Subgroups — #2283×C₂₁₁ #41009903

Order 27008.

568 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#383×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#382×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#381×C₂₁₁	C₂	1	2	yes	maximal
5	#380×C₂₁₁	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal
8	#376×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
14	C₂×C₄²×C₂₁₁	C₄	1	8	yes	complemented
15	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
20	#95×C₂₁₁	C₂²	1	4	yes	—
21	#95×C₂₁₁	C₂²	1	4	yes	—
22	#95×C₂₁₁	C₂²	1	4	yes	—
23	#95×C₂₁₁	C₂²	1	4	yes	—
24	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₁₁	—	2	4	no	—
26	C₂²×C₄×C₂₁₁	—	2	4	no	—
27	C₂²×C₄×C₂₁₁	—	2	4	no	—
28	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	8	yes	—
32	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₁₁	—	2	8	no	—
39	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
40	C₄²×C₂₁₁	—	2	8	no	—
41	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
42	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
43	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
44	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
45	C₂³×C₂₁₁	—	2	2	no	—
46	C₂³×C₂₁₁	—	2	2	no	—
47	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
48	C₂³×C₂₁₁	#41	1	4	yes	—
49	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	—	2	8	no	—
51	C₂×C₄×C₂₁₁	—	2	2	no	—
52	C₂×C₄×C₂₁₁	—	2	2	no	—
53	C₂×C₄×C₂₁₁	—	2	2	no	—
54	C₂×C₄×C₂₁₁	—	2	2	no	—
55	C₂×C₄×C₂₁₁	—	2	8	no	—
56	C₂×C₄×C₂₁₁	—	2	8	no	—
57	C₂×C₄×C₂₁₁	—	2	8	no	—
58	C₂×C₄×C₂₁₁	—	2	8	no	—
59	C₂×C₄×C₂₁₁	—	2	8	no	—
60	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	#41	1	4	yes	—
67	C₂×C₄×C₂₁₁	#41	1	4	yes	—
68	C₂²×C₂₁₁	—	2	2	no	—
69	C₂²×C₂₁₁	—	2	2	no	—
70	C₂²×C₂₁₁	—	2	8	no	—
71	C₂²×C₂₁₁	#117	1	4	yes	—
72	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
74	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
75	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
76	C₄×C₂₁₁	—	2	2	no	—
77	C₄×C₂₁₁	—	2	2	no	—
78	C₄×C₂₁₁	—	2	2	no	—
79	C₄×C₂₁₁	—	2	2	no	—
80	C₄×C₂₁₁	—	4	8	no	—
81	C₄×C₂₁₁	#117	1	4	yes	—
82	C₄×C₂₁₁	#117	1	4	yes	—
83	C₂×C₂₁₁	—	2	4	no	—
84	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
85	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
86	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
87	C₂×C₂₁₁	#529	1	4	yes	—
88	C₂₁₁	#2283	1	1	yes	characteristic, direct product factor, Sylow
89	#2283	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic
91	#383	C₂×C₂₁₁	1	2	yes	complemented
92	#382	C₂×C₂₁₁	1	2	yes	complemented
93	#381	C₂×C₂₁₁	1	2	yes	—
94	#380	C₂×C₂₁₁	1	2	yes	—
95	C₄×C₄⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
96	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	—
97	#376	C₂×C₂₁₁	1	2	yes	complemented
98	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
100	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
103	C₂×C₄²	C₄×C₂₁₁	1	8	yes	complemented
104	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	—
105	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
108	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
109	#95	C₂²×C₂₁₁	1	4	yes	—
110	#95	C₂²×C₂₁₁	1	4	yes	—
111	#95	C₂²×C₂₁₁	1	4	yes	—
112	#95	C₂²×C₂₁₁	1	4	yes	—
113	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
118	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
119	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
120	C₂²×C₄	C₂³×C₂₁₁	1	8	yes	—
121	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
127	C₄⋊C₄	—	2	8	no	—
128	C₂²⋊C₄	—	2	8	no	—
129	C₄²	—	2	8	no	—
130	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
131	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
132	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
133	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
137	C₂³	#41×C₂₁₁	1	4	yes	—
138	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
150	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
151	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
152	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
153	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
154	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
155	C₂×C₄	#41×C₂₁₁	1	4	yes	—
156	C₂×C₄	#41×C₂₁₁	1	4	yes	—
157	C₂²	—	2	8	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#117×C₂₁₁	1	4	yes	—
161	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
162	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
163	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
164	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	4	8	no	—
170	C₄	#117×C₂₁₁	1	4	yes	—
171	C₄	#117×C₂₁₁	1	4	yes	—
172	C₂	—	2	4	no	—
173	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
174	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
175	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
176	C₂	#529×C₂₁₁	1	4	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	C₂⁴×C₂₁₁	C₂³
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. #383×C₂₁₁	C₂	semidirect
3. #382×C₂₁₁	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
8. #376×C₂₁₁	C₂	semidirect
14. C₂×C₄²×C₂₁₁	C₄	semidirect
88. C₂₁₁	#2283	direct (N × H)
89. #2283	C₂₁₁	direct (N × H)
91. #383	C₂×C₂₁₁	semidirect
92. #382	C₂×C₂₁₁	semidirect
95. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
97. #376	C₂×C₂₁₁	semidirect
103. C₂×C₄²	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₂₁₁	2
#383×C₂₁₁	2
#382×C₂₁₁	2
#381×C₂₁₁	2
#380×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#376×C₂₁₁	2
#2283	211

Subgroups — #2283×C211 #41009903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2283×C₂₁₁ #41009903