Subgroups — C₂²×C₄₁×#381 #23810206

Order 10496.

8062 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₄₁	C₂	1	2	yes	maximal
3	C₂×C₄₁×#381	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95×C₄₁	C₂	1	4	yes	maximal, complemented
5	C₂⁶×C₄₁	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₄₁	—	2	16	no	—
7	C₂⁴×C₄×C₄₁	C₄	1	16	yes	complemented
8	C₂⁴×C₄×C₄₁	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₄₁	C₂²	1	12	yes	—
11	C₂²×C₂²⋊C₄×C₄₁	—	2	16	no	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	24	yes	—
13	C₄₁×#381	C₂²	1	64	yes	direct product factor
14	C₂×#95×C₄₁	C₂²	1	48	yes	complemented
15	C₂⁵×C₄₁	—	2	16	no	—
16	C₂⁵×C₄₁	C₂×C₄	1	8	yes	—
17	C₂⁵×C₄₁	D₄	1	16	yes	—
18	C₂⁵×C₄₁	C₂³	1	6	yes	—
19	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₄₁	—	2	128	no	—
21	C₂³×C₄×C₄₁	—	2	16	no	—
22	C₂³×C₄×C₄₁	—	2	96	no	—
23	C₂³×C₄×C₄₁	—	2	16	no	—
24	C₂³×C₄×C₄₁	—	2	32	no	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	16	yes	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	96	yes	complemented
27	C₂³×C₄×C₄₁	C₂³	1	24	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	12	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	16	yes	—
30	C₂×C₂²⋊C₄×C₄₁	—	2	96	no	—
31	C₂×C₂²⋊C₄×C₄₁	C₂³	1	32	yes	—
32	#95×C₄₁	C₂³	1	64	yes	complemented
33	C₂⁴×C₄₁	—	2	64	no	—
34	C₂⁴×C₄₁	—	2	128	no	—
35	C₂⁴×C₄₁	—	2	48	no	—
36	C₂⁴×C₄₁	C₂²⋊C₄	1	64	yes	—
37	C₂⁴×C₄₁	C₂²×C₄	1	24	yes	—
38	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
39	C₂⁴×C₄₁	C₂×D₄	1	48	yes	—
40	C₂⁴×C₄₁	C₂×D₄	1	8	yes	—
41	C₂⁴×C₄₁	#41	1	16	yes	—
42	C₂⁴×C₄₁	C₂⁴	1	4	yes	—
43	C₂⁴×C₄₁	C₂⁴	1	3	yes	—
44	C₂²×C₄×C₄₁	—	2	64	no	—
45	C₂²×C₄×C₄₁	—	2	384	no	—
46	C₂²×C₄×C₄₁	—	2	48	no	—
47	C₂²×C₄×C₄₁	—	2	64	no	—
48	C₂²×C₄×C₄₁	—	2	48	no	—
49	C₂²×C₄×C₄₁	—	2	96	no	—
50	C₂²×C₄×C₄₁	—	4	64	no	—
51	C₂²×C₄×C₄₁	C₂²×C₄	1	48	yes	—
52	C₂²×C₄×C₄₁	C₂²×C₄	1	64	yes	complemented
53	C₂²×C₄×C₄₁	C₂⁴	1	16	yes	—
54	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
55	C₂²⋊C₄×C₄₁	—	2	64	no	—
56	C₂³×C₄₁	—	2	96	no	—
57	C₂³×C₄₁	—	2	192	no	—
58	C₂³×C₄₁	—	2	16	no	—
59	C₂³×C₄₁	—	2	256	no	—
60	C₂³×C₄₁	C₂×C₂²⋊C₄	1	96	yes	—
61	C₂³×C₄₁	C₂×C₂²⋊C₄	1	16	yes	—
62	C₂³×C₄₁	#117	1	32	yes	—
63	C₂³×C₄₁	#123	1	64	yes	—
64	C₂³×C₄₁	C₂³×C₄	1	8	yes	—
65	C₂³×C₄₁	C₂³×C₄	1	6	yes	—
66	C₂³×C₄₁	C₂²×D₄	1	16	yes	—
67	C₂³×C₄₁	C₂²×D₄	1	12	yes	—
68	C₂³×C₄₁	C₂×#41	1	24	yes	—
69	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
70	C₂×C₄×C₄₁	—	2	96	no	—
71	C₂×C₄×C₄₁	—	2	128	no	—
72	C₂×C₄×C₄₁	—	2	16	no	—
73	C₂×C₄×C₄₁	—	2	16	no	—
