Subgroups — #2346×C₂₁₁ #41009966

Order 27008.

504 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#389×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#388×C₂₁₁	C₂	1	4	yes	maximal
4	#382×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#380×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	#376×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	#117×C₂₁₁	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
21	#95×C₂₁₁	C₂²	1	1	yes	characteristic
22	#95×C₂₁₁	C₂²	1	2	yes	—
23	#95×C₂₁₁	C₂²	1	4	yes	—
24	#95×C₂₁₁	C₂²	1	1	yes	characteristic
25	#95×C₂₁₁	C₂²	1	4	yes	—
26	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₂₁₁	—	2	4	no	—
28	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₁₁	—	2	4	no	—
37	C₄⋊C₄×C₂₁₁	—	4	4	no	—
38	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
39	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
40	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
41	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
42	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
43	C₄²×C₂₁₁	—	2	8	no	—
44	C₄²×C₂₁₁	—	2	4	no	—
45	C₄²×C₂₁₁	C₂×C₄	1	8	yes	—
46	C₂³×C₂₁₁	—	2	2	no	—
47	C₂³×C₂₁₁	—	2	2	no	—
48	C₂³×C₂₁₁	—	2	1	no	—
49	C₂³×C₂₁₁	—	2	1	no	—
50	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
51	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	—	2	4	no	—
53	C₂×C₄×C₂₁₁	—	2	2	no	—
54	C₂×C₄×C₂₁₁	—	2	2	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	—	2	2	no	—
57	C₂×C₄×C₂₁₁	—	2	2	no	—
58	C₂×C₄×C₂₁₁	—	2	8	no	—
59	C₂×C₄×C₂₁₁	—	2	2	no	—
60	C₂×C₄×C₂₁₁	—	2	4	no	—
61	C₂×C₄×C₂₁₁	—	2	2	no	—
62	C₂×C₄×C₂₁₁	—	2	4	no	—
63	C₂×C₄×C₂₁₁	—	2	4	no	—
64	C₂×C₄×C₂₁₁	—	2	2	no	—
65	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
67	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₁₁	#41	1	4	yes	—
70	C₂×C₄×C₂₁₁	#41	1	4	yes	—
71	C₂²×C₂₁₁	—	2	2	no	—
72	C₂²×C₂₁₁	—	2	2	no	—
73	C₂²×C₂₁₁	—	2	2	no	—
74	C₂²×C₂₁₁	—	4	2	no	—
75	C₂²×C₂₁₁	—	4	2	no	—
76	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
77	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
78	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
79	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₂₁₁	—	2	2	no	—
82	C₄×C₂₁₁	—	2	4	no	—
83	C₄×C₂₁₁	—	2	4	no	—
84	C₄×C₂₁₁	—	2	4	no	—
85	C₄×C₂₁₁	—	4	2	no	—
86	C₄×C₂₁₁	—	4	4	no	—
87	C₄×C₂₁₁	#117	1	4	yes	—
88	C₂×C₂₁₁	—	4	2	no	—
89	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
90	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
91	C₂×C₂₁₁	#520	1	1	yes	characteristic
92	C₂×C₂₁₁	#552	1	2	yes	—
93	C₂×C₂₁₁	#555	1	2	yes	—
94	C₂₁₁	#2346	1	1	yes	characteristic, direct product factor, Sylow
95	#2346	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
97	#389	C₂×C₂₁₁	1	2	yes	complemented
98	#388	C₂×C₂₁₁	1	4	yes	—
99	#382	C₂×C₂₁₁	1	2	yes	complemented
100	#380	C₂×C₂₁₁	1	1	yes	characteristic
101	C₄×C₄⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
102	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic
103	#376	C₂×C₂₁₁	1	2	yes	complemented
104	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
105	#117	C₄×C₂₁₁	1	8	yes	complemented
106	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
108	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
113	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	—
114	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
115	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
116	#95	C₂²×C₂₁₁	1	1	yes	characteristic
117	#95	C₂²×C₂₁₁	1	2	yes	—
118	#95	C₂²×C₂₁₁	1	4	yes	—
119	#95	C₂²×C₂₁₁	1	1	yes	characteristic
120	#95	C₂²×C₂₁₁	1	4	yes	—
121	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
124	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
125	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
126	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
128	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	4	no	—
133	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
134	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
138	C₄²	—	2	8	no	—
139	C₄²	—	2	4	no	—
140	C₄²	C₂×C₄×C₂₁₁	1	8	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
146	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	8	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
161	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
162	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
163	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
164	C₂×C₄	#41×C₂₁₁	1	4	yes	—
165	C₂×C₄	#41×C₂₁₁	1	4	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	4	2	no	—
170	C₂²	—	4	2	no	—
171	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
172	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
173	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
174	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
175	C₂²	2₋¹⁺⁴×C₂₁₁	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	4	no	—
179	C₄	—	2	4	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	4	no	—
182	C₄	#117×C₂₁₁	1	4	yes	—
183	C₂	—	4	2	no	—
184	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
185	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
186	C₂	#520×C₂₁₁	1	1	yes	characteristic
187	C₂	#552×C₂₁₁	1	2	yes	—
188	C₂	#555×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂⁴×C₂₁₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #389×C₂₁₁	C₂	semidirect
4. #382×C₂₁₁	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
8. #376×C₂₁₁	C₂	semidirect
10. #117×C₂₁₁	C₄	semidirect
94. C₂₁₁	#2346	direct (N × H)
95. #2346	C₂₁₁	direct (N × H)
97. #389	C₂×C₂₁₁	semidirect
99. #382	C₂×C₂₁₁	semidirect
101. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
103. #376	C₂×C₂₁₁	semidirect
105. #117	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₂₁₁	2
#389×C₂₁₁	2
#388×C₂₁₁	2
#382×C₂₁₁	2
#380×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#376×C₂₁₁	2
#2346	211

Subgroups — #2346×C211 #41009966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2346×C₂₁₁ #41009966