Subgroups — #36469807

Order 22784.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022025	C₂	1	2	yes	maximal
2	#26020131	C₂	1	2	yes	maximal
3	#26018356	C₂	1	1	yes	maximal, characteristic
4	#26018354	C₂	1	1	yes	maximal, characteristic
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	C₂²	1	2	yes	—
7	#19151636	C₂²	1	2	yes	—
8	#19151553	—	2	2	no	—
9	#19151421	C₄	1	8	yes	complemented
10	#19151358	—	2	2	no	—
11	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#346×C₈₉	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₈₉	C₄	1	2	yes	complemented
16	C₂×#13083558	—	2	2	no	—
17	#15667971	—	2	2	no	—
18	#15667948	—	2	2	no	—
19	#15667948	C₂×C₄	1	4	yes	—
20	#15667946	—	2	2	no	—
21	#15667944	C₂×C₄	1	4	yes	—
22	#15667922	—	4	2	no	—
23	C₈×Dic₈₉	—	4	2	no	—
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
27	M₃₂×C₈₉	—	2	2	no	—
28	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
35	#13083558	—	2	4	no	—
36	#13083558	—	4	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₄⋊Dic₈₉	—	2	4	no	—
39	C₂²⋊Dic₈₉	—	4	2	no	—
40	C₄×Dic₈₉	—	4	2	no	—
41	C₂²×C₄×C₈₉	—	2	1	no	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₈₉	#41	1	2	yes	—
51	C₄²×C₈₉	—	2	1	no	—
52	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₈₉	—	2	2	no	—
55	C₁₆×C₈₉	C₄²	1	4	yes	—
56	C₂³×C₈₉	#135	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
64	C₂×Dic₈₉	—	4	4	no	—
65	C₈×C₈₉	#106	1	2	yes	—
66	C₈×C₈₉	#106	1	1	yes	characteristic
67	C₈×C₈₉	#106	1	1	yes	characteristic
68	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₈₉	#117	1	1	yes	characteristic
70	C₈×C₈₉	C₄×Q₈	1	2	yes	—
71	#10623494	—	4	4	no	—
72	C₂²×C₈₉	—	4	1	no	—
73	C₂²×C₈₉	#427	1	1	yes	characteristic
74	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	4	1	no	—
77	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
79	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
80	Dic₈₉	—	8	4	no	—
81	#12668	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	2	no	—
83	C₂×C₈₉	—	2	1	no	—
84	C₂×C₈₉	C₄×#106	1	1	yes	characteristic
85	#2155	—	89	2	no	—
86	C₄×M₃₂	D₈₉	1	1	yes	characteristic
87	#1772	—	89	2	no	—
88	#1385	—	89	1	no	—
89	#1384	—	89	1	no	—
90	C₈₉	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
92	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
93	C₂×#108	—	89	2	no	—
94	#427	—	89	2	no	—
95	#405	—	178	2	no	—
96	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
97	#365	—	89	8	no	—
98	#346	Dic₈₉	1	2	yes	complemented
99	C₄×C₁₆	Dic₈₉	1	2	yes	complemented
100	#339	—	178	2	no	—
101	C₂×M₁₆	—	178	2	no	—
102	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	#117	—	178	2	no	—
105	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₄×D₈₉	1	4	yes	—
108	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
110	#108	—	89	4	no	—
111	#108	—	178	2	no	—
112	#107	—	178	2	no	—
113	#106	—	89	4	no	—
114	#97	—	356	2	no	—
115	C₄×C₈	—	356	2	no	—
116	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
117	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
122	M₁₆	—	356	2	no	—
123	M₁₆	—	178	4	no	—
124	C₂×C₈	—	356	2	no	—
125	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
127	C₂×C₈	#13083563	1	2	yes	—
128	C₂×C₈	#13083563	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
130	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
132	C₂×C₈	#13083574	1	2	yes	—
133	C₄⋊C₄	—	178	4	no	—
134	C₂²⋊C₄	—	356	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	356	2	no	—
137	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄×Dic₈₉	1	4	yes	—
141	C₂³	#15668034	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	356	4	no	—
146	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
147	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
148	C₂×C₄	C₂×#13083563	1	2	yes	—
149	C₂×C₄	#15668034	1	1	yes	characteristic
150	C₈	—	356	4	no	—
151	C₈	#15667944	1	2	yes	—
152	C₈	#15667944	1	1	yes	characteristic
153	C₈	#15667944	1	1	yes	characteristic
154	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
155	C₈	#15667971	1	1	yes	characteristic
156	C₈	C₄×Dic₁₇₈	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#19151636	1	1	yes	characteristic
159	C₂²	C₄×#13083563	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	712	4	no	—
163	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
164	C₄	C₂×#15667944	1	1	yes	characteristic
165	C₄	C₄×#13083563	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#15667944	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667944
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂×C₈₉	C₄×#106
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#12668
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
9. #19151421	C₄	semidirect
14. #346×C₈₉	C₄	semidirect
15. C₄×C₁₆×C₈₉	C₄	semidirect
90. C₈₉	#12668	semidirect
98. #346	Dic₈₉	semidirect
99. C₄×C₁₆	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022025	2
#26020131	2
#26018356	2
#26018354	2
C₄×M₃₂×C₈₉	2
#12668	89