Subgroups — #36470875

Order 22784.

314 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151428	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151427	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#26020422	C₂	1	2	yes	maximal
5	C₂×#368×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667948	C₂²	1	2	yes	—
8	C₂×#15667947	—	2	2	no	—
9	#19151621	—	2	2	no	—
10	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
11	#19151428	—	2	2	no	—
12	#19151427	—	2	4	no	—
13	C₁₆×Dic₈₉	C₄	1	2	yes	—
14	C₁₆×Dic₈₉	C₄	1	2	yes	—
15	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₈₉	C₂²	1	2	yes	—
17	#368×C₈₉	—	2	2	no	—
18	C₂×#13083558	—	2	2	no	—
19	C₂²×#10623494	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
21	C₂×#13083549	C₂×C₄	1	2	yes	—
22	C₂×#13083549	D₄	1	2	yes	—
23	C₂×#13083549	Q₈	1	2	yes	—
24	#15667948	—	2	2	no	—
25	#15667948	—	2	2	no	—
26	#15667947	—	4	4	no	—
27	#15667941	—	4	4	no	—
28	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₈₉	Q₈	1	1	yes	characteristic
31	C₈×Dic₈₉	Q₈	1	1	yes	characteristic
32	C₂×M₁₆×C₈₉	—	2	2	no	—
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
36	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
37	#108×C₈₉	—	2	2	no	—
38	#108×C₈₉	—	2	2	no	—
39	C₂²×Dic₈₉	Q₁₆	1	2	yes	—
40	#13083558	—	4	2	no	—
41	#13083558	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#10623494	D₈	1	2	yes	—
46	C₂×#10623494	Q₁₆	1	2	yes	—
47	C₄×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄×Dic₈₉	D₈	1	2	yes	—
50	#13083549	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₈₉	—	4	2	no	—
53	M₁₆×C₈₉	—	4	2	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
61	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	—
62	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₈₉	—	2	4	no	—
70	C₂×Dic₈₉	#107	1	2	yes	—
71	C₂×Dic₈₉	#107	1	2	yes	—
72	C₈×C₈₉	—	8	2	no	—
73	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
77	#10623494	—	8	2	no	—
78	#10623494	#107	1	2	yes	—
79	#10623494	#107	1	2	yes	—
80	C₂²×C₈₉	—	2	2	no	—
81	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
82	C₂²×C₈₉	#494	1	1	yes	characteristic
83	C₂²×C₈₉	#500	1	1	yes	characteristic
84	C₄×C₈₉	#384	1	1	yes	characteristic
85	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
86	C₄×C₈₉	#493	1	1	yes	characteristic
87	C₄×C₈₉	#500	1	1	yes	characteristic
88	Dic₈₉	—	2	2	no	—
89	Dic₈₉	#368	1	4	yes	complemented
90	#12909	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	—	2	1	no	—
92	C₂×C₈₉	—	2	1	no	—
93	C₂×C₈₉	#1845	1	1	yes	characteristic
94	C₂×C₈₉	C₂×#368	1	2	yes	—
95	C₂×#368	—	89	1	no	—
96	C₂×#368	—	89	2	no	—
97	C₂×#368	D₈₉	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	89	1	no	—
99	#1848	—	89	2	no	—
100	C₈₉	#12909	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	89	1	no	—
102	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
103	C₂×#108	—	89	2	no	—
104	C₂×#108	—	178	2	no	—
105	C₂×#108	C₂×D₈₉	1	2	yes	—
106	#423	—	178	2	no	—
107	C₂×C₄×C₈	—	89	1	no	—
108	#368	—	2	2	no	—
109	#368	—	178	4	no	—
110	#368	—	178	2	no	—
111	C₄×C₁₆	—	89	2	no	—
112	C₄×C₁₆	—	89	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	178	2	no	—
115	C₂²×C₈	—	89	1	no	—
116	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₂×C₄²	—	89	1	no	—
118	C₂×C₁₆	—	89	2	no	—
119	C₂×C₁₆	—	89	2	no	—
120	C₂×C₁₆	—	89	2	no	—
121	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
122	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	356	4	no	—
127	#108	—	178	2	no	—
128	#108	—	178	2	no	—
129	C₄⋊C₈	—	356	4	no	—
130	C₄×C₈	—	89	1	no	—
131	C₄×C₈	—	89	1	no	—
132	C₄×C₈	—	89	1	no	—
133	C₄×C₈	—	89	1	no	—
134	C₂²×C₄	—	89	2	no	—
135	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	356	2	no	—
139	M₁₆	—	356	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	89	2	no	—
142	C₂×C₈	—	89	2	no	—
143	C₂×C₈	—	89	1	no	—
144	C₂×C₈	—	89	1	no	—
145	C₂×C₈	—	356	2	no	—
146	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
147	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
152	C₄²	—	89	2	no	—
153	C₄²	—	89	1	no	—
154	C₄²	—	89	1	no	—
155	C₁₆	—	89	4	no	—
156	C₁₆	#13083553	1	4	yes	—
157	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
158	C₂×C₄	—	89	2	no	—
159	C₂×C₄	—	89	2	no	—
160	C₂×C₄	—	178	4	no	—
161	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
162	C₂×C₄	#15668008	1	1	yes	characteristic
163	C₂×C₄	#15668011	1	1	yes	characteristic
164	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
165	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	89	2	no	—
169	C₈	—	89	2	no	—
170	C₈	—	712	2	no	—
171	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
172	C₈	C₂×#13083553	1	1	yes	characteristic
173	C₈	#15668010	1	1	yes	characteristic
174	C₈	#15668011	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	#107×D₈₉	1	1	yes	characteristic
177	C₂²	#19151976	1	1	yes	characteristic
178	C₂²	#19152002	1	1	yes	characteristic
179	C₄	—	89	4	no	—
180	C₄	—	178	2	no	—
181	C₄	#19151467	1	1	yes	characteristic
182	C₄	#107×D₈₉	1	1	yes	characteristic
183	C₄	#19151973	1	1	yes	characteristic
184	C₄	#19152002	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#26020410	1	1	yes	characteristic
188	C₂	#368×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#368×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#368	D₈₉
hypocenter	C₈₉	#12909
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
89. Dic₈₉	#368	semidirect
100. C₈₉	#12909	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151428	2
C₂×#19151427	2
C₂×C₁₆×Dic₈₉	2
#26020422	2
C₂×#368×C₈₉	2
#12909	89