74	C₂×C₄×C₄₁	—	2	32	no	—
75	C₂×C₄×C₄₁	—	4	96	no	—
76	C₂×C₄×C₄₁	C₂³×C₄	1	16	yes	—
77	C₂²×#381	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₄₁	—	2	16	no	—
79	C₂²×C₄₁	—	2	32	no	—
80	C₂²×C₄₁	—	2	192	no	—
81	C₂²×C₄₁	#381	1	64	yes	direct product factor
82	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	16	yes	—
83	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	12	yes	—
84	C₂²×C₄₁	C₂×#117	1	24	yes	—
85	C₂²×C₄₁	C₂×#123	1	48	yes	—
86	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
87	C₂²×C₄₁	C₂³×D₄	1	2	yes	—
88	C₂²×C₄₁	C₂²×#41	1	4	yes	—
89	C₄×C₄₁	—	2	16	no	—
90	C₄×C₄₁	—	4	16	no	—
91	C₂⁵×C₄	C₂×C₄₁	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	C₂×C₄₁	1	2	yes	—
93	C₂×#381	C₂×C₄₁	1	24	yes	direct product factor
94	C₂²×#95	C₂×C₄₁	1	4	yes	complemented
95	C₂×C₄₁	—	2	16	no	—
96	C₂×C₄₁	C₂×#381	1	24	yes	direct product factor
97	C₂×C₄₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄₁	C₂²×#117	1	2	yes	—
99	C₂×C₄₁	C₂²×#123	1	4	yes	—
100	C₂⁶	C₂²×C₄₁	1	1	yes	characteristic
101	C₂⁴×C₄	—	2	16	no	—
102	C₂⁴×C₄	C₄×C₄₁	1	16	yes	complemented
103	C₂⁴×C₄	C₂²×C₄₁	1	4	yes	—
104	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
105	C₂⁴×C₄	C₂²×C₄₁	1	12	yes	—
106	C₂²×C₂²⋊C₄	—	2	16	no	—
107	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	24	yes	—
108	#381	C₂²×C₄₁	1	64	yes	direct product factor
109	C₂×#95	C₂²×C₄₁	1	48	yes	complemented
110	C₄₁	C₂²×#381	1	1	yes	characteristic, direct product factor, Sylow
111	C₂⁵	—	2	16	no	—
112	C₂⁵	C₂×C₄×C₄₁	1	8	yes	—
113	C₂⁵	D₄×C₄₁	1	16	yes	—
114	C₂⁵	C₂³×C₄₁	1	6	yes	—
115	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
116	C₂³×C₄	—	2	128	no	—
117	C₂³×C₄	—	2	16	no	—
118	C₂³×C₄	—	2	96	no	—
119	C₂³×C₄	—	2	16	no	—
120	C₂³×C₄	—	2	32	no	—
121	C₂³×C₄	C₂×C₄×C₄₁	1	16	yes	—
122	C₂³×C₄	C₂×C₄×C₄₁	1	96	yes	complemented
123	C₂³×C₄	C₂³×C₄₁	1	24	yes	—
124	C₂³×C₄	C₂³×C₄₁	1	12	yes	—
125	C₂³×C₄	C₂³×C₄₁	1	16	yes	—
126	C₂×C₂²⋊C₄	—	2	96	no	—
127	C₂×C₂²⋊C₄	C₂³×C₄₁	1	32	yes	—
128	#95	C₂³×C₄₁	1	64	yes	complemented
129	C₂⁴	—	2	64	no	—
130	C₂⁴	—	2	128	no	—
131	C₂⁴	—	2	48	no	—
132	C₂⁴	C₂²⋊C₄×C₄₁	1	64	yes	—
133	C₂⁴	C₂²×C₄×C₄₁	1	24	yes	—
134	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
135	C₂⁴	C₂×D₄×C₄₁	1	48	yes	—
136	C₂⁴	C₂×D₄×C₄₁	1	8	yes	—
137	C₂⁴	#41×C₄₁	1	16	yes	—
138	C₂⁴	C₂⁴×C₄₁	1	4	yes	—
139	C₂⁴	C₂⁴×C₄₁	1	3	yes	—
140	C₂²×C₄	—	2	64	no	—
141	C₂²×C₄	—	2	384	no	—
142	C₂²×C₄	—	2	48	no	—
143	C₂²×C₄	—	2	64	no	—
144	C₂²×C₄	—	2	48	no	—
145	C₂²×C₄	—	2	96	no	—
146	C₂²×C₄	—	4	64	no	—
