Subgroups — C₂×#390×C₂₁₁ #41009912

Order 27008.

1360 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂²⋊C₄×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₂₁₁	C₂	1	8	yes	maximal, direct product factor
6	C₂⁵×C₂₁₁	C₂²	1	2	yes	—
7	C₂²×D₄×C₂₁₁	—	2	4	no	—
8	C₂²×D₄×C₂₁₁	—	2	4	no	—
9	C₂²×D₄×C₂₁₁	C₄	1	8	yes	complemented
10	C₂²×D₄×C₂₁₁	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₂₁₁	—	2	16	no	—
17	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	16	yes	complemented
18	C₂×C₄²×C₂₁₁	—	2	4	no	—
19	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	complemented
20	C₂⁴×C₂₁₁	—	2	8	no	—
21	C₂⁴×C₂₁₁	—	2	4	no	—
22	C₂⁴×C₂₁₁	—	2	4	no	—
23	C₂⁴×C₂₁₁	—	2	8	no	—
24	C₂⁴×C₂₁₁	C₂×C₄	1	8	yes	—
25	C₂⁴×C₂₁₁	C₂³	1	4	yes	—
26	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂×D₄×C₂₁₁	—	2	32	no	—
28	C₂×D₄×C₂₁₁	—	2	8	no	—
29	C₂×D₄×C₂₁₁	—	2	8	no	—
30	C₂×D₄×C₂₁₁	C₂×C₄	1	16	yes	complemented
31	C₂²×C₄×C₂₁₁	—	2	4	no	—
32	C₂²×C₄×C₂₁₁	—	2	4	no	—
33	C₂²×C₄×C₂₁₁	—	2	8	no	—
34	C₂²×C₄×C₂₁₁	—	2	4	no	—
35	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
37	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
38	C₂²×C₄×C₂₁₁	D₄	1	8	yes	complemented
39	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
42	C₄⋊C₄×C₂₁₁	—	2	8	no	—
43	C₂²⋊C₄×C₂₁₁	—	2	32	no	—
44	C₄²×C₂₁₁	—	2	8	no	—
45	C₂³×C₂₁₁	—	2	16	no	—
46	C₂³×C₂₁₁	—	2	32	no	—
47	C₂³×C₂₁₁	—	2	8	no	—
48	C₂³×C₂₁₁	—	2	4	no	—
49	C₂³×C₂₁₁	—	2	4	no	—
50	C₂³×C₂₁₁	—	2	8	no	—
51	C₂³×C₂₁₁	—	4	16	no	—
52	C₂³×C₂₁₁	—	4	8	no	—
53	C₂³×C₂₁₁	—	4	8	no	—
54	C₂³×C₂₁₁	C₂²×C₄	1	8	yes	—
55	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
56	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
57	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
58	C₂³×C₂₁₁	C₂×D₄	1	4	yes	—
59	C₂³×C₂₁₁	#41	1	4	yes	—
60	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
61	D₄×C₂₁₁	—	2	32	no	—
62	C₂×C₄×C₂₁₁	—	2	16	no	—
63	C₂×C₄×C₂₁₁	—	2	4	no	—
64	C₂×C₄×C₂₁₁	—	2	4	no	—
65	C₂×C₄×C₂₁₁	—	2	8	no	—
66	C₂×C₄×C₂₁₁	—	2	4	no	—
67	C₂×C₄×C₂₁₁	—	4	8	no	—
68	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	16	yes	complemented
69	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₂₁₁	C₂×D₄	1	8	yes	complemented
73	C₂²×C₂₁₁	—	2	8	no	—
74	C₂²×C₂₁₁	—	2	16	no	—
75	C₂²×C₂₁₁	—	4	8	no	—
76	C₂²×C₂₁₁	—	4	4	no	—
77	C₂²×C₂₁₁	—	4	4	no	—
78	C₂²×C₂₁₁	—	4	32	no	—
79	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
80	C₂²×C₂₁₁	C₄×D₄	1	8	yes	—
81	C₂²×C₂₁₁	#121	1	8	yes	—
82	C₂²×C₂₁₁	#124	1	4	yes	—
83	C₂²×C₂₁₁	#127	1	4	yes	—
84	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₂₁₁	C₂²×D₄	1	2	yes	—
88	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
89	C₄×C₂₁₁	—	2	8	no	—
90	C₄×C₂₁₁	—	4	4	no	—
91	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	8	yes	complemented
92	C₂×C₂₁₁	—	4	8	no	—
93	C₂×C₂₁₁	#390	1	8	yes	direct product factor
94	C₂×C₂₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₂₁₁	C₂×C₄×D₄	1	2	yes	—
96	C₂×C₂₁₁	C₂×#121	1	2	yes	—
97	C₂×C₂₁₁	C₂×#124	1	1	yes	characteristic
98	C₂×C₂₁₁	C₂×#127	1	1	yes	characteristic
99	C₂₁₁	C₂×#390	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×#390	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×D₄	C₂×C₂₁₁	1	1	yes	characteristic
102	C₂²×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
103	C₂²×C₂²⋊C₄	C₂×C₂₁₁	1	4	yes	complemented
104	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic, complemented
105	#390	C₂×C₂₁₁	1	8	yes	direct product factor
106	C₂⁵	C₂²×C₂₁₁	1	2	yes	—
107	C₂²×D₄	—	2	4	no	—
108	C₂²×D₄	—	2	4	no	—
109	C₂²×D₄	C₄×C₂₁₁	1	8	yes	complemented
110	C₂²×D₄	C₂²×C₂₁₁	1	4	yes	—
111	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
112	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
113	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	complemented
116	C₂×C₂²⋊C₄	—	2	16	no	—
117	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	16	yes	complemented
118	C₂×C₄²	—	2	4	no	—
