Subgroups — #36469819

Order 22784.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 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×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26020133	C₂	1	2	yes	maximal
4	#26018347	C₂	1	2	yes	maximal
5	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667948	C₂²	1	2	yes	—
8	#19151554	—	2	2	no	—
9	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
10	#19151428	C₄	1	4	yes	complemented
11	C₁₆×Dic₈₉	C₄	1	4	yes	complemented
12	#19151357	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	#427×C₈₉	C₂²	1	2	yes	—
15	#367×C₈₉	C₄	1	2	yes	complemented
16	#366×C₈₉	C₄	1	2	yes	complemented
17	C₂×#13083558	—	2	2	no	—
18	C₂²×#10623494	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₂×#13083549	C₂×C₄	1	2	yes	—
23	#15667948	—	2	2	no	—
24	#15667948	C₂×C₄	1	4	yes	—
25	#15667923	—	4	2	no	—
26	C₄×#10623494	—	4	2	no	—
27	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
28	C₈×Dic₈₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	#117×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
35	#107×C₈₉	C₂×C₄	1	4	yes	—
36	#106×C₈₉	—	2	2	no	—
37	C₂²×Dic₈₉	—	2	1	no	—
38	#13083558	—	2	4	no	—
39	#13083558	—	4	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	C₄⋊C₄	1	2	yes	—
42	C₂×#10623494	D₈	1	1	yes	characteristic
43	C₂×#10623494	D₈	1	1	yes	characteristic
44	C₂×#10623494	Q₁₆	1	2	yes	—
45	C₄×Dic₈₉	—	2	1	no	—
46	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
47	#13083549	—	2	2	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	2	4	no	—
56	C₂²⋊C₄×C₈₉	—	4	2	no	—
57	C₄²×C₈₉	—	4	2	no	—
58	C₁₆×C₈₉	C₄²	1	2	yes	—
59	C₁₆×C₈₉	C₄²	1	2	yes	—
60	C₂³×C₈₉	#135	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	4	4	no	—
62	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
68	C₂×Dic₈₉	—	2	4	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₈₉	#117	1	1	yes	characteristic
72	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
74	#10623494	—	4	4	no	—
75	#10623494	#107	1	2	yes	—
76	#10623494	#107	1	2	yes	—
77	C₂²×C₈₉	—	2	2	no	—
78	C₂²×C₈₉	#427	1	1	yes	characteristic
79	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₈₉	—	8	4	no	—
82	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
84	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
85	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
86	Dic₈₉	—	2	4	no	—
87	#12670	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	—	2	1	no	—
90	C₂×C₈₉	C₄×#107	1	1	yes	characteristic
91	C₂×C₈₉	#2178	1	2	yes	—
92	C₂×#368	—	89	1	no	—
93	#2158	D₈₉	1	1	yes	characteristic
94	C₂×C₄×C₁₆	—	89	1	no	—
95	#1772	—	89	2	no	—
96	#1382	—	89	2	no	—
97	C₈₉	#12670	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	89	1	no	—
99	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
100	C₂×#108	—	89	2	no	—
101	#427	C₂×D₈₉	1	2	yes	—
102	#405	—	178	2	no	—
103	C₂×C₄×C₈	—	89	1	no	—
104	#368	—	89	4	no	—
105	#367	Dic₈₉	1	2	yes	complemented
106	#366	Dic₈₉	1	2	yes	complemented
107	C₄×C₁₆	—	89	4	no	—
108	#339	—	178	2	no	—
109	C₂×M₁₆	—	178	2	no	—
110	C₂²×C₈	—	89	1	no	—
111	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
112	#117	—	2	2	no	—
113	C₂×C₄²	—	89	1	no	—
114	C₂×C₁₆	—	89	2	no	—
115	C₂×C₁₆	—	178	1	no	—
116	C₂×C₁₆	—	178	1	no	—
117	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
118	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
120	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
121	#108	—	89	4	no	—
122	#108	—	178	2	no	—
123	#107	C₂×Dic₈₉	1	4	yes	—
124	#106	—	2	2	no	—
125	#97	—	356	2	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	356	2	no	—
129	C₂²×C₄	—	178	1	no	—
130	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
131	M₁₆	—	356	2	no	—
132	M₁₆	—	178	4	no	—
133	C₂×C₈	—	89	1	no	—
134	C₂×C₈	—	89	1	no	—
135	C₂×C₈	—	89	2	no	—
136	C₂×C₈	—	89	2	no	—
137	C₂×C₈	—	356	2	no	—
138	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
142	C₂×C₈	#13083573	1	1	yes	characteristic
143	C₂×C₈	#13083575	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	4	2	no	—
147	C₄²	—	89	2	no	—
148	C₄²	—	178	1	no	—
149	C₁₆	—	178	2	no	—
150	C₁₆	C₄×Dic₈₉	1	2	yes	—
151	C₁₆	C₄×Dic₈₉	1	2	yes	—
152	C₂³	#15668031	1	1	yes	characteristic
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	178	4	no	—
155	C₂×C₄	—	178	2	no	—
156	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
157	C₂×C₄	#15667974	1	1	yes	characteristic
158	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
159	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
160	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
161	C₂×C₄	#15668035	1	1	yes	characteristic
162	C₈	—	89	2	no	—
163	C₈	—	89	2	no	—
164	C₈	—	356	4	no	—
165	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
166	C₈	#15667970	1	1	yes	characteristic
167	C₈	#15667977	1	1	yes	characteristic
168	C₈	Q₈×Dic₈₉	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	#19151635	1	1	yes	characteristic
171	C₂²	D₈×Dic₈₉	1	1	yes	characteristic
172	C₂²	#19151692	1	1	yes	characteristic
173	C₄	—	8	4	no	—
174	C₄	—	178	4	no	—
175	C₄	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
176	C₄	#107×D₈₉	1	1	yes	characteristic
177	C₄	#19151680	1	1	yes	characteristic
178	C₄	Q₁₆×Dic₈₉	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#107×Dic₈₉	1	1	yes	characteristic
182	C₂	#26022133	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2158	C₂
socle	C₂²×C₈₉	C₄×D₈
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2158	D₈₉
hypocenter	C₈₉	#12670
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
10. #19151428	C₄	semidirect
11. C₁₆×Dic₈₉	C₄	semidirect
15. #367×C₈₉	C₄	semidirect
16. #366×C₈₉	C₄	semidirect
97. C₈₉	#12670	semidirect
105. #367	Dic₈₉	semidirect
106. #366	Dic₈₉	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×C₁₆×Dic₈₉	2
#26020133	2
#26018347	2
C₈₉×#2158	2
#12670	89