Subgroups — #2727×C₂₄₁ #44504394

Order 30848.

476 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126×C₂₄₁	C₂	1	2	yes	maximal, complemented
2	#399×C₂₄₁	C₂	1	2	yes	maximal, complemented
3	#396×C₂₄₁	C₂	1	2	yes	maximal, complemented
4	#394×C₂₄₁	C₂	1	2	yes	maximal
5	#390×C₂₄₁	C₂	1	1	yes	maximal, characteristic
6	#389×C₂₄₁	C₂	1	2	yes	maximal, complemented
7	#389×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#389×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#388×C₂₄₁	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₂₄₁	C₂²	1	1	yes	characteristic
11	#126×C₂₄₁	—	2	8	no	—
12	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
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	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₄₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
29	#95×C₂₄₁	C₂²	1	2	yes	—
30	#95×C₂₄₁	C₂²	1	2	yes	—
31	#95×C₂₄₁	C₂²	1	2	yes	—
32	C₂⁴×C₂₄₁	C₂³	1	2	yes	—
33	C₂×D₄×C₂₄₁	—	2	2	no	—
34	C₂×D₄×C₂₄₁	—	4	1	no	—
35	C₂×D₄×C₂₄₁	—	4	1	no	—
36	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₂₄₁	—	2	4	no	—
45	C₄⋊C₄×C₂₄₁	—	2	4	no	—
46	C₄⋊C₄×C₂₄₁	—	2	4	no	—
47	C₄⋊C₄×C₂₄₁	—	2	2	no	—
48	C₄⋊C₄×C₂₄₁	—	4	1	no	—
49	C₄⋊C₄×C₂₄₁	—	4	1	no	—
50	C₄⋊C₄×C₂₄₁	—	4	1	no	—
51	C₄⋊C₄×C₂₄₁	—	4	2	no	—
52	C₄⋊C₄×C₂₄₁	D₄	1	4	yes	complemented
53	C₂²⋊C₄×C₂₄₁	—	2	4	no	—
54	C₂²⋊C₄×C₂₄₁	—	2	4	no	—
55	C₂²⋊C₄×C₂₄₁	—	2	4	no	—
56	C₂²⋊C₄×C₂₄₁	—	2	4	no	—
57	C₂²⋊C₄×C₂₄₁	—	4	2	no	—
58	C₂²⋊C₄×C₂₄₁	—	4	2	no	—
59	C₄²×C₂₄₁	—	2	4	no	—
60	C₄²×C₂₄₁	—	4	1	no	—
61	C₂³×C₂₄₁	—	2	2	no	—
62	C₂³×C₂₄₁	—	2	2	no	—
63	C₂³×C₂₄₁	—	2	2	no	—
64	C₂³×C₂₄₁	—	2	2	no	—
65	C₂³×C₂₄₁	—	2	2	no	—
66	C₂³×C₂₄₁	—	2	2	no	—
67	C₂³×C₂₄₁	—	2	2	no	—
68	C₂³×C₂₄₁	C₂⁴	1	1	yes	characteristic
69	D₄×C₂₄₁	—	4	4	no	—
70	C₂×C₄×C₂₄₁	—	2	2	no	—
71	C₂×C₄×C₂₄₁	—	2	1	no	—
72	C₂×C₄×C₂₄₁	—	2	1	no	—
73	C₂×C₄×C₂₄₁	—	2	1	no	—
74	C₂×C₄×C₂₄₁	—	2	1	no	—
75	C₂×C₄×C₂₄₁	—	2	1	no	—
76	C₂×C₄×C₂₄₁	—	2	1	no	—
77	C₂×C₄×C₂₄₁	—	2	2	no	—
78	C₂×C₄×C₂₄₁	—	2	2	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	2	no	—
85	C₂×C₄×C₂₄₁	—	2	2	no	—
86	C₂×C₄×C₂₄₁	—	2	2	no	—
87	C₂×C₄×C₂₄₁	—	2	2	no	—
88	C₂×C₄×C₂₄₁	—	2	2	no	—
89	C₂×C₄×C₂₄₁	—	2	2	no	—
90	C₂×C₄×C₂₄₁	C₂×D₄	1	4	yes	—
91	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₄₁	#41	1	2	yes	—
93	C₂×C₄×C₂₄₁	#41	1	2	yes	—
94	C₂²×C₂₄₁	—	4	2	no	—
95	C₂²×C₂₄₁	—	4	2	no	—
96	C₂²×C₂₄₁	—	4	2	no	—
97	C₂²×C₂₄₁	—	4	2	no	—
98	C₂²×C₂₄₁	—	4	2	no	—
99	C₂²×C₂₄₁	—	4	2	no	—
100	C₂²×C₂₄₁	—	4	2	no	—
101	C₂²×C₂₄₁	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
103	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
104	C₂²×C₂₄₁	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²×C₂₄₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂²×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
108	C₄×C₂₄₁	—	2	2	no	—
109	C₄×C₂₄₁	—	2	2	no	—
110	C₄×C₂₄₁	—	2	2	no	—
111	C₄×C₂₄₁	—	2	4	no	—
112	C₄×C₂₄₁	—	4	2	no	—
113	C₄×C₂₄₁	—	4	2	no	—
114	C₄×C₂₄₁	—	4	2	no	—
115	C₄×C₂₄₁	—	4	2	no	—
116	C₂×C₂₄₁	—	8	2	no	—
117	C₂×C₂₄₁	#537	1	1	yes	characteristic
118	C₂×C₂₄₁	#538	1	1	yes	characteristic
119	C₂×C₂₄₁	#539	1	1	yes	characteristic
120	C₂×C₂₄₁	#547	1	1	yes	characteristic
121	C₂×C₂₄₁	#550	1	1	yes	characteristic
122	C₂×C₂₄₁	#556	1	1	yes	characteristic
123	C₂×C₂₄₁	#563	1	1	yes	characteristic
