Subgroups — #1877×C₂₄₁ #44503544

Order 30848.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#102×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	#391×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#340×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₂₄₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₄₁	—	2	1	no	—
11	C₂×SD₁₆×C₂₄₁	—	2	2	no	—
12	C₂×SD₁₆×C₂₄₁	—	2	2	no	—
13	C₂×SD₁₆×C₂₄₁	C₄	1	4	yes	complemented
14	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₄₁	—	2	2	no	—
17	C₄×D₄×C₂₄₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₄₁	—	2	1	no	—
21	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
22	#103×C₂₄₁	—	2	2	no	—
23	#102×C₂₄₁	—	2	2	no	—
24	#95×C₂₄₁	—	2	1	no	—
25	#95×C₂₄₁	—	2	1	no	—
26	C₂⁴×C₂₄₁	—	2	1	no	—
27	C₂×Q₈×C₂₄₁	—	2	1	no	—
28	C₂×Q₈×C₂₄₁	—	2	1	no	—
29	C₂×Q₈×C₂₄₁	—	2	1	no	—
30	C₂×Q₈×C₂₄₁	—	2	1	no	—
31	C₂×Q₈×C₂₄₁	—	2	1	no	—
32	C₂×Q₈×C₂₄₁	C₂×C₄	1	2	yes	complemented
33	C₂×D₄×C₂₄₁	—	2	1	no	—
34	C₂×D₄×C₂₄₁	—	2	1	no	—
35	C₂×D₄×C₂₄₁	—	2	1	no	—
36	C₂×D₄×C₂₄₁	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₂₄₁	D₄	1	2	yes	—
38	C₂×D₄×C₂₄₁	D₄	1	2	yes	—
39	C₂²×C₄×C₂₄₁	—	2	4	no	—
40	C₂²×C₄×C₂₄₁	—	2	1	no	—
41	C₂²×C₄×C₂₄₁	—	2	1	no	—
42	C₂²×C₄×C₂₄₁	—	2	1	no	—
43	C₂²×C₄×C₂₄₁	—	4	2	no	—
44	C₂²×C₄×C₂₄₁	—	4	2	no	—
45	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
48	SD₁₆×C₂₄₁	—	2	8	no	—
49	C₂×C₈×C₂₄₁	—	2	1	no	—
50	C₂×C₈×C₂₄₁	—	2	1	no	—
51	C₂×C₈×C₂₄₁	—	2	1	no	—
52	C₂×C₈×C₂₄₁	—	2	1	no	—
53	C₂×C₈×C₂₄₁	—	2	1	no	—
54	C₂×C₈×C₂₄₁	C₂×C₄	1	2	yes	complemented
55	C₄⋊C₄×C₂₄₁	—	2	1	no	—
56	C₄⋊C₄×C₂₄₁	—	2	1	no	—
57	C₄⋊C₄×C₂₄₁	—	2	1	no	—
58	C₄⋊C₄×C₂₄₁	D₄	1	2	yes	—
59	C₂²⋊C₄×C₂₄₁	—	2	4	no	—
60	C₄²×C₂₄₁	—	2	1	no	—
61	C₄²×C₂₄₁	—	2	1	no	—
62	C₂³×C₂₄₁	—	2	2	no	—
63	C₂³×C₂₄₁	—	2	2	no	—
64	C₂³×C₂₄₁	—	2	2	no	—
65	C₂³×C₂₄₁	—	4	1	no	—
66	C₂³×C₂₄₁	—	4	1	no	—
67	C₂³×C₂₄₁	—	4	1	no	—
68	C₂³×C₂₄₁	—	4	1	no	—
69	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₂₄₁	—	2	2	no	—
71	Q₈×C₂₄₁	—	2	2	no	—
72	Q₈×C₂₄₁	—	4	1	no	—
73	Q₈×C₂₄₁	—	4	1	no	—
74	D₄×C₂₄₁	—	2	2	no	—
75	D₄×C₂₄₁	—	2	2	no	—
76	D₄×C₂₄₁	—	2	2	no	—
77	D₄×C₂₄₁	C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₂₄₁	—	2	1	no	—
79	C₂×C₄×C₂₄₁	—	2	1	no	—
80	C₂×C₄×C₂₄₁	—	2	2	no	—
81	C₂×C₄×C₂₄₁	—	2	1	no	—
82	C₂×C₄×C₂₄₁	—	2	2	no	—
83	C₂×C₄×C₂₄₁	—	2	1	no	—
84	C₂×C₄×C₂₄₁	—	2	1	no	—
85	C₂×C₄×C₂₄₁	—	4	8	no	—
86	C₂×C₄×C₂₄₁	—	4	1	no	—
87	C₂×C₄×C₂₄₁	—	4	1	no	—
88	C₂×C₄×C₂₄₁	—	4	1	no	—
89	C₂×C₄×C₂₄₁	—	4	1	no	—
90	C₂×C₄×C₂₄₁	—	4	1	no	—
91	C₂×C₄×C₂₄₁	—	4	1	no	—
92	C₂×C₄×C₂₄₁	—	4	1	no	—
93	C₂×C₄×C₂₄₁	SD₁₆	1	2	yes	complemented
94	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
