Subgroups — #2973×C₂₁₁ #41010593

Order 27008.

456 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	C₂×C₄×D₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#435×C₂₁₁	C₂	1	8	yes	maximal, complemented
3	#432×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#407×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₂₁₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₁₁	C₂²	1	4	yes	complemented
10	C₂²×C₈×C₂₁₁	C₂²	1	4	yes	complemented
11	C₄×D₄×C₂₁₁	C₄	1	4	yes	—
12	C₄×D₄×C₂₁₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₂₁₁	C₂²	1	4	yes	complemented
18	C₂²⋊C₈×C₂₁₁	C₂²	1	4	yes	—
19	C₂²⋊C₈×C₂₁₁	C₂²	1	4	yes	complemented
20	C₂²⋊C₈×C₂₁₁	C₂²	1	4	yes	complemented
21	#97×C₂₁₁	C₂²	1	4	yes	complemented
22	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₂₁₁	—	2	4	no	—
25	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₁₁	—	4	2	no	—
27	C₂²×C₄×C₂₁₁	—	4	2	no	—
28	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
37	M₁₆×C₂₁₁	—	4	4	no	—
38	C₂×C₈×C₂₁₁	—	4	4	no	—
39	C₂×C₈×C₂₁₁	C₂³	1	8	yes	—
40	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
42	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
43	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
44	C₄²×C₂₁₁	C₂³	1	2	yes	—
45	C₂³×C₂₁₁	—	2	2	no	—
46	C₂³×C₂₁₁	—	4	1	no	—
47	C₂³×C₂₁₁	—	4	1	no	—
48	C₂³×C₂₁₁	—	4	1	no	—
49	C₂³×C₂₁₁	—	4	1	no	—
50	C₂³×C₂₁₁	M₁₆	1	4	yes	complemented
51	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
52	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
53	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
54	D₄×C₂₁₁	—	2	8	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	—	2	4	no	—
57	C₂×C₄×C₂₁₁	—	2	4	no	—
58	C₂×C₄×C₂₁₁	—	2	2	no	—
59	C₂×C₄×C₂₁₁	—	4	2	no	—
60	C₂×C₄×C₂₁₁	—	4	2	no	—
61	C₂×C₄×C₂₁₁	—	4	1	no	—
62	C₂×C₄×C₂₁₁	—	4	1	no	—
63	C₂×C₄×C₂₁₁	—	4	1	no	—
64	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
70	C₈×C₂₁₁	—	2	8	no	—
71	C₂²×C₂₁₁	—	2	1	no	—
72	C₂²×C₂₁₁	—	2	2	no	—
73	C₂²×C₂₁₁	—	2	1	no	—
74	C₂²×C₂₁₁	—	2	2	no	—
75	C₂²×C₂₁₁	—	2	2	no	—
76	C₂²×C₂₁₁	—	2	2	no	—
77	C₂²×C₂₁₁	—	2	2	no	—
78	C₂²×C₂₁₁	—	4	4	no	—
79	C₂²×C₂₁₁	—	4	4	no	—
80	C₂²×C₂₁₁	C₂×M₁₆	1	2	yes	—
81	C₂²×C₂₁₁	C₂×M₁₆	1	4	yes	—
82	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₂₁₁	—	2	1	no	—
84	C₄×C₂₁₁	—	2	2	no	—
85	C₄×C₂₁₁	—	2	2	no	—
86	C₄×C₂₁₁	—	2	2	no	—
87	C₄×C₂₁₁	—	2	2	no	—
88	C₄×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
89	C₂×C₂₁₁	—	2	2	no	—
90	C₂×C₂₁₁	—	2	4	no	—
91	C₂×C₂₁₁	—	4	2	no	—
92	C₂×C₂₁₁	#518	1	1	yes	characteristic
93	C₂×C₂₁₁	C₂²×M₁₆	1	1	yes	characteristic
94	C₂×C₂₁₁	#568	1	1	yes	characteristic
95	C₂₁₁	#2973	1	1	yes	characteristic, direct product factor, Sylow
96	#2973	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×C₄×D₄	C₂×C₂₁₁	1	1	yes	characteristic
98	#435	C₂×C₂₁₁	1	8	yes	complemented
99	#432	C₂×C₂₁₁	1	2	yes	complemented
100	#407	C₂×C₂₁₁	1	2	yes	complemented
101	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	2	yes	complemented
102	C₂²×D₄	C₄×C₂₁₁	1	1	yes	characteristic
103	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
104	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
