Subgroups — #2317×C₂₁₁ #41009937

Order 27008.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#389×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#385×C₂₁₁	C₂	1	4	yes	maximal
6	#384×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	#382×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
9	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	#117×C₂₁₁	—	2	4	no	—
11	#117×C₂₁₁	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₂₁₁	—	2	4	no	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	8	yes	—
18	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
23	#95×C₂₁₁	C₂²	1	8	yes	—
24	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₁₁	—	2	4	no	—
26	C₂²×C₄×C₂₁₁	—	4	4	no	—
27	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
29	C₂²×C₄×C₂₁₁	Q₈	1	4	yes	—
30	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₁₁	—	2	4	no	—
37	C₄⋊C₄×C₂₁₁	—	2	4	no	—
38	C₄⋊C₄×C₂₁₁	—	2	2	no	—
39	C₄⋊C₄×C₂₁₁	—	2	2	no	—
40	C₄⋊C₄×C₂₁₁	—	4	8	no	—
41	C₄⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
42	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
43	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
44	C₄²×C₂₁₁	—	2	4	no	—
45	C₄²×C₂₁₁	C₂×C₄	1	8	yes	complemented
46	C₂³×C₂₁₁	—	2	2	no	—
47	C₂³×C₂₁₁	—	2	2	no	—
48	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
50	C₂³×C₂₁₁	C₂×Q₈	1	2	yes	—
51	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	—	2	4	no	—
53	C₂×C₄×C₂₁₁	—	2	4	no	—
54	C₂×C₄×C₂₁₁	—	2	2	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	8	no	—
59	C₂×C₄×C₂₁₁	—	2	4	no	—
60	C₂×C₄×C₂₁₁	—	4	8	no	—
61	C₂×C₄×C₂₁₁	C₄⋊C₄	1	16	yes	complemented
62	C₂×C₄×C₂₁₁	C₂²×C₄	1	8	yes	—
63	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₂₁₁	—	2	4	no	—
70	C₂²×C₂₁₁	—	2	2	no	—
71	C₂²×C₂₁₁	—	2	2	no	—
72	C₂²×C₂₁₁	—	2	4	no	—
73	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
79	C₄×C₂₁₁	—	2	8	no	—
80	C₄×C₂₁₁	—	4	2	no	—
81	C₄×C₂₁₁	—	4	4	no	—
82	C₄×C₂₁₁	—	4	2	no	—
83	C₄×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
84	C₄×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₂₁₁	—	2	4	no	—
86	C₂×C₂₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₂₁₁	#520	1	2	yes	—
88	C₂×C₂₁₁	#535	1	1	yes	characteristic
89	C₂×C₂₁₁	#536	1	1	yes	characteristic
90	C₂×C₂₁₁	#544	1	2	yes	—
91	C₂₁₁	#2317	1	1	yes	characteristic, direct product factor, Sylow
92	#2317	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
94	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
95	C₂²×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic
96	#389	C₂×C₂₁₁	1	4	yes	complemented
97	#385	C₂×C₂₁₁	1	4	yes	—
98	#384	C₂×C₂₁₁	1	2	yes	complemented
99	#382	C₂×C₂₁₁	1	2	yes	complemented
100	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
101	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
102	#117	—	2	4	no	—
103	#117	C₄×C₂₁₁	1	8	yes	complemented
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	8	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
113	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
114	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
115	#95	C₂²×C₂₁₁	1	8	yes	—
116	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
120	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
121	C₂²×C₄	Q₈×C₂₁₁	1	4	yes	—
122	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
123	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
124	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
125	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
126	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	4	8	no	—
133	C₄⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
136	C₄²	—	2	4	no	—
137	C₄²	C₂×C₄×C₂₁₁	1	8	yes	complemented
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
141	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
142	C₂³	C₂×Q₈×C₂₁₁	1	2	yes	—
143	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	C₄⋊C₄×C₂₁₁	1	16	yes	complemented
154	C₂×C₄	C₂²×C₄×C₂₁₁	1	8	yes	—
155	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
156	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
157	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
159	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
160	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
166	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
167	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
168	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
169	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
170	C₂²	2₋¹⁺⁴×C₂₁₁	1	2	yes	—
171	C₄	—	2	8	no	—
172	C₄	—	4	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
176	C₄	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
179	C₂	#520×C₂₁₁	1	2	yes	—
180	C₂	#535×C₂₁₁	1	1	yes	characteristic
181	C₂	#536×C₂₁₁	1	1	yes	characteristic
182	C₂	#544×C₂₁₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #389×C₂₁₁	C₂	semidirect
6. #384×C₂₁₁	C₂	semidirect
7. #382×C₂₁₁	C₂	semidirect
11. #117×C₂₁₁	C₄	semidirect
41. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
45. C₄²×C₂₁₁	C₂×C₄	semidirect
61. C₂×C₄×C₂₁₁	C₄⋊C₄	semidirect
91. C₂₁₁	#2317	direct (N × H)
92. #2317	C₂₁₁	direct (N × H)
96. #389	C₂×C₂₁₁	semidirect
98. #384	C₂×C₂₁₁	semidirect
99. #382	C₂×C₂₁₁	semidirect
103. #117	C₄×C₂₁₁	semidirect
133. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
137. C₄²	C₂×C₄×C₂₁₁	semidirect
153. C₂×C₄	C₄⋊C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₂₁₁	2
C₂²×C₄⋊C₄×C₂₁₁	2
#389×C₂₁₁	2
#385×C₂₁₁	2
#384×C₂₁₁	2
#382×C₂₁₁	2
#2317	211

Subgroups — #2317×C211 #41009937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2317×C₂₁₁ #41009937