99	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
100	C₈×C₂₄₁	—	2	2	no	—
101	C₈×C₂₄₁	—	4	1	no	—
102	C₂²×C₂₄₁	—	2	4	no	—
103	C₂²×C₂₄₁	—	4	2	no	—
104	C₂²×C₂₄₁	—	4	2	no	—
105	C₂²×C₂₄₁	—	4	2	no	—
106	C₂²×C₂₄₁	—	4	2	no	—
107	C₂²×C₂₄₁	—	4	2	no	—
108	C₂²×C₂₄₁	—	4	2	no	—
109	C₂²×C₂₄₁	C₄×D₄	1	1	yes	characteristic
110	C₂²×C₂₄₁	#120	1	1	yes	characteristic
111	C₂²×C₂₄₁	#121	1	1	yes	characteristic
112	C₂²×C₂₄₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₂₄₁	#135	1	1	yes	characteristic
114	C₂²×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂²×C₂₄₁	#137	1	1	yes	characteristic
116	C₄×C₂₄₁	—	2	2	no	—
117	C₄×C₂₄₁	—	4	2	no	—
118	C₄×C₂₄₁	—	4	2	no	—
119	C₄×C₂₄₁	—	4	1	no	—
120	C₄×C₂₄₁	—	8	1	no	—
121	C₄×C₂₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₂₄₁	C₄×D₄	1	1	yes	characteristic
123	C₄×C₂₄₁	#121	1	1	yes	characteristic
124	C₄×C₂₄₁	#123	1	1	yes	characteristic
125	C₂×C₂₄₁	—	4	4	no	—
126	C₂×C₂₄₁	#386	1	1	yes	characteristic
127	C₂×C₂₄₁	C₄×SD₁₆	1	1	yes	characteristic
128	C₂×C₂₄₁	#440	1	1	yes	characteristic
129	C₂×C₂₄₁	#450	1	1	yes	characteristic
130	C₂×C₂₄₁	#452	1	1	yes	characteristic
131	C₂×C₂₄₁	#461	1	1	yes	characteristic
132	C₂×C₂₄₁	#463	1	1	yes	characteristic
133	C₂₄₁	#1877	1	1	yes	characteristic, direct product factor, Sylow
134	#1877	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×SD₁₆	C₂×C₂₄₁	1	1	yes	characteristic
136	C₂×C₄×D₄	C₂×C₂₄₁	1	1	yes	characteristic
137	C₂×#103	C₂×C₂₄₁	1	1	yes	characteristic, complemented
138	C₂×#102	C₂×C₂₄₁	1	1	yes	characteristic
139	#391	C₂×C₂₄₁	1	1	yes	characteristic, complemented
140	#340	C₂×C₂₄₁	1	1	yes	characteristic, complemented
141	#336	C₂×C₂₄₁	1	1	yes	characteristic, complemented
142	C₂²×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
143	C₂²×D₄	C₂²×C₂₄₁	1	1	yes	characteristic
144	C₂³×C₄	—	2	1	no	—
145	C₂×SD₁₆	—	2	2	no	—
146	C₂×SD₁₆	—	2	2	no	—
147	C₂×SD₁₆	C₄×C₂₄₁	1	4	yes	complemented
148	C₂²×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
149	C₂²×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	—	2	1	no	—
155	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
156	#103	—	2	2	no	—
157	#102	—	2	2	no	—
158	#95	—	2	1	no	—
159	#95	—	2	1	no	—
160	C₂⁴	—	2	1	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	C₂×C₄×C₂₄₁	1	2	yes	complemented
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂×C₄×C₂₄₁	1	2	yes	—
171	C₂×D₄	D₄×C₂₄₁	1	2	yes	—
172	C₂×D₄	D₄×C₂₄₁	1	2	yes	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	D₄×C₂₄₁	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₂₄₁	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
182	SD₁₆	—	2	8	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	C₂×C₄×C₂₄₁	1	2	yes	complemented
