Subgroups — #2293×C₂₁₁ #41009913

Order 27008.

652 conjugacy classes of subgroups.

The subgroup lattice has 206 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	C₂×#117×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#390×C₂₁₁	C₂	1	2	yes	maximal
4	#389×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#388×C₂₁₁	C₂	1	4	yes	maximal, complemented
6	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	C₄³×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₄×D₄×C₂₁₁	C₄	1	8	yes	complemented
12	#117×C₂₁₁	—	2	8	no	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
25	#95×C₂₁₁	C₂²	1	4	yes	—
26	#95×C₂₁₁	C₂²	1	2	yes	—
27	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
28	C₂×D₄×C₂₁₁	—	2	4	no	—
29	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₁₁	—	2	4	no	—
31	C₂²×C₄×C₂₁₁	—	2	4	no	—
32	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	—
33	C₂²×C₄×C₂₁₁	C₂³	1	4	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	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₂₁₁	—	2	8	no	—
41	C₄⋊C₄×C₂₁₁	—	2	4	no	—
42	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
43	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
44	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
45	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
46	C₄²×C₂₁₁	—	2	4	no	—
47	C₄²×C₂₁₁	—	2	4	no	—
48	C₄²×C₂₁₁	—	2	4	no	—
49	C₄²×C₂₁₁	—	2	2	no	—
50	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
51	C₄²×C₂₁₁	D₄	1	4	yes	complemented
52	C₂³×C₂₁₁	—	2	2	no	—
53	C₂³×C₂₁₁	—	2	4	no	—
54	C₂³×C₂₁₁	—	2	2	no	—
55	C₂³×C₂₁₁	—	2	4	no	—
56	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
57	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
58	D₄×C₂₁₁	—	2	8	no	—
59	C₂×C₄×C₂₁₁	—	2	2	no	—
60	C₂×C₄×C₂₁₁	—	2	2	no	—
61	C₂×C₄×C₂₁₁	—	2	2	no	—
62	C₂×C₄×C₂₁₁	—	2	2	no	—
63	C₂×C₄×C₂₁₁	—	2	2	no	—
64	C₂×C₄×C₂₁₁	—	2	2	no	—
65	C₂×C₄×C₂₁₁	—	2	2	no	—
66	C₂×C₄×C₂₁₁	—	2	16	no	—
67	C₂×C₄×C₂₁₁	—	2	4	no	—
68	C₂×C₄×C₂₁₁	—	2	4	no	—
69	C₂×C₄×C₂₁₁	—	2	4	no	—
70	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₂₁₁	#41	1	4	yes	—
77	C₂×C₄×C₂₁₁	#41	1	4	yes	—
78	C₂×C₄×C₂₁₁	#41	1	2	yes	—
79	C₂²×C₂₁₁	—	2	4	no	—
80	C₂²×C₂₁₁	—	2	4	no	—
81	C₂²×C₂₁₁	—	2	4	no	—
82	C₂²×C₂₁₁	—	4	4	no	—
83	C₂²×C₂₁₁	—	4	4	no	—
84	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
87	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
88	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
89	C₄×C₂₁₁	—	2	4	no	—
90	C₄×C₂₁₁	—	2	2	no	—
91	C₄×C₂₁₁	—	2	4	no	—
92	C₄×C₂₁₁	—	2	4	no	—
93	C₄×C₂₁₁	—	4	4	no	—
94	C₄×C₂₁₁	#117	1	4	yes	complemented
95	C₄×C₂₁₁	C₄×D₄	1	4	yes	—
96	C₂×C₂₁₁	—	4	4	no	—
97	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
98	C₂×C₂₁₁	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
100	C₂×C₂₁₁	#529	1	2	yes	—
101	C₂×C₂₁₁	#532	1	2	yes	—
102	C₂₁₁	#2293	1	1	yes	characteristic, direct product factor, Sylow
103	#2293	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₄×D₄	C₂×C₂₁₁	1	1	yes	characteristic
105	C₂×#117	C₂×C₂₁₁	1	2	yes	complemented
106	#390	C₂×C₂₁₁	1	2	yes	—
107	#389	C₂×C₂₁₁	1	2	yes	complemented
108	#388	C₂×C₂₁₁	1	4	yes	complemented
109	C₄×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
110	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
111	C₄³	C₂×C₂₁₁	1	1	yes	characteristic, complemented
112	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
114	C₄×D₄	C₄×C₂₁₁	1	8	yes	complemented
115	#117	—	2	8	no	—
116	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
117	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
122	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
123	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
125	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
126	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
128	#95	C₂²×C₂₁₁	1	4	yes	—
129	#95	C₂²×C₂₁₁	1	2	yes	—
130	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
131	C₂×D₄	—	2	4	no	—
132	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	—
136	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
137	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
139	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
140	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
141	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
146	C₂²⋊C₄	—	2	8	no	—
147	C₂²⋊C₄	—	2	8	no	—
148	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
154	C₄²	D₄×C₂₁₁	1	4	yes	complemented
155	C₂³	—	2	2	no	—
156	C₂³	—	2	4	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	4	no	—
159	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
160	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
161	D₄	—	2	8	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
175	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
176	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
177	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
178	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
179	C₂×C₄	#41×C₂₁₁	1	4	yes	—
180	C₂×C₄	#41×C₂₁₁	1	4	yes	—
181	C₂×C₄	#41×C₂₁₁	1	2	yes	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	4	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
188	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
189	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
190	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
191	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	4	no	—
197	C₄	#117×C₂₁₁	1	4	yes	complemented
198	C₄	C₄×D₄×C₂₁₁	1	4	yes	—
199	C₂	—	4	4	no	—
200	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
201	C₂	C₂×C₄×D₄×C₂₁₁	1	1	yes	characteristic
202	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
203	C₂	#529×C₂₁₁	1	2	yes	—
204	C₂	#532×C₂₁₁	1	2	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	C₂²×D₄×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
4. #389×C₂₁₁	C₂	semidirect
5. #388×C₂₁₁	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
7. C₄×C₂²⋊C₄×C₂₁₁	C₂	semidirect
8. C₄³×C₂₁₁	C₂	semidirect
11. C₄×D₄×C₂₁₁	C₄	semidirect
51. C₄²×C₂₁₁	D₄	semidirect
94. C₄×C₂₁₁	#117	semidirect
102. C₂₁₁	#2293	direct (N × H)
103. #2293	C₂₁₁	direct (N × H)
105. C₂×#117	C₂×C₂₁₁	semidirect
107. #389	C₂×C₂₁₁	semidirect
108. #388	C₂×C₂₁₁	semidirect
109. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
110. C₄×C₂²⋊C₄	C₂×C₂₁₁	semidirect
111. C₄³	C₂×C₂₁₁	semidirect
114. C₄×D₄	C₄×C₂₁₁	semidirect
154. C₄²	D₄×C₂₁₁	semidirect
197. C₄	#117×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
C₂×#117×C₂₁₁	2
#390×C₂₁₁	2
#389×C₂₁₁	2
#388×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
C₄³×C₂₁₁	2
#2293	211

Subgroups — #2293×C211 #41009913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2293×C₂₁₁ #41009913