Subgroups — #2280×C₂₁₁ #41009900

Order 27008.

1088 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#381×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#380×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#379×C₂₁₁	C₂	1	2	yes	maximal
6	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	4	yes	maximal, complemented
7	C₂⁵×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₁₁	—	2	8	no	—
9	C₂³×C₄×C₂₁₁	C₄	1	8	yes	complemented
10	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	#117×C₂₁₁	—	2	8	no	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	8	yes	—
17	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
18	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	complemented
19	#95×C₂₁₁	C₂²	1	8	yes	complemented
20	C₂⁴×C₂₁₁	—	2	4	no	—
21	C₂⁴×C₂₁₁	—	2	8	no	—
22	C₂⁴×C₂₁₁	C₂×C₄	1	4	yes	—
23	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
25	C₂²×C₄×C₂₁₁	—	2	4	no	—
26	C₂²×C₄×C₂₁₁	—	2	32	no	—
27	C₂²×C₄×C₂₁₁	—	2	8	no	—
28	C₂²×C₄×C₂₁₁	—	2	8	no	—
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	—
32	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₂₁₁	D₄	1	8	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	8	yes	—
35	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₁₁	—	2	8	no	—
39	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
40	C₂²⋊C₄×C₂₁₁	—	2	16	no	—
41	C₄²×C₂₁₁	—	2	8	no	—
42	C₂³×C₂₁₁	—	2	32	no	—
43	C₂³×C₂₁₁	—	2	8	no	—
44	C₂³×C₂₁₁	—	2	16	no	—
45	C₂³×C₂₁₁	—	2	4	no	—
46	C₂³×C₂₁₁	—	2	4	no	—
47	C₂³×C₂₁₁	—	2	4	no	—
48	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
50	C₂³×C₂₁₁	C₂×D₄	1	4	yes	—
51	C₂³×C₂₁₁	#41	1	8	yes	—
52	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₂₁₁	—	2	8	no	—
54	C₂×C₄×C₂₁₁	—	2	4	no	—
55	C₂×C₄×C₂₁₁	—	2	16	no	—
56	C₂×C₄×C₂₁₁	—	2	16	no	—
57	C₂×C₄×C₂₁₁	—	2	8	no	—
58	C₂×C₄×C₂₁₁	—	2	16	no	—
59	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	16	yes	complemented
60	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
65	C₂²×C₂₁₁	—	2	32	no	—
66	C₂²×C₂₁₁	—	2	4	no	—
67	C₂²×C₂₁₁	—	2	4	no	—
68	C₂²×C₂₁₁	—	2	4	no	—
69	C₂²×C₂₁₁	—	2	8	no	—
70	C₂²×C₂₁₁	—	2	16	no	—
71	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
72	C₂²×C₂₁₁	#117	1	8	yes	complemented
73	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₂₁₁	C₂²×D₄	1	2	yes	—
75	C₂²×C₂₁₁	C₂×#41	1	4	yes	—
76	C₄×C₂₁₁	—	2	4	no	—
77	C₄×C₂₁₁	—	4	8	no	—
78	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
79	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
80	C₂×C₂₁₁	—	2	4	no	—
81	C₂×C₂₁₁	—	2	8	no	—
82	C₂×C₂₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₂₁₁	C₂×#117	1	2	yes	—
84	C₂×C₂₁₁	#525	1	4	yes	—
85	C₂₁₁	#2280	1	1	yes	characteristic, direct product factor, Sylow
86	#2280	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂⁴×C₄	C₂×C₂₁₁	1	1	yes	characteristic
88	C₂×#117	C₂×C₂₁₁	1	2	yes	complemented
89	#381	C₂×C₂₁₁	1	2	yes	complemented
90	#380	C₂×C₂₁₁	1	4	yes	complemented
91	#379	C₂×C₂₁₁	1	2	yes	—
92	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	4	yes	complemented
93	C₂⁵	C₂²×C₂₁₁	1	1	yes	characteristic
94	C₂³×C₄	—	2	8	no	—
95	C₂³×C₄	C₄×C₂₁₁	1	8	yes	complemented
96	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
97	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
98	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
100	#117	—	2	8	no	—
101	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	complemented