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	D₄×C₂₄₁	1	2	yes	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	4	1	no	—
200	C₂³	—	4	1	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	C₂×D₄×C₂₄₁	1	1	yes	characteristic
204	Q₈	—	2	2	no	—
205	Q₈	—	2	2	no	—
206	Q₈	—	4	1	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	2	2	no	—
210	D₄	—	2	2	no	—
211	D₄	C₂²⋊C₄×C₂₄₁	1	4	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	SD₁₆×C₂₄₁	1	2	yes	complemented
228	C₂×C₄	C₂²×C₄×C₂₄₁	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
232	C₂×C₄	#41×C₂₄₁	1	1	yes	characteristic
233	C₂×C₄	#41×C₂₄₁	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	4	1	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₄×D₄×C₂₄₁	1	1	yes	characteristic
244	C₂²	#120×C₂₄₁	1	1	yes	characteristic
245	C₂²	#121×C₂₄₁	1	1	yes	characteristic
246	C₂²	C₂×SD₁₆×C₂₄₁	1	1	yes	characteristic
247	C₂²	#135×C₂₄₁	1	1	yes	characteristic
248	C₂²	AGL(1,ℤ/8)×C₂₄₁	1	1	yes	characteristic
249	C₂²	#137×C₂₄₁	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	8	1	no	—
255	C₄	C₂×C₂²⋊C₄×C₂₄₁	1	1	yes	characteristic
256	C₄	C₄×D₄×C₂₄₁	1	1	yes	characteristic
257	C₄	#121×C₂₄₁	1	1	yes	characteristic
258	C₄	#123×C₂₄₁	1	1	yes	characteristic
259	C₂	—	4	4	no	—
260	C₂	#386×C₂₄₁	1	1	yes	characteristic
261	C₂	C₄×SD₁₆×C₂₄₁	1	1	yes	characteristic
262	C₂	#440×C₂₄₁	1	1	yes	characteristic
263	C₂	#450×C₂₄₁	1	1	yes	characteristic
264	C₂	#452×C₂₄₁	1	1	yes	characteristic
265	C₂	#461×C₂₄₁	1	1	yes	characteristic
266	C₂	#463×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₄₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₄₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₄₁
prime radical	C₂²×D₄×C₂₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₄₁ ◁ C₂×C₄²×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#103×C₂₄₁	C₂	semidirect
5. #391×C₂₄₁	C₂	semidirect
6. #340×C₂₄₁	C₂	semidirect
7. #336×C₂₄₁	C₂	semidirect
13. C₂×SD₁₆×C₂₄₁	C₄	semidirect
32. C₂×Q₈×C₂₄₁	C₂×C₄	semidirect
54. C₂×C₈×C₂₄₁	C₂×C₄	semidirect
93. C₂×C₄×C₂₄₁	SD₁₆	semidirect
133. C₂₄₁	#1877	direct (N × H)
134. #1877	C₂₄₁	direct (N × H)
137. C₂×#103	C₂×C₂₄₁	semidirect
139. #391	C₂×C₂₄₁	semidirect
140. #340	C₂×C₂₄₁	semidirect
141. #336	C₂×C₂₄₁	semidirect
147. C₂×SD₁₆	C₄×C₂₄₁	semidirect
166. C₂×Q₈	C₂×C₄×C₂₄₁	semidirect
188. C₂×C₈	C₂×C₄×C₂₄₁	semidirect
227. C₂×C₄	SD₁₆×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₂₄₁	2
C₂×C₄×D₄×C₂₄₁	2
C₂×#103×C₂₄₁	2
C₂×#102×C₂₄₁	2
#391×C₂₄₁	2
#340×C₂₄₁	2
#336×C₂₄₁	2
#1877	241

Subgroups — #1877×C241 #44503544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1877×C₂₄₁ #44503544