Subgroups — C₂×C₄₁×#1992 #23797520

Order 10496.

1294 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1992	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#340	C₂	1	4	yes	maximal
6	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
7	C₂×#125×C₄₁	—	2	2	no	—
8	C₂×#125×C₄₁	—	2	4	no	—
9	C₂×#125×C₄₁	C₄	1	8	yes	complemented
10	C₂×#125×C₄₁	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#97×C₄₁	—	2	4	no	—
17	C₂×#97×C₄₁	C₂²	1	4	yes	—
18	C₄₁×#384	—	2	2	no	—
19	C₄₁×#384	C₂²	1	4	yes	—
20	C₄₁×#340	C₂²	1	16	yes	—
21	C₂³×C₄×C₄₁	—	2	2	no	—
22	C₂³×C₄×C₄₁	—	2	2	no	—
23	C₂³×C₄×C₄₁	D₄	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₄₁	—	2	4	no	—
26	C₂²×C₈×C₄₁	—	2	4	no	—
27	C₂²×C₈×C₄₁	Q₈	1	8	yes	—
28	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
29	#125×C₄₁	—	2	16	no	—
30	#125×C₄₁	—	2	8	no	—
31	#125×C₄₁	C₂×C₄	1	16	yes	complemented
32	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₁	—	4	8	no	—
40	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₁	D₄	1	2	yes	—
45	C₂×C₄²×C₄₁	D₄	1	4	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
47	#97×C₄₁	—	2	8	no	—
48	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	2	8	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	4	no	—
54	C₂²×C₄×C₄₁	—	4	4	no	—
55	C₂²×C₄×C₄₁	—	4	4	no	—
56	C₂²×C₄×C₄₁	—	4	4	no	—
57	C₂²×C₄×C₄₁	—	4	8	no	—
58	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
59	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₄₁	#41	1	4	yes	—
65	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₄₁	—	2	16	no	—
67	C₂×C₈×C₄₁	—	2	4	no	—
68	C₂×C₈×C₄₁	—	2	4	no	—
69	C₂×C₈×C₄₁	C₂×Q₈	1	8	yes	—
70	C₄⋊C₄×C₄₁	—	2	32	no	—
71	C₄⋊C₄×C₄₁	—	2	8	no	—
72	C₄⋊C₄×C₄₁	—	2	4	no	—
73	C₄⋊C₄×C₄₁	—	2	8	no	—
74	C₄⋊C₄×C₄₁	—	4	16	no	—
75	C₄⋊C₄×C₄₁	—	4	4	no	—
76	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₁	C₂²⋊C₄	1	8	yes	—
78	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₁	C₂×D₄	1	4	yes	—
80	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
81	C₂³×C₄₁	#124	1	2	yes	—
82	C₂³×C₄₁	C₄⋊Q₈	1	2	yes	—
83	C₂³×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
84	C₂³×C₄₁	#137	1	4	yes	—
85	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	—	2	4	no	—
88	C₂×C₄×C₄₁	—	2	16	no	—
89	C₂×C₄×C₄₁	—	2	8	no	—
90	C₂×C₄×C₄₁	—	4	4	no	—
91	C₂×C₄×C₄₁	—	4	4	no	—
92	C₂×C₄×C₄₁	—	4	16	no	—
93	C₂×C₄×C₄₁	—	4	4	no	—
94	C₂×C₄×C₄₁	—	4	8	no	—
95	C₂×C₄×C₄₁	—	8	8	no	—
96	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₁	C₄×Q₈	1	8	yes	—
99	C₂×C₄×C₄₁	C₂²⋊Q₈	1	8	yes	—
100	C₂×C₄×C₄₁	#124	1	2	yes	—
101	C₂×C₄×C₄₁	C₄⋊Q₈	1	2	yes	—
102	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₁	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
106	C₈×C₄₁	—	2	8	no	—
107	C₂×#1992	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₄₁	#391	1	4	yes	—
109	C₂²×C₄₁	#417	1	8	yes	—
110	C₂²×C₄₁	#490	1	4	yes	—
111	C₂²×C₄₁	#491	1	4	yes	—
112	C₂²×C₄₁	#501	1	8	yes	—
113	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
115	C₂²×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
117	C₂²×C₄₁	C₂×#137	1	2	yes	—
118	C₄×C₄₁	—	2	8	no	—
119	C₄×C₄₁	—	4	8	no	—
120	C₄×C₄₁	—	8	4	no	—
121	C₄×C₄₁	#391	1	4	yes	—
122	C₄×C₄₁	C₂×C₄×Q₈	1	2	yes	—
123	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
124	C₂²×#125	C₂×C₄₁	1	1	yes	characteristic
125	C₂²×#97	C₂×C₄₁	1	1	yes	characteristic
126	C₂×#384	C₂×C₄₁	1	1	yes	characteristic
127	#1992	C₂×C₄₁	1	8	yes	direct product factor
128	C₂×#340	C₂×C₄₁	1	4	yes	—
129	C₂×C₄₁	#1992	1	8	yes	direct product factor
130	C₂×C₄₁	C₂×#391	1	1	yes	characteristic
131	C₂×C₄₁	C₂×#417	1	2	yes	—
132	C₂×C₄₁	C₂×#490	1	1	yes	characteristic
133	C₂×C₄₁	C₂×#491	1	1	yes	characteristic
134	C₂×C₄₁	C₂×#501	1	2	yes	—
135	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
136	C₂×#125	—	2	2	no	—
137	C₂×#125	—	2	4	no	—
138	C₂×#125	C₄×C₄₁	1	8	yes	complemented
139	C₂×#125	C₂²×C₄₁	1	4	yes	—