119	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	complemented
120	C₂⁴	—	2	8	no	—
121	C₂⁴	—	2	4	no	—
122	C₂⁴	—	2	4	no	—
123	C₂⁴	—	2	8	no	—
124	C₂⁴	C₂×C₄×C₂₁₁	1	8	yes	—
125	C₂⁴	C₂³×C₂₁₁	1	4	yes	—
126	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
127	C₂×D₄	—	2	32	no	—
128	C₂×D₄	—	2	8	no	—
129	C₂×D₄	—	2	8	no	—
130	C₂×D₄	C₂×C₄×C₂₁₁	1	16	yes	complemented
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
136	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
137	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
138	C₂²×C₄	D₄×C₂₁₁	1	8	yes	complemented
139	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
140	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
141	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
142	C₄⋊C₄	—	2	8	no	—
143	C₂²⋊C₄	—	2	32	no	—
144	C₄²	—	2	8	no	—
145	C₂³	—	2	16	no	—
146	C₂³	—	2	32	no	—
147	C₂³	—	2	8	no	—
148	C₂³	—	2	4	no	—
149	C₂³	—	2	4	no	—
150	C₂³	—	2	8	no	—
151	C₂³	—	4	16	no	—
152	C₂³	—	4	8	no	—
153	C₂³	—	4	8	no	—
154	C₂³	C₂²×C₄×C₂₁₁	1	8	yes	—
155	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
156	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
157	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
158	C₂³	C₂×D₄×C₂₁₁	1	4	yes	—
159	C₂³	#41×C₂₁₁	1	4	yes	—
160	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
161	D₄	—	2	32	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	16	yes	complemented
169	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
170	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
171	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
172	C₂×C₄	C₂×D₄×C₂₁₁	1	8	yes	complemented
173	C₂²	—	2	8	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	4	8	no	—
176	C₂²	—	4	4	no	—
177	C₂²	—	4	4	no	—
178	C₂²	—	4	32	no	—
179	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
180	C₂²	C₄×D₄×C₂₁₁	1	8	yes	—
181	C₂²	#121×C₂₁₁	1	8	yes	—
182	C₂²	#124×C₂₁₁	1	4	yes	—
183	C₂²	#127×C₂₁₁	1	4	yes	—
184	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
185	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
186	C₂²	C₂²×D₄×C₂₁₁	1	2	yes	—
187	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
188	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
189	C₄	—	2	8	no	—
190	C₄	—	4	4	no	—
191	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	8	yes	complemented
192	C₂	—	4	8	no	—
193	C₂	#390×C₂₁₁	1	8	yes	direct product factor
194	C₂	C₂²×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
195	C₂	C₂×C₄×D₄×C₂₁₁	1	2	yes	—
196	C₂	C₂×#121×C₂₁₁	1	2	yes	—
197	C₂	C₂×#124×C₂₁₁	1	1	yes	characteristic
198	C₂	C₂×#127×C₂₁₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₂₁₁	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₂₁₁	C₂	semidirect
4. C₂²×C₄²×C₂₁₁	C₂	semidirect
5. #390×C₂₁₁	C₂	direct (N × H)
9. C₂²×D₄×C₂₁₁	C₄	semidirect
15. C₂×C₄⋊C₄×C₂₁₁	C₂²	semidirect
17. C₂×C₂²⋊C₄×C₂₁₁	C₂²	semidirect
19. C₂×C₄²×C₂₁₁	C₂²	semidirect
30. C₂×D₄×C₂₁₁	C₂×C₄	semidirect
38. C₂²×C₄×C₂₁₁	D₄	semidirect
68. C₂×C₄×C₂₁₁	C₂²⋊C₄	semidirect
72. C₂×C₄×C₂₁₁	C₂×D₄	semidirect
91. C₄×C₂₁₁	C₂×C₂²⋊C₄	semidirect
93. C₂×C₂₁₁	#390	direct (N × H)
99. C₂₁₁	C₂×#390	direct (N × H)
100. C₂×#390	C₂₁₁	direct (N × H)
102. C₂²×C₄⋊C₄	C₂×C₂₁₁	semidirect
103. C₂²×C₂²⋊C₄	C₂×C₂₁₁	semidirect
104. C₂²×C₄²	C₂×C₂₁₁	semidirect
105. #390	C₂×C₂₁₁	direct (N × H)
109. C₂²×D₄	C₄×C₂₁₁	semidirect
115. C₂×C₄⋊C₄	C₂²×C₂₁₁	semidirect
117. C₂×C₂²⋊C₄	C₂²×C₂₁₁	semidirect
119. C₂×C₄²	C₂²×C₂₁₁	semidirect
130. C₂×D₄	C₂×C₄×C₂₁₁	semidirect
138. C₂²×C₄	D₄×C₂₁₁	semidirect
168. C₂×C₄	C₂²⋊C₄×C₂₁₁	semidirect
172. C₂×C₄	C₂×D₄×C₂₁₁	semidirect
191. C₄	C₂×C₂²⋊C₄×C₂₁₁	semidirect
193. C₂	#390×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
C₂³×D₄×C₂₁₁	2
C₂²×C₄⋊C₄×C₂₁₁	2
C₂²×C₂²⋊C₄×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
#390×C₂₁₁	2
C₂×#390	211

Subgroups — C2×#390×C211 #41009912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#390×C₂₁₁ #41009912