—	101	4	no	—
177	C₂²×C₄	—	101	4	no	—
178	C₂²×C₄	—	202	4	no	—
179	C₂²×C₄	—	202	32	no	—
180	C₂²×C₄	—	202	4	no	—
181	C₂²×C₄	—	202	4	no	—
182	C₂²×C₄	—	202	4	no	—
183	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
184	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
185	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₄⋊C₄	—	202	8	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	202	8	no	—
190	C₂²⋊C₄	—	202	8	no	—
191	C₂²⋊C₄	—	202	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	202	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	101	2	no	—
197	C₂³	—	101	2	no	—
198	C₂³	—	101	4	no	—
199	C₂³	—	101	4	no	—
200	C₂³	—	202	8	no	—
201	C₂³	—	202	8	no	—
202	C₂³	—	202	32	no	—
203	C₂³	—	202	2	no	—
204	C₂³	—	202	2	no	—
205	C₂³	—	202	2	no	—
206	C₂³	—	202	2	no	—
207	C₂³	—	202	8	no	—
208	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
209	C₂³	#13108074	1	4	yes	—
210	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	101	2	no	—
217	C₂×C₄	—	101	4	no	—
218	C₂×C₄	—	101	16	no	—
219	C₂×C₄	—	101	2	no	—
220	C₂×C₄	—	202	8	no	—
221	C₂×C₄	—	202	8	no	—
222	C₂×C₄	—	202	8	no	—
223	C₂×C₄	—	202	4	no	—
224	C₂×C₄	—	202	4	no	—
225	C₂×C₄	—	202	4	no	—
226	C₂×C₄	—	202	2	no	—
227	C₂×C₄	—	202	8	no	—
228	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
230	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	101	4	no	—
235	C₂²	—	101	4	no	—
236	C₂²	—	202	32	no	—
237	C₂²	—	202	2	no	—
238	C₂²	—	202	2	no	—
239	C₂²	—	202	2	no	—
240	C₂²	—	202	2	no	—
241	C₂²	—	202	8	no	—
242	C₂²	—	202	8	no	—
243	C₂²	#15692212	1	4	yes	complemented
244	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
245	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
246	C₂²	#41×D₁₀₁	1	2	yes	—
247	C₂²	C₂×#13108074	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	4	4	no	—
250	C₄	—	101	4	no	—
251	C₄	—	404	4	no	—
252	C₄	—	202	2	no	—
253	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
254	C₂	—	2	4	no	—
255	C₂	—	202	4	no	—
256	C₂	—	202	4	no	—
257	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
258	C₂	#117×D₁₀₁	1	1	yes	characteristic
259	C₂	C₂×#15692212	1	1	yes	characteristic
260	C₂	#20265686	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2280
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#15692212	C₂	semidirect
3. #20265039	C₂	semidirect
4. #20265036	C₂	semidirect
5. #20265035	C₂	semidirect
7. C₄×#13108051	C₂	semidirect
8. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
9. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
13. C₂²×C₄×D₁₀₁	C₄	semidirect
19. C₂×#13108053	C₂²	semidirect
23. C₂×C₄×Dic₁₀₁	C₂²	semidirect
24. #15692162	C₂²	semidirect
25. #15692162	C₂²	semidirect
28. C₂×C₄²×C₁₀₁	C₂²	semidirect
29. #95×C₁₀₁	C₂²	semidirect
99. C₄×D₁₀₁	C₂²⋊C₄	semidirect
103. C₂²×C₁₀₁	#117	semidirect
116. C₂×D₁₀₁	#117	semidirect
131. C₁₀₁	#2280	semidirect
140. C₄×C₂²⋊C₄	D₁₀₁	semidirect
157. C₂×C₄²	C₂×D₁₀₁	semidirect
160. #95	C₂×D₁₀₁	semidirect
243. C₂²	#15692212	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂×#15692212	2
#20265039	2
#20265036	2
#20265035	2
#20265032	2
C₄×#13108051	2
C₄×C₂²⋊Dic₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
#2280	101