Subgroups — #36474920

Order 22784.

198 conjugacy classes of subgroups.

The subgroup lattice has 154 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	#26021848	C₂	1	1	yes	maximal, characteristic
3	#26018449	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₄	1	1	yes	characteristic
6	#19151553	C₄	1	1	yes	characteristic
7	#19151432	—	2	1	no	—
8	#19151432	C₂²	1	2	yes	—
9	C₂×#15667918	—	2	1	no	—
10	#19151373	C₄	1	2	yes	—
11	C₁₆×Dic₈₉	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	M₆₄×C₈₉	—	2	1	no	—
14	M₆₄×C₈₉	C₂²	1	2	yes	—
15	C₂×C₃₂×C₈₉	—	2	1	no	—
16	C₂×#13083558	C₈	1	1	yes	characteristic
17	#15667971	C₈	1	1	yes	characteristic
18	#15667952	—	2	1	no	—
19	#15667952	—	2	1	no	—
20	C₂×#13083549	C₂×C₄	1	2	yes	—
21	#15667922	C₈	1	2	yes	—
22	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
23	#15667918	—	2	2	no	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	C₂×C₄	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	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
31	C₃₂×C₈₉	—	2	2	no	—
32	#13083558	C₁₆	1	2	yes	—
33	#13083558	C₁₆	1	2	yes	—
34	C₂×#10623494	C₂×C₈	1	2	yes	—
35	C₄⋊Dic₈₉	—	2	1	no	—
36	C₂²⋊Dic₈₉	—	2	1	no	—
37	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
38	#13083549	—	2	2	no	—
39	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₈₉	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₈₉	—	2	2	no	—
58	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
62	#10623494	C₂×C₁₆	1	4	yes	—
63	C₂²×C₈₉	—	4	1	no	—
64	C₂²×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
67	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
68	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
69	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
71	Dic₈₉	—	4	2	no	—
72	#13630	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	—	2	1	no	—
74	C₂×C₈₉	—	2	1	no	—
75	C₂×C₈₉	—	2	1	no	—
76	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
77	C₂×M₆₄	—	89	1	no	—
78	C₂×M₆₄	D₈₉	1	1	yes	characteristic
79	#2108	—	89	1	no	—
80	#1422	—	89	4	no	—
81	C₈₉	#13630	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₃₂	—	89	1	no	—
83	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
84	#405	—	89	1	no	—
85	M₆₄	—	2	1	no	—
86	M₆₄	—	89	2	no	—
87	M₆₄	—	178	1	no	—
88	M₆₄	C₂×D₈₉	1	2	yes	—
89	C₂×C₃₂	—	2	1	no	—
90	C₂×C₃₂	—	178	1	no	—
91	#346	—	89	2	no	—
92	C₄×C₁₆	—	89	2	no	—
93	C₂×M₁₆	—	89	1	no	—
94	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
95	#117	—	89	1	no	—
96	M₃₂	—	178	1	no	—
97	M₃₂	—	178	1	no	—
98	C₂×C₁₆	—	89	2	no	—
99	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
100	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
103	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
105	#97	—	89	2	no	—
106	C₄×C₈	—	89	2	no	—
107	C₃₂	—	2	2	no	—
108	C₃₂	—	178	2	no	—
109	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
110	M₁₆	—	89	2	no	—
111	M₁₆	—	89	2	no	—
112	C₂×C₈	—	89	2	no	—
113	C₂×C₈	#13083553	1	1	yes	characteristic
114	C₂×C₈	#13083553	1	1	yes	characteristic
115	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
116	C₂×C₈	#13083557	1	1	yes	characteristic
117	C₂×C₈	#13083557	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
119	C₄⋊C₄	—	178	1	no	—
120	C₂²⋊C₄	—	178	1	no	—
121	C₄²	—	89	2	no	—
122	C₁₆	—	178	2	no	—
123	C₁₆	#13083553	1	1	yes	characteristic
124	C₁₆	#13083553	1	1	yes	characteristic
125	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
126	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
127	C₂³	#15667951	1	1	yes	characteristic
128	C₂×C₄	—	178	2	no	—
129	C₂×C₄	#15667941	1	1	yes	characteristic
130	C₂×C₄	#15667941	1	1	yes	characteristic
131	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
132	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
133	C₂×C₄	#15667951	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
135	C₈	—	89	4	no	—
136	C₈	#15667941	1	1	yes	characteristic
137	C₈	#15667941	1	1	yes	characteristic
138	C₈	C₂×#13083553	1	1	yes	characteristic
139	C₈	C₂×#13083557	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	#19151416	1	1	yes	characteristic
142	C₂²	#19151416	1	1	yes	characteristic
143	C₂²	C₂×#15667951	1	1	yes	characteristic
144	C₄	—	356	2	no	—
145	C₄	#19151416	1	1	yes	characteristic
146	C₄	#19151416	1	1	yes	characteristic
147	C₄	C₂×#15667941	1	1	yes	characteristic
148	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×#19151416	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂×M₆₄×C₈₉	C₂
socle	C₂×C₈₉	C₂×C₄⋊C₁₆
prime radical	C₂³×C₈₉	M₃₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂×M₆₄	D₈₉
hypocenter	C₈₉	#13630
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #26018449	C₂	semidirect
81. C₈₉	#13630	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
#26021848	2
#26018449	2
C₂×M₆₄×C₈₉	2
#13630	89