124	C₂₄₁	#2727	1	1	yes	characteristic, direct product factor, Sylow
125	#2727	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×#126	C₂×C₂₄₁	1	2	yes	complemented
127	#399	C₂×C₂₄₁	1	2	yes	complemented
128	#396	C₂×C₂₄₁	1	2	yes	complemented
129	#394	C₂×C₂₄₁	1	2	yes	—
130	#390	C₂×C₂₄₁	1	1	yes	characteristic
131	#389	C₂×C₂₄₁	1	2	yes	complemented
132	#389	C₂×C₂₄₁	1	1	yes	characteristic, complemented
133	#389	C₂×C₂₄₁	1	1	yes	characteristic, complemented
134	#388	C₂×C₂₄₁	1	2	yes	complemented
135	C₂²×D₄	C₂²×C₂₄₁	1	1	yes	characteristic
136	#126	—	2	8	no	—
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	1	yes	characteristic
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	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
152	C₂×C₄²	C₂²×C₂₄₁	1	2	yes	—
153	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
154	#95	C₂²×C₂₄₁	1	2	yes	—
155	#95	C₂²×C₂₄₁	1	2	yes	—
156	#95	C₂²×C₂₄₁	1	2	yes	—
157	C₂⁴	C₂³×C₂₄₁	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	4	1	no	—
160	C₂×D₄	—	4	1	no	—
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	2	yes	—
165	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
166	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	D₄×C₂₄₁	1	4	yes	complemented
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	4	1	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₂⁴×C₂₄₁	1	1	yes	characteristic
194	D₄	—	4	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	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	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	C₂×D₄×C₂₄₁	1	4	yes	—
216	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
217	C₂×C₄	#41×C₂₄₁	1	2	yes	—
218	C₂×C₄	#41×C₂₄₁	1	2	yes	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×D₄×C₂₄₁	1	1	yes	characteristic
227	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
228	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
229	C₂²	H(5,𝔽₂)×C₂₄₁	1	1	yes	characteristic
230	C₂²	H(5,𝔽₂)×C₂₄₁	1	1	yes	characteristic
231	C₂²	2₋¹⁺⁴×C₂₄₁	1	1	yes	characteristic
232	C₂²	2₋¹⁺⁴×C₂₄₁	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₂	—	8	2	no	—
242	C₂	#537×C₂₄₁	1	1	yes	characteristic
243	C₂	#538×C₂₄₁	1	1	yes	characteristic
244	C₂	#539×C₂₄₁	1	1	yes	characteristic
245	C₂	#547×C₂₄₁	1	1	yes	characteristic
246	C₂	#550×C₂₄₁	1	1	yes	characteristic
247	C₂	#556×C₂₄₁	1	1	yes	characteristic
248	C₂	#563×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₄₁
prime radical	C₂²×D₄×C₂₄₁	C₂²
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
1. C₂×#126×C₂₄₁	C₂	semidirect
2. #399×C₂₄₁	C₂	semidirect
3. #396×C₂₄₁	C₂	semidirect
6. #389×C₂₄₁	C₂	semidirect
7. #389×C₂₄₁	C₂	semidirect
8. #389×C₂₄₁	C₂	semidirect
9. #388×C₂₄₁	C₂	semidirect
52. C₄⋊C₄×C₂₄₁	D₄	semidirect
124. C₂₄₁	#2727	direct (N × H)
125. #2727	C₂₄₁	direct (N × H)
126. C₂×#126	C₂×C₂₄₁	semidirect
127. #399	C₂×C₂₄₁	semidirect
128. #396	C₂×C₂₄₁	semidirect
131. #389	C₂×C₂₄₁	semidirect
132. #389	C₂×C₂₄₁	semidirect
133. #389	C₂×C₂₄₁	semidirect
134. #388	C₂×C₂₄₁	semidirect
177. C₄⋊C₄	D₄×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#126×C₂₄₁	2
#399×C₂₄₁	2
#396×C₂₄₁	2
#394×C₂₄₁	2
#390×C₂₄₁	2
#389×C₂₄₁	2
#388×C₂₄₁	2
#2727	241

Subgroups — #2727×C241 #44504394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2727×C₂₄₁ #44504394