147	C₂²×C₄	C₂²×C₄×C₄₁	1	48	yes	—
148	C₂²×C₄	C₂²×C₄×C₄₁	1	64	yes	complemented
149	C₂²×C₄	C₂⁴×C₄₁	1	16	yes	—
150	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
151	C₂²⋊C₄	—	2	64	no	—
152	C₂³	—	2	96	no	—
153	C₂³	—	2	192	no	—
154	C₂³	—	2	16	no	—
155	C₂³	—	2	256	no	—
156	C₂³	C₂×C₂²⋊C₄×C₄₁	1	96	yes	—
157	C₂³	C₂×C₂²⋊C₄×C₄₁	1	16	yes	—
158	C₂³	#117×C₄₁	1	32	yes	—
159	C₂³	#123×C₄₁	1	64	yes	—
160	C₂³	C₂³×C₄×C₄₁	1	8	yes	—
161	C₂³	C₂³×C₄×C₄₁	1	6	yes	—
162	C₂³	C₂²×D₄×C₄₁	1	16	yes	—
163	C₂³	C₂²×D₄×C₄₁	1	12	yes	—
164	C₂³	C₂×#41×C₄₁	1	24	yes	—
165	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
166	C₂×C₄	—	2	96	no	—
167	C₂×C₄	—	2	128	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	32	no	—
171	C₂×C₄	—	4	96	no	—
172	C₂×C₄	C₂³×C₄×C₄₁	1	16	yes	—
173	C₂²	—	2	192	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	2	32	no	—
176	C₂²	C₄₁×#381	1	64	yes	direct product factor
177	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	16	yes	—
178	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	12	yes	—
179	C₂²	C₂×#117×C₄₁	1	24	yes	—
180	C₂²	C₂×#123×C₄₁	1	48	yes	—
181	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
182	C₂²	C₂³×D₄×C₄₁	1	2	yes	—
183	C₂²	C₂²×#41×C₄₁	1	4	yes	—
184	C₄	—	2	16	no	—
185	C₄	—	4	16	no	—
186	C₂	—	2	16	no	—
187	C₂	C₂×C₄₁×#381	1	24	yes	direct product factor
188	C₂	C₂³×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
189	C₂	C₂²×#117×C₄₁	1	2	yes	—
190	C₂	C₂²×#123×C₄₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₄₁	C₂³
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂⁶×C₄₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄₁×#381	C₂	direct (N × H)
4. C₂²×#95×C₄₁	C₂	semidirect
7. C₂⁴×C₄×C₄₁	C₄	semidirect
13. C₄₁×#381	C₂²	direct (N × H)
14. C₂×#95×C₄₁	C₂²	semidirect
26. C₂³×C₄×C₄₁	C₂×C₄	semidirect
32. #95×C₄₁	C₂³	semidirect
52. C₂²×C₄×C₄₁	C₂²×C₄	semidirect
77. C₂²×#381	C₄₁	direct (N × H)
81. C₂²×C₄₁	#381	direct (N × H)
93. C₂×#381	C₂×C₄₁	direct (N × H)
94. C₂²×#95	C₂×C₄₁	semidirect
96. C₂×C₄₁	C₂×#381	direct (N × H)
102. C₂⁴×C₄	C₄×C₄₁	semidirect
108. #381	C₂²×C₄₁	direct (N × H)
109. C₂×#95	C₂²×C₄₁	semidirect
110. C₄₁	C₂²×#381	direct (N × H)
122. C₂³×C₄	C₂×C₄×C₄₁	semidirect
128. #95	C₂³×C₄₁	semidirect
148. C₂²×C₄	C₂²×C₄×C₄₁	semidirect
176. C₂²	C₄₁×#381	direct (N × H)
187. C₂	C₂×C₄₁×#381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₄₁	2
C₂³×C₂²⋊C₄×C₄₁	2
C₂×C₄₁×#381	2
C₂²×#95×C₄₁	2
C₂²×#381	41

Subgroups — C22×C41×#381 #23810206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#381 #23810206