105	C₂×M₁₆	C₂²×C₂₁₁	1	4	yes	complemented
106	C₂²×C₈	C₂²×C₂₁₁	1	4	yes	complemented
107	C₄×D₄	C₄×C₂₁₁	1	4	yes	—
108	C₄×D₄	C₂²×C₂₁₁	1	4	yes	—
109	C₂×C₄⋊C₄	C₄×C₂₁₁	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	C₄×C₂₁₁	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	C₄×C₂₁₁	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
113	C₄⋊C₈	C₂²×C₂₁₁	1	4	yes	complemented
114	C₂²⋊C₈	C₂²×C₂₁₁	1	4	yes	—
115	C₂²⋊C₈	C₂²×C₂₁₁	1	4	yes	complemented
116	C₂²⋊C₈	C₂²×C₂₁₁	1	4	yes	complemented
117	#97	C₂²×C₂₁₁	1	4	yes	complemented
118	C₂⁴	C₂×C₄×C₂₁₁	1	1	yes	characteristic
119	C₂⁴	C₂×C₄×C₂₁₁	1	1	yes	characteristic
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
129	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
132	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
133	M₁₆	—	4	4	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	C₂³×C₂₁₁	1	8	yes	—
136	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
137	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
138	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
139	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
140	C₄²	C₂³×C₂₁₁	1	2	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	1	no	—
146	C₂³	M₁₆×C₂₁₁	1	4	yes	complemented
147	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
148	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
149	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
150	D₄	—	2	8	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
162	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
163	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
164	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
165	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
166	C₈	—	2	8	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂×M₁₆×C₂₁₁	1	2	yes	—
177	C₂²	C₂×M₁₆×C₂₁₁	1	4	yes	—
178	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	2	no	—
188	C₂	#518×C₂₁₁	1	1	yes	characteristic
189	C₂	C₂²×M₁₆×C₂₁₁	1	1	yes	characteristic
190	C₂	#568×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₁₁
socle	C₂²×C₂₁₁	C₂³×C₄
prime radical	C₂²×D₄×C₂₁₁	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #435×C₂₁₁	C₂	semidirect
3. #432×C₂₁₁	C₂	semidirect
4. #407×C₂₁₁	C₂	semidirect
5. C₂×C₂²⋊C₈×C₂₁₁	C₂	semidirect
9. C₂×M₁₆×C₂₁₁	C₂²	semidirect
10. C₂²×C₈×C₂₁₁	C₂²	semidirect
17. C₄⋊C₈×C₂₁₁	C₂²	semidirect
19. C₂²⋊C₈×C₂₁₁	C₂²	semidirect
20. C₂²⋊C₈×C₂₁₁	C₂²	semidirect
21. #97×C₂₁₁	C₂²	semidirect
50. C₂³×C₂₁₁	M₁₆	semidirect
95. C₂₁₁	#2973	direct (N × H)
96. #2973	C₂₁₁	direct (N × H)
98. #435	C₂×C₂₁₁	semidirect
99. #432	C₂×C₂₁₁	semidirect
100. #407	C₂×C₂₁₁	semidirect
101. C₂×C₂²⋊C₈	C₂×C₂₁₁	semidirect
105. C₂×M₁₆	C₂²×C₂₁₁	semidirect
106. C₂²×C₈	C₂²×C₂₁₁	semidirect
113. C₄⋊C₈	C₂²×C₂₁₁	semidirect
115. C₂²⋊C₈	C₂²×C₂₁₁	semidirect
116. C₂²⋊C₈	C₂²×C₂₁₁	semidirect
117. #97	C₂²×C₂₁₁	semidirect
146. C₂³	M₁₆×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₁₁	2
#435×C₂₁₁	2
#432×C₂₁₁	2
#407×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
#2973	211

Subgroups — #2973×C211 #41010593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2973×C₂₁₁ #41010593