Subgroups — C₄₁×#21799 #23798000

Order 10496.

1258 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2920	C₂	1	2	yes	maximal
3	C₄₁×#1966	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1784	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₄₁	C₂²	1	2	yes	—
6	C₂×C₄⋊Q₈×C₄₁	—	2	4	no	—
7	C₂×C₄⋊Q₈×C₄₁	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₄₁	C₄	1	4	yes	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₄	1	4	yes	—
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#432	—	2	4	no	—
13	C₄₁×#431	—	2	4	no	—
14	C₄₁×#423	C₂²	1	4	yes	—
15	C₂×#101×C₄₁	C₂²	1	16	yes	—
16	C₄₁×#407	C₂²	1	4	yes	—
17	C₄×M₁₆×C₄₁	C₂²	1	4	yes	—
18	C₂²×Q₈×C₄₁	—	2	8	no	—
19	C₂²×Q₈×C₄₁	—	4	8	no	—
20	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
21	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
26	C₄⋊Q₈×C₄₁	—	2	4	no	—
27	C₄⋊Q₈×C₄₁	—	2	8	no	—
28	C₄⋊Q₈×C₄₁	—	2	4	no	—
29	C₄⋊Q₈×C₄₁	—	2	4	no	—
30	C₄⋊Q₈×C₄₁	—	2	4	no	—
31	C₄⋊Q₈×C₄₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
36	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₁	D₄	1	4	yes	—
38	C₂×C₄²×C₄₁	D₄	1	4	yes	—
39	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
40	C₄⋊C₈×C₄₁	—	2	8	no	—
41	#101×C₄₁	—	4	16	no	—
42	C₂²⋊C₈×C₄₁	—	2	8	no	—
43	#97×C₄₁	—	2	4	no	—
44	C₄×C₈×C₄₁	—	2	4	no	—
45	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×Q₈×C₄₁	—	2	8	no	—
47	C₂×Q₈×C₄₁	—	2	8	no	—
48	C₂×Q₈×C₄₁	—	2	16	no	—
49	C₂×Q₈×C₄₁	—	4	8	no	—
50	C₂×Q₈×C₄₁	—	4	32	no	—
51	C₂²×C₄×C₄₁	—	2	8	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	8	no	—
54	C₂²×C₄×C₄₁	—	4	8	no	—
55	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₄₁	—	2	16	no	—
64	C₂×C₈×C₄₁	—	2	8	no	—
65	C₄⋊C₄×C₄₁	—	2	16	no	—
66	C₄⋊C₄×C₄₁	—	2	16	no	—
67	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₄₁	C₂²⋊C₄	1	4	yes	—
69	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₄₁	C₂×D₄	1	2	yes	—
71	C₄²×C₄₁	C₂×D₄	1	2	yes	—
72	C₄²×C₄₁	C₂×D₄	1	4	yes	—
73	C₂³×C₄₁	—	2	2	no	—
74	C₂³×C₄₁	—	2	4	no	—
75	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
77	Q₈×C₄₁	—	4	16	no	—
78	Q₈×C₄₁	—	4	16	no	—
79	C₂×C₄×C₄₁	—	2	2	no	—
80	C₂×C₄×C₄₁	—	2	4	no	—
81	C₂×C₄×C₄₁	—	2	8	no	—
82	C₂×C₄×C₄₁	—	2	8	no	—
83	C₂×C₄×C₄₁	—	2	8	no	—
84	C₂×C₄×C₄₁	—	2	8	no	—
85	C₂×C₄×C₄₁	—	2	16	no	—
86	C₂×C₄×C₄₁	—	4	16	no	—
87	C₂×C₄×C₄₁	#101	1	8	yes	complemented
88	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
90	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
93	C₈×C₄₁	—	4	8	no	—
94	#21799	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₄₁	—	2	8	no	—
96	C₂²×C₄₁	—	2	2	no	—
97	C₂²×C₄₁	—	2	2	no	—
98	C₂²×C₄₁	—	2	2	no	—
99	C₂²×C₄₁	C₂×#101	1	4	yes	—
100	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₄₁	#518	1	2	yes	—
102	C₄×C₄₁	—	2	2	no	—
103	C₄×C₄₁	—	2	4	no	—
104	C₄×C₄₁	—	4	8	no	—
105	C₄×C₄₁	C₂×#101	1	8	yes	—
106	C₄×C₄₁	#535	1	4	yes	—
107	C₂²×C₄⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
108	#2920	C₂×C₄₁	1	2	yes	—
109	#1966	C₂×C₄₁	1	8	yes	complemented
110	#1784	C₂×C₄₁	1	4	yes	—
111	C₂×C₄₁	—	2	2	no	—
112	C₂×C₄₁	—	2	4	no	—
113	C₂×C₄₁	#2309	1	1	yes	characteristic
114	C₂×C₄₁	C₂²×#101	1	2	yes	—
115	C₂³×Q₈	C₂²×C₄₁	1	2	yes	—
116	C₂×C₄⋊Q₈	—	2	4	no	—
117	C₂×C₄⋊Q₈	—	2	4	no	—
118	C₂×C₄⋊Q₈	C₄×C₄₁	1	4	yes	—
119	C₂×C₄⋊Q₈	C₂²×C₄₁	1	4	yes	—
120	C₂²×C₄⋊C₄	C₄×C₄₁	1	4	yes	—
