Subgroups — #36442126

Order 22784.

178 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151432	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667918	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic
4	#19151432	—	2	1	no	—
5	#19151432	C₄	1	2	yes	complemented
6	C₂×#15667918	—	2	1	no	—
7	C₂×#15667918	—	2	1	no	—
8	C₂×#15667918	—	2	1	no	—
9	C₂×#15667918	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₈₉	C₄	1	1	yes	characteristic
11	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	#405×C₈₉	C₄	1	1	yes	characteristic
13	#346×C₈₉	—	2	1	no	—
14	C₄×C₁₆×C₈₉	—	2	1	no	—
15	#15667918	—	2	2	no	—
16	#15667918	—	2	2	no	—
17	C₂×M₁₆×C₈₉	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
19	#117×C₈₉	C₈	1	1	yes	characteristic
20	M₃₂×C₈₉	—	2	1	no	—
21	M₃₂×C₈₉	C₈	1	2	yes	—
22	C₂×C₁₆×C₈₉	—	2	1	no	—
23	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
29	#97×C₈₉	—	2	1	no	—
30	C₄×C₈×C₈₉	—	2	1	no	—
31	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₈₉	—	2	1	no	—
33	M₁₆×C₈₉	C₁₆	1	2	yes	—
34	C₂×C₈×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	C₁₆	1	2	yes	—
42	C₂²⋊C₄×C₈₉	—	2	1	no	—
43	C₄²×C₈₉	—	2	1	no	—
44	C₁₆×C₈₉	—	2	2	no	—
45	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₈₉	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
57	C₈×C₈₉	—	2	2	no	—
58	C₈×C₈₉	#95	1	1	yes	characteristic
59	C₈×C₈₉	#97	1	1	yes	characteristic
60	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	#346	1	1	yes	characteristic
64	C₂²×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
66	C₄×C₈₉	—	4	2	no	—
67	C₄×C₈₉	#336	1	1	yes	characteristic
68	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
69	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
71	#7470	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	—	2	1	no	—
73	C₂×C₈₉	—	2	1	no	—
74	C₂×C₈₉	#1374	1	1	yes	characteristic
75	C₂×C₈₉	#1401	1	2	yes	—
76	C₂×M₆₄	—	89	1	no	—
77	C₂²×C₃₂	—	89	1	no	—
78	#2108	D₈₉	1	1	yes	characteristic
79	C₈₉	#7470	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	Dic₈₉	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
82	#405	Dic₈₉	1	1	yes	characteristic
83	M₆₄	—	89	2	no	—
84	M₆₄	—	178	1	no	—
85	C₂×C₃₂	—	89	2	no	—
86	C₂×C₃₂	—	178	1	no	—
87	C₂×C₃₂	—	178	1	no	—
88	C₂×C₃₂	—	178	1	no	—
89	#346	—	2	1	no	—
90	C₄×C₁₆	—	2	1	no	—
91	C₂×M₁₆	#10623494	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
93	#117	#10623494	1	1	yes	characteristic
94	M₃₂	—	2	1	no	—
95	M₃₂	#10623494	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
100	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
101	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₃₂	—	178	2	no	—
106	C₃₂	—	178	2	no	—
107	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	#13083549	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
113	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
114	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
115	C₂×C₈	#13083558	1	1	yes	characteristic
116	C₂×C₈	#13083558	1	1	yes	characteristic
117	C₄⋊C₄	#13083549	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	#13083551	1	1	yes	characteristic
122	C₁₆	#13083551	1	1	yes	characteristic
123	C₁₆	#13083553	1	1	yes	characteristic
124	C₁₆	#13083553	1	1	yes	characteristic
125	C₂³	#15667952	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄×#10623494	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
129	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
130	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
131	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
132	C₂×C₄	#15667952	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#15667919	1	1	yes	characteristic
135	C₈	#15667923	1	1	yes	characteristic
136	C₈	C₂²⋊#10623494	1	1	yes	characteristic
137	C₈	C₄⋊#10623494	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#19151374	1	1	yes	characteristic
140	C₂²	C₂²⋊#13083549	1	1	yes	characteristic
141	C₂²	C₄⋊#13083549	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	#19151353	1	1	yes	characteristic
144	C₄	C₄×#13083549	1	1	yes	characteristic
145	C₄	C₂²⋊#13083549	1	1	yes	characteristic
146	C₄	C₄⋊#13083549	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#26018330	1	1	yes	characteristic
150	C₂	#26018394	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₄×C₁₆
Frattini	C₂²×C₁₆	C₂×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	C₂²⋊C₁₆
prime radical	C₂³×C₈₉	M₃₂
cosocle	C₂²×C₁₆×C₈₉	C₂²
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#7470
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
5. #19151432	C₄	semidirect
9. C₂×#15667918	C₄	semidirect
79. C₈₉	#7470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151432	2
C₂²×#15667918	2
C₈₉×#2108	2
#7470	89