102	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	8	yes	—
104	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	complemented
105	#95	C₂²×C₂₁₁	1	8	yes	complemented
106	C₂⁴	—	2	4	no	—
107	C₂⁴	—	2	8	no	—
108	C₂⁴	C₂×C₄×C₂₁₁	1	4	yes	—
109	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
110	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	32	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
116	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	—
117	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	—
118	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
119	C₂²×C₄	D₄×C₂₁₁	1	8	yes	—
120	C₂²×C₄	C₂³×C₂₁₁	1	8	yes	—
121	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
122	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
124	C₄⋊C₄	—	2	8	no	—
125	C₂²⋊C₄	—	2	8	no	—
126	C₂²⋊C₄	—	2	16	no	—
127	C₄²	—	2	8	no	—
128	C₂³	—	2	32	no	—
129	C₂³	—	2	8	no	—
130	C₂³	—	2	16	no	—
131	C₂³	—	2	4	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	2	4	no	—
134	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
135	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
136	C₂³	C₂×D₄×C₂₁₁	1	4	yes	—
137	C₂³	#41×C₂₁₁	1	8	yes	—
138	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	16	no	—
142	C₂×C₄	—	2	16	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	16	no	—
145	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	16	yes	complemented
146	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
147	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
148	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
149	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
150	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
151	C₂²	—	2	32	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
158	C₂²	#117×C₂₁₁	1	8	yes	complemented
159	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
160	C₂²	C₂²×D₄×C₂₁₁	1	2	yes	—
161	C₂²	C₂×#41×C₂₁₁	1	4	yes	—
162	C₄	—	2	4	no	—
163	C₄	—	4	8	no	—
164	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
165	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
166	C₂	—	2	4	no	—
167	C₂	—	2	8	no	—
168	C₂	C₂²×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
169	C₂	C₂×#117×C₂₁₁	1	2	yes	—
170	C₂	#525×C₂₁₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
socle	C₂³×C₂₁₁	C₂⁴
prime radical	C₂⁵×C₂₁₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₂₁₁	C₂	semidirect
3. #381×C₂₁₁	C₂	semidirect
4. #380×C₂₁₁	C₂	semidirect
6. C₄×C₂²⋊C₄×C₂₁₁	C₂	semidirect
9. C₂³×C₄×C₂₁₁	C₄	semidirect
15. C₂×C₄⋊C₄×C₂₁₁	C₂²	semidirect
18. C₂×C₄²×C₂₁₁	C₂²	semidirect
19. #95×C₂₁₁	C₂²	semidirect
59. C₂×C₄×C₂₁₁	C₂²⋊C₄	semidirect
72. C₂²×C₂₁₁	#117	semidirect
85. C₂₁₁	#2280	direct (N × H)
86. #2280	C₂₁₁	direct (N × H)
88. C₂×#117	C₂×C₂₁₁	semidirect
89. #381	C₂×C₂₁₁	semidirect
90. #380	C₂×C₂₁₁	semidirect
92. C₄×C₂²⋊C₄	C₂×C₂₁₁	semidirect
95. C₂³×C₄	C₄×C₂₁₁	semidirect
101. C₂×C₄⋊C₄	C₂²×C₂₁₁	semidirect
104. C₂×C₄²	C₂²×C₂₁₁	semidirect
105. #95	C₂²×C₂₁₁	semidirect
145. C₂×C₄	C₂²⋊C₄×C₂₁₁	semidirect
158. C₂²	#117×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₂₁₁	2
C₂×#117×C₂₁₁	2
#381×C₂₁₁	2
#380×C₂₁₁	2
#379×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#2280	211

Subgroups — #2280×C211 #41009900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2280×C₂₁₁ #41009900