Subgroups — C₂×#339×C₂₁₁ #41009352

Order 27008.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#339×C₂₁₁	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₄×C₂₁₁	—	2	1	no	—
6	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₁₁	—	2	2	no	—
8	C₂×M₁₆×C₂₁₁	—	2	1	no	—
9	C₂×M₁₆×C₂₁₁	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₁₁	—	2	1	no	—
12	#117×C₂₁₁	—	2	1	no	—
13	#117×C₂₁₁	—	2	1	no	—
14	#117×C₂₁₁	C₄	1	2	yes	complemented
15	#117×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	complemented
17	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
18	C₂×C₄²×C₂₁₁	—	2	1	no	—
19	C₂×C₄²×C₂₁₁	—	2	2	no	—
20	C₂×C₄²×C₂₁₁	—	2	1	no	—
21	C₂×C₄²×C₂₁₁	C₄	1	4	yes	complemented
22	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
23	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₁₁	—	2	8	no	—
25	C₂²×C₄×C₂₁₁	—	2	4	no	—
26	C₂²×C₄×C₂₁₁	—	2	2	no	—
27	C₂²×C₄×C₂₁₁	—	2	2	no	—
28	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
34	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
37	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
38	M₁₆×C₂₁₁	—	2	4	no	—
39	M₁₆×C₂₁₁	—	2	2	no	—
40	M₁₆×C₂₁₁	C₂×C₄	1	4	yes	complemented
41	C₂×C₈×C₂₁₁	—	2	4	no	—
42	C₂×C₈×C₂₁₁	—	2	2	no	—
43	C₄⋊C₄×C₂₁₁	—	2	2	no	—
44	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
45	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
46	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
47	C₄²×C₂₁₁	—	2	4	no	—
48	C₄²×C₂₁₁	—	2	2	no	—
49	C₄²×C₂₁₁	—	2	2	no	—
50	C₄²×C₂₁₁	—	2	2	no	—
51	C₄²×C₂₁₁	C₂×C₄	1	4	yes	complemented
52	C₂³×C₂₁₁	—	2	4	no	—
53	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₂₁₁	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
57	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₁₁	—	2	16	no	—
59	C₂×C₄×C₂₁₁	—	2	8	no	—
60	C₂×C₄×C₂₁₁	—	2	4	no	—
61	C₂×C₄×C₂₁₁	—	2	4	no	—
62	C₂×C₄×C₂₁₁	—	4	4	no	—
63	C₂×C₄×C₂₁₁	C₄²	1	2	yes	—
64	C₂×C₄×C₂₁₁	C₄²	1	2	yes	—
65	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
81	C₈×C₂₁₁	—	2	4	no	—
82	C₂²×C₂₁₁	—	2	4	no	—
83	C₂²×C₂₁₁	—	2	4	no	—
84	C₂²×C₂₁₁	—	2	2	no	—
85	C₂²×C₂₁₁	—	2	2	no	—
86	C₂²×C₂₁₁	#95	1	2	yes	—
87	C₂²×C₂₁₁	#95	1	2	yes	—
88	C₂²×C₂₁₁	#104	1	4	yes	—
89	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₂₁₁	—	2	8	no	—
93	C₄×C₂₁₁	—	4	4	no	—
94	C₄×C₂₁₁	#95	1	2	yes	—
95	C₄×C₂₁₁	#95	1	2	yes	—
96	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
97	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₂₁₁	—	2	1	no	—
101	C₂×C₂₁₁	—	2	2	no	—
102	C₂×C₂₁₁	—	2	1	no	—
103	C₂×C₂₁₁	#339	1	4	yes	direct product factor
104	C₂×C₂₁₁	C₂×#95	1	1	yes	characteristic
105	C₂×C₂₁₁	C₂×#104	1	2	yes	—
106	C₂₁₁	C₂×#339	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×#339	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×M₁₆	C₂×C₂₁₁	1	1	yes	characteristic
109	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
110	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic
111	#339	C₂×C₂₁₁	1	4	yes	direct product factor
112	C₂³×C₄	—	2	1	no	—
113	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	C₄×C₂₁₁	1	4	yes	complemented
117	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
118	C₂²×C₈	—	2	1	no	—
119	#117	—	2	1	no	—
120	#117	—	2	1	no	—
121	#117	C₄×C₂₁₁	1	2	yes	complemented
122	#117	C₂²×C₂₁₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₄×C₂₁₁	1	2	yes	complemented
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₄×C₂₁₁	1	4	yes	complemented
129	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
130	C₂⁴	D₄×C₂₁₁	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
136	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
137	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
139	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
140	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
141	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
142	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
143	C₂²×C₄	Q₈×C₂₁₁	1	2	yes	—
144	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	C₂×C₄×C₂₁₁	1	4	yes	complemented
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	C₂×C₄×C₂₁₁	1	4	yes	complemented
159	C₂³	—	2	4	no	—
160	C₂³	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
161	C₂³	C₂²⋊C₄×C₂₁₁	1	2	yes	—
162	C₂³	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
163	C₂³	C₄⋊C₄×C₂₁₁	1	2	yes	—
164	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₄²×C₂₁₁	1	2	yes	—
171	C₂×C₄	C₄²×C₂₁₁	1	2	yes	—
172	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
173	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
174	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
175	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
176	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
177	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
178	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
179	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
180	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
181	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
182	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
187	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	#95×C₂₁₁	1	2	yes	—
194	C₂²	#95×C₂₁₁	1	2	yes	—
195	C₂²	#104×C₂₁₁	1	4	yes	—
196	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
197	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
198	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
199	C₄	—	2	8	no	—
200	C₄	—	4	4	no	—
201	C₄	#95×C₂₁₁	1	2	yes	—
202	C₄	#95×C₂₁₁	1	2	yes	—
203	C₄	C₂×C₄²×C₂₁₁	1	1	yes	characteristic
204	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
205	C₄	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
206	C₄	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#339×C₂₁₁	1	4	yes	direct product factor
211	C₂	C₂×#95×C₂₁₁	1	1	yes	characteristic
212	C₂	C₂×#104×C₂₁₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	D₄
commutator	C₄	C₂×C₄²×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂³×C₂₁₁	C₂²⋊C₄
prime radical	C₂⁴×C₂₁₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #339×C₂₁₁	C₂	direct (N × H)
9. C₂×M₁₆×C₂₁₁	C₄	semidirect
14. #117×C₂₁₁	C₄	semidirect
16. C₂×C₄⋊C₄×C₂₁₁	C₄	semidirect
21. C₂×C₄²×C₂₁₁	C₄	semidirect
40. M₁₆×C₂₁₁	C₂×C₄	semidirect
44. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
51. C₄²×C₂₁₁	C₂×C₄	semidirect
103. C₂×C₂₁₁	#339	direct (N × H)
106. C₂₁₁	C₂×#339	direct (N × H)
107. C₂×#339	C₂₁₁	direct (N × H)
111. #339	C₂×C₂₁₁	direct (N × H)
116. C₂×M₁₆	C₄×C₂₁₁	semidirect
121. #117	C₄×C₂₁₁	semidirect
123. C₂×C₄⋊C₄	C₄×C₂₁₁	semidirect
128. C₂×C₄²	C₄×C₂₁₁	semidirect
147. M₁₆	C₂×C₄×C₂₁₁	semidirect
151. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
158. C₄²	C₂×C₄×C₂₁₁	semidirect
210. C₂	#339×C₂₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₂₁₁	2
C₂×#117×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
#339×C₂₁₁	2
C₂×#339	211

Subgroups — C2×#339×C211 #41009352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#339×C₂₁₁ #41009352