Subgroups — #2506×C₂₄₁ #44504173

Order 30848.

436 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₂₄₁	C₂	1	2	yes	maximal
2	#400×C₂₄₁	C₂	1	2	yes	maximal
3	#398×C₂₄₁	C₂	1	2	yes	maximal
4	#389×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	#389×C₂₄₁	C₂	1	1	yes	maximal, characteristic
6	#389×C₂₄₁	C₂	1	1	yes	maximal, characteristic
7	#389×C₂₄₁	C₂	1	1	yes	maximal, characteristic
8	#387×C₂₄₁	C₂	1	2	yes	maximal
9	#384×C₂₄₁	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
12	#125×C₂₄₁	—	2	8	no	—
13	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₄₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
29	#95×C₂₄₁	C₂²	1	1	yes	characteristic
30	#95×C₂₄₁	C₂²	1	2	yes	—
31	#95×C₂₄₁	C₂²	1	1	yes	characteristic
32	#95×C₂₄₁	C₂²	1	2	yes	—
33	#95×C₂₄₁	C₂²	1	2	yes	—
34	#95×C₂₄₁	C₂²	1	2	yes	—
35	#95×C₂₄₁	C₂²	1	2	yes	—
36	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₄₁	—	2	4	no	—
48	C₄⋊C₄×C₂₄₁	—	2	4	no	—
49	C₄⋊C₄×C₂₄₁	—	2	4	no	—
50	C₄⋊C₄×C₂₄₁	—	2	4	no	—
51	C₄⋊C₄×C₂₄₁	—	2	4	no	—
52	C₄⋊C₄×C₂₄₁	—	2	2	no	—
53	C₄⋊C₄×C₂₄₁	—	2	4	no	—
54	C₄⋊C₄×C₂₄₁	—	2	2	no	—
55	C₄⋊C₄×C₂₄₁	—	2	2	no	—
56	C₄⋊C₄×C₂₄₁	—	2	2	no	—
57	C₄⋊C₄×C₂₄₁	D₄	1	4	yes	—
58	C₄⋊C₄×C₂₄₁	Q₈	1	4	yes	—
59	C₄²×C₂₄₁	—	2	4	no	—
60	C₄²×C₂₄₁	—	2	2	no	—
61	C₄²×C₂₄₁	—	2	2	no	—
62	C₄²×C₂₄₁	—	2	2	no	—
63	C₂³×C₂₄₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₂₄₁	—	2	2	no	—
65	C₂×C₄×C₂₄₁	—	2	1	no	—
66	C₂×C₄×C₂₄₁	—	2	1	no	—
67	C₂×C₄×C₂₄₁	—	2	1	no	—
68	C₂×C₄×C₂₄₁	—	2	1	no	—
69	C₂×C₄×C₂₄₁	—	2	1	no	—
70	C₂×C₄×C₂₄₁	—	2	1	no	—
71	C₂×C₄×C₂₄₁	—	2	1	no	—
72	C₂×C₄×C₂₄₁	—	2	2	no	—
73	C₂×C₄×C₂₄₁	—	2	1	no	—
74	C₂×C₄×C₂₄₁	—	2	2	no	—
75	C₂×C₄×C₂₄₁	—	2	2	no	—
76	C₂×C₄×C₂₄₁	—	2	2	no	—
77	C₂×C₄×C₂₄₁	—	2	1	no	—
78	C₂×C₄×C₂₄₁	—	2	1	no	—
79	C₂×C₄×C₂₄₁	—	2	2	no	—
80	C₂×C₄×C₂₄₁	—	2	2	no	—
81	C₂×C₄×C₂₄₁	—	2	2	no	—
82	C₂×C₄×C₂₄₁	—	2	2	no	—
83	C₂×C₄×C₂₄₁	—	2	2	no	—
84	C₂×C₄×C₂₄₁	—	2	1	no	—
85	C₂×C₄×C₂₄₁	—	2	1	no	—
86	C₂×C₄×C₂₄₁	—	2	1	no	—
87	C₂×C₄×C₂₄₁	—	2	2	no	—
88	C₂×C₄×C₂₄₁	—	2	1	no	—
89	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₄₁	C₂×Q₈	1	4	yes	—
93	C₂×C₄×C₂₄₁	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₂₄₁	#41	1	2	yes	—
95	C₂×C₄×C₂₄₁	#41	1	2	yes	—
96	C₂×C₄×C₂₄₁	#41	1	2	yes	—
97	C₂²×C₂₄₁	C₂²×D₄	1	1	yes	characteristic
98	C₂²×C₂₄₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂²×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₂₄₁	—	2	2	no	—
105	C₄×C₂₄₁	—	2	2	no	—
106	C₄×C₂₄₁	—	2	2	no	—
107	C₄×C₂₄₁	—	2	4	no	—
108	C₄×C₂₄₁	—	2	2	no	—
109	C₄×C₂₄₁	—	4	2	no	—
110	C₄×C₂₄₁	—	4	2	no	—
111	C₄×C₂₄₁	—	4	1	no	—
112	C₄×C₂₄₁	—	4	2	no	—
113	C₄×C₂₄₁	—	4	1	no	—
114	C₄×C₂₄₁	#125	1	4	yes	complemented
115	C₂×C₂₄₁	C₂×#125	1	1	yes	characteristic
116	C₂×C₂₄₁	#532	1	1	yes	characteristic
117	C₂×C₂₄₁	#541	1	1	yes	characteristic
118	C₂×C₂₄₁	#547	1	1	yes	characteristic