121	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
122	#432	—	2	4	no	—
123	#431	—	2	4	no	—
124	#423	C₂²×C₄₁	1	4	yes	—
125	C₂×#101	C₂²×C₄₁	1	16	yes	—
126	#407	C₂²×C₄₁	1	4	yes	—
127	C₄×M₁₆	C₂²×C₄₁	1	4	yes	—
128	C₄₁	#21799	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×Q₈	—	2	8	no	—
130	C₂²×Q₈	—	4	8	no	—
131	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
132	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
133	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
134	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
135	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
136	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	2	8	no	—
139	C₄⋊Q₈	—	2	4	no	—
140	C₄⋊Q₈	—	2	4	no	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	—	2	8	no	—
143	C₂×C₄⋊C₄	—	2	4	no	—
144	C₂×C₄⋊C₄	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
147	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
148	C₂×C₄²	D₄×C₄₁	1	4	yes	—
149	C₂×C₄²	D₄×C₄₁	1	4	yes	—
150	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
151	C₄⋊C₈	—	2	8	no	—
152	#101	—	4	16	no	—
153	C₂²⋊C₈	—	2	8	no	—
154	#97	—	2	4	no	—
155	C₄×C₈	—	2	4	no	—
156	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
157	C₂×Q₈	—	2	8	no	—
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	—	2	16	no	—
160	C₂×Q₈	—	4	8	no	—
161	C₂×Q₈	—	4	32	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	4	8	no	—
166	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
167	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
168	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
170	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
172	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
173	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
174	M₁₆	—	2	16	no	—
175	C₂×C₈	—	2	8	no	—
176	C₄⋊C₄	—	2	16	no	—
177	C₄⋊C₄	—	2	16	no	—
178	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
179	C₄²	C₂²⋊C₄×C₄₁	1	4	yes	—
180	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
181	C₄²	C₂×D₄×C₄₁	1	2	yes	—
182	C₄²	C₂×D₄×C₄₁	1	2	yes	—
183	C₄²	C₂×D₄×C₄₁	1	4	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
187	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
188	Q₈	—	4	16	no	—
189	Q₈	—	4	16	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	#101×C₄₁	1	8	yes	complemented
199	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
202	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
203	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
204	C₈	—	4	8	no	—
205	C₂²	—	2	8	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	C₂×#101×C₄₁	1	4	yes	—
210	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
211	C₂²	C₄₁×#518	1	2	yes	—
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	4	8	no	—
215	C₄	C₂×#101×C₄₁	1	8	yes	—
216	C₄	C₄₁×#535	1	4	yes	—
217	C₂	—	2	2	no	—
218	C₂	—	2	4	no	—
219	C₂	C₄₁×#2309	1	1	yes	characteristic
220	C₂	C₂²×#101×C₄₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1966	C₂	semidirect
87. C₂×C₄×C₄₁	#101	semidirect
94. #21799	C₄₁	direct (N × H)
109. #1966	C₂×C₄₁	semidirect
128. C₄₁	#21799	direct (N × H)
198. C₂×C₄	#101×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₄₁	2
C₄₁×#2920	2
C₄₁×#1966	2
C₄₁×#1784	2
#21799	41

Subgroups — C41×#21799 #23798000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21799 #23798000