140	C₂²×C₄⋊C₄	—	2	2	no	—
141	C₂²×C₄⋊C₄	—	2	2	no	—
142	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
144	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
145	C₂×#97	—	2	4	no	—
146	C₂×#97	C₂²×C₄₁	1	4	yes	—
147	#384	—	2	2	no	—
148	#384	C₂²×C₄₁	1	4	yes	—
149	#340	C₂²×C₄₁	1	16	yes	—
150	C₄₁	C₂×#1992	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	—	2	2	no	—
153	C₂³×C₄	D₄×C₄₁	1	2	yes	—
154	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	Q₈×C₄₁	1	8	yes	—
158	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
159	#125	—	2	16	no	—
160	#125	—	2	8	no	—
161	#125	C₂×C₄×C₄₁	1	16	yes	complemented
162	C₂×C₄⋊C₄	—	2	16	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	8	no	—
167	C₂×C₄⋊C₄	—	2	8	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	4	8	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
171	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
172	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
174	C₂×C₄²	D₄×C₄₁	1	2	yes	—
175	C₂×C₄²	D₄×C₄₁	1	4	yes	—
176	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
177	#97	—	2	8	no	—
178	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
189	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
191	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
193	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
194	C₂²×C₄	#41×C₄₁	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
196	C₂×C₈	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂×Q₈×C₄₁	1	8	yes	—
200	C₄⋊C₄	—	2	32	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	4	16	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
207	C₄²	C₂²⋊C₄×C₄₁	1	8	yes	—
208	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
209	C₄²	C₂×D₄×C₄₁	1	4	yes	—
210	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
211	C₂³	#124×C₄₁	1	2	yes	—
212	C₂³	C₄⋊Q₈×C₄₁	1	2	yes	—
213	C₂³	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
214	C₂³	#137×C₄₁	1	4	yes	—
215	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	16	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	16	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	8	8	no	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
228	C₂×C₄	C₄×Q₈×C₄₁	1	8	yes	—
229	C₂×C₄	C₂²⋊Q₈×C₄₁	1	8	yes	—
230	C₂×C₄	#124×C₄₁	1	2	yes	—
231	C₂×C₄	C₄⋊Q₈×C₄₁	1	2	yes	—
232	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
234	C₂×C₄	C₂²×Q₈×C₄₁	1	2	yes	—
235	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
236	C₈	—	2	8	no	—
237	C₂²	C₄₁×#391	1	4	yes	—
238	C₂²	C₄₁×#417	1	8	yes	—
239	C₂²	C₄₁×#490	1	4	yes	—
240	C₂²	C₄₁×#491	1	4	yes	—
241	C₂²	C₄₁×#501	1	8	yes	—
242	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
243	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
244	C₂²	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
245	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
246	C₂²	C₂×#137×C₄₁	1	2	yes	—
247	C₄	—	2	8	no	—
248	C₄	—	4	8	no	—
249	C₄	—	8	4	no	—
250	C₄	C₄₁×#391	1	4	yes	—
251	C₄	C₂×C₄×Q₈×C₄₁	1	2	yes	—
252	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
253	C₂	C₄₁×#1992	1	8	yes	direct product factor
254	C₂	C₂×C₄₁×#391	1	1	yes	characteristic
255	C₂	C₂×C₄₁×#417	1	2	yes	—
256	C₂	C₂×C₄₁×#490	1	1	yes	characteristic
257	C₂	C₂×C₄₁×#491	1	1	yes	characteristic
258	C₂	C₂×C₄₁×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1992	C₂	direct (N × H)
9. C₂×#125×C₄₁	C₄	semidirect
31. #125×C₄₁	C₂×C₄	semidirect
107. C₂×#1992	C₄₁	direct (N × H)
127. #1992	C₂×C₄₁	direct (N × H)
129. C₂×C₄₁	#1992	direct (N × H)
138. C₂×#125	C₄×C₄₁	semidirect
150. C₄₁	C₂×#1992	direct (N × H)
161. #125	C₂×C₄×C₄₁	semidirect
253. C₂	C₄₁×#1992	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₄₁	2
C₂²×#97×C₄₁	2
C₂×C₄₁×#384	2
C₄₁×#1992	2
C₂×C₄₁×#340	2
C₂×#1992	41

Subgroups — C2×C41×#1992 #23797520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1992 #23797520