119	C₂×C₂₄₁	D₄×Q₈	1	1	yes	characteristic
120	C₂×C₂₄₁	#552	1	1	yes	characteristic
121	C₂×C₂₄₁	#557	1	1	yes	characteristic
122	C₂₄₁	#2506	1	1	yes	characteristic, direct product factor, Sylow
123	#2506	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×#125	C₂×C₂₄₁	1	2	yes	—
125	#400	C₂×C₂₄₁	1	2	yes	—
126	#398	C₂×C₂₄₁	1	2	yes	—
127	#389	C₂×C₂₄₁	1	1	yes	characteristic
128	#389	C₂×C₂₄₁	1	1	yes	characteristic
129	#389	C₂×C₂₄₁	1	1	yes	characteristic
130	#389	C₂×C₂₄₁	1	1	yes	characteristic
131	#387	C₂×C₂₄₁	1	2	yes	—
132	#384	C₂×C₂₄₁	1	1	yes	characteristic
133	C₄×C₄⋊C₄	C₂×C₂₄₁	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂×C₂₄₁	1	1	yes	characteristic
135	#125	—	2	8	no	—
136	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
144	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₂₄₁	1	2	yes	—
149	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
150	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
151	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
152	#95	C₂²×C₂₄₁	1	1	yes	characteristic
153	#95	C₂²×C₂₄₁	1	2	yes	—
154	#95	C₂²×C₂₄₁	1	1	yes	characteristic
155	#95	C₂²×C₂₄₁	1	2	yes	—
156	#95	C₂²×C₂₄₁	1	2	yes	—
157	#95	C₂²×C₂₄₁	1	2	yes	—
158	#95	C₂²×C₂₄₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
163	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	D₄×C₂₄₁	1	4	yes	—
181	C₄⋊C₄	Q₈×C₂₄₁	1	4	yes	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₂³	C₂⁴×C₂₄₁	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
215	C₂×C₄	C₂×Q₈×C₂₄₁	1	4	yes	—
216	C₂×C₄	C₂×Q₈×C₂₄₁	1	2	yes	—
217	C₂×C₄	#41×C₂₄₁	1	2	yes	—
218	C₂×C₄	#41×C₂₄₁	1	2	yes	—
219	C₂×C₄	#41×C₂₄₁	1	2	yes	—
220	C₂²	C₂²×D₄×C₂₄₁	1	1	yes	characteristic
221	C₂²	C₂²×Q₈×C₂₄₁	1	1	yes	characteristic
222	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
223	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
224	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
225	C₂²	2₋¹⁺⁴×C₂₄₁	1	1	yes	characteristic
226	C₂²	2₋¹⁺⁴×C₂₄₁	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	#125×C₂₄₁	1	4	yes	complemented
238	C₂	C₂×#125×C₂₄₁	1	1	yes	characteristic
239	C₂	#532×C₂₄₁	1	1	yes	characteristic
240	C₂	#541×C₂₄₁	1	1	yes	characteristic
241	C₂	#547×C₂₄₁	1	1	yes	characteristic
242	C₂	D₄×Q₈×C₂₄₁	1	1	yes	characteristic
243	C₂	#552×C₂₄₁	1	1	yes	characteristic
244	C₂	#557×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₄₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
114. C₄×C₂₄₁	#125	semidirect
122. C₂₄₁	#2506	direct (N × H)
123. #2506	C₂₄₁	direct (N × H)
237. C₄	#125×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₂₄₁	2
#400×C₂₄₁	2
#398×C₂₄₁	2
#389×C₂₄₁	2
#387×C₂₄₁	2
#384×C₂₄₁	2
C₄×C₄⋊C₄×C₂₄₁	2
#2506	241

Subgroups — #2506×C241 #44504173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2506×C₂₄₁ #44504173