Subgroups — #41013118

Order 27008.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269936	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212909	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₁₁	—	2	1	no	—
6	#20269936	—	2	2	no	—
7	#20269936	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₂₁₁	—	2	2	no	—
9	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
10	C₂×C₄⋊Dic₂₁₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₂₁₁	—	2	1	no	—
14	C₂×C₄×Dic₂₁₁	—	2	1	no	—
15	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₁₁	—	2	2	no	—
17	C₂×M₁₆×C₂₁₁	—	2	2	no	—
18	C₂×M₁₆×C₂₁₁	C₄	1	4	yes	complemented
19	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
20	C₂²×Dic₂₁₁	—	2	4	no	—
21	C₂²×Dic₂₁₁	—	2	2	no	—
22	C₂²×Dic₂₁₁	—	2	2	no	—
23	C₂²×Dic₂₁₁	—	4	4	no	—
24	C₄⋊Dic₂₁₁	—	2	4	no	—
25	C₄⋊Dic₂₁₁	—	2	2	no	—
26	C₄⋊Dic₂₁₁	—	2	2	no	—
27	C₄⋊Dic₂₁₁	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₂₁₁	C₂×C₄	1	4	yes	complemented
29	C₂²⋊Dic₂₁₁	—	2	4	no	—
30	C₄×Dic₂₁₁	—	2	4	no	—
31	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
36	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
37	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
39	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₂₁₁	—	2	8	no	—
41	C₂×C₈×C₂₁₁	—	2	2	no	—
42	C₂×C₈×C₂₁₁	—	2	2	no	—
43	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₁₁	—	2	4	no	—
45	C₂×Dic₂₁₁	—	2	4	no	—
46	C₂×Dic₂₁₁	—	4	8	no	—
47	C₂×Dic₂₁₁	—	4	2	no	—
48	C₂×Dic₂₁₁	—	4	4	no	—
49	C₂×Dic₂₁₁	—	4	2	no	—
50	C₂³×C₂₁₁	—	2	2	no	—
51	C₂³×C₂₁₁	—	2	2	no	—
52	C₂³×C₂₁₁	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₂₁₁	C₂²⋊C₄	1	2	yes	—
54	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
55	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	C₄²	1	4	yes	—
57	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
60	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
69	C₈×C₂₁₁	—	2	4	no	—
70	C₂²×C₂₁₁	—	2	4	no	—
71	C₂²×C₂₁₁	—	2	2	no	—
72	C₂²×C₂₁₁	—	2	2	no	—
73	C₂²×C₂₁₁	—	2	4	no	—
74	C₂²×C₂₁₁	#95	1	4	yes	—
75	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²×C₂₁₁	#137	1	2	yes	—
80	C₄×C₂₁₁	#95	1	4	yes	—
81	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
82	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	Dic₂₁₁	—	4	4	no	—
86	Dic₂₁₁	—	4	4	no	—
87	C₂×C₂₁₁	—	2	4	no	—
88	C₂×C₂₁₁	C₂×#95	1	1	yes	characteristic
89	C₂×C₂₁₁	#417	1	2	yes	—
90	C₂×C₂₁₁	#418	1	1	yes	characteristic
91	C₂×C₂₁₁	#419	1	1	yes	characteristic
92	C₂×C₂₁₁	#428	1	2	yes	—
93	C₂₁₁	#1736	1	1	yes	characteristic, complemented, Sylow
94	#1736	—	211	1	no	maximal, Sylow
95	C₂²×M₁₆	D₂₁₁	1	1	yes	characteristic
96	C₂×#117	—	211	1	no	—
97	C₂²×C₄⋊C₄	—	211	1	no	—
98	#340	—	211	4	no	—
99	C₂³×C₄	—	422	1	no	—
100	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	Dic₂₁₁	1	4	yes	complemented
104	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
105	#117	—	422	2	no	—
106	#117	—	211	4	no	—
107	C₂×C₄⋊C₄	—	422	2	no	—
108	C₂×C₄⋊C₄	—	422	1	no	—
109	C₂×C₄⋊C₄	—	211	4	no	—
110	C₂×C₄⋊C₄	—	211	2	no	—
111	C₂×C₄⋊C₄	—	211	2	no	—
112	C₂×C₂²⋊C₄	—	422	1	no	—
113	C₂×C₄²	—	422	1	no	—
114	C₂⁴	D₈₄₄	1	1	yes	characteristic
115	C₂²×C₄	—	844	4	no	—
116	C₂²×C₄	—	422	4	no	—
117	C₂²×C₄	—	422	2	no	—
118	C₂²×C₄	—	422	2	no	—
119	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
120	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
121	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
122	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
123	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
124	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
125	C₂²×C₄	Dic₄₂₂	1	2	yes	—
126	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₂×Dic₂₁₁	1	4	yes	complemented
131	C₄⋊C₄	—	422	4	no	—
132	C₄⋊C₄	—	422	2	no	—
133	C₄⋊C₄	—	422	2	no	—
134	C₄⋊C₄	—	211	4	no	—
135	C₄⋊C₄	—	211	4	no	—
136	C₂²⋊C₄	—	422	4	no	—
137	C₄²	—	422	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
141	C₂³	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
142	C₂³	C₄⋊Dic₂₁₁	1	2	yes	—
143	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
144	C₂×C₄	—	844	8	no	—
145	C₂×C₄	—	844	4	no	—
146	C₂×C₄	—	844	2	no	—
147	C₂×C₄	—	844	2	no	—
148	C₂×C₄	—	422	4	no	—
149	C₂×C₄	—	422	4	no	—
150	C₂×C₄	C₄×Dic₂₁₁	1	4	yes	—
151	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
152	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
153	C₂×C₄	C₂²⋊Dic₂₁₁	1	4	yes	—
154	C₂×C₄	Dic₂₁₁⋊C₄	1	4	yes	—
155	C₂×C₄	Dic₂₁₁⋊C₄	1	4	yes	—
156	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
157	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	C₂₁₁⋊#95	1	4	yes	—
169	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
170	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
171	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
172	C₂²	#20270005	1	2	yes	—
173	C₂²	#20270010	1	2	yes	—
174	C₄	—	844	4	no	—
175	C₄	—	844	4	no	—
176	C₄	C₂₁₁⋊#95	1	4	yes	—
177	C₄	C₂×C₄×Dic₂₁₁	1	1	yes	characteristic
178	C₄	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
179	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
180	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×C₂₁₁⋊#95	1	1	yes	characteristic
183	C₂	#27213148	1	2	yes	—
184	C₂	#27213152	1	1	yes	characteristic
185	C₂	#27213156	1	1	yes	characteristic
186	C₂	#27213188	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator	C₄×C₂₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂²×M₁₆×C₂₁₁	C₂
socle	C₂³×C₂₁₁	C₂×D₄
prime radical	C₂⁴×C₂₁₁	D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂²×M₁₆	D₂₁₁
hypocenter	C₂₁₁	#1736
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27212909	C₂	semidirect
7. #20269936	C₄	semidirect
10. C₂×C₄⋊Dic₂₁₁	C₄	semidirect
18. C₂×M₁₆×C₂₁₁	C₄	semidirect
28. C₄⋊Dic₂₁₁	C₂×C₄	semidirect
43. C₂×C₈×C₂₁₁	C₂×C₄	semidirect
93. C₂₁₁	#1736	semidirect
103. C₂×M₁₆	Dic₂₁₁	semidirect
130. C₂×C₈	C₂×Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269936	2
C₂²×C₄⋊Dic₂₁₁	2
#27212909	2
C₂²×M₁₆×C₂₁₁	2
#1736	211