Subgroups — #36470817

Order 22784.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
2	#26020424	C₂	1	1	yes	maximal, characteristic
3	#26018345	C₂	1	4	yes	maximal
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₂²	1	2	yes	—
6	C₂×#15667948	—	2	1	no	—
7	C₂×#15667948	C₂²	1	2	yes	—
8	#19151622	—	2	1	no	—
9	C₄×#13083549	C₄	1	2	yes	—
10	C₄×#13083549	C₄	1	2	yes	—
11	C₂×#108×C₈₉	—	2	1	no	—
12	C₂×#108×C₈₉	C₂²	1	2	yes	—
13	#423×C₈₉	—	2	1	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	2	no	—
16	C₂×#13083549	—	2	2	no	—
17	C₂×#13083549	—	2	2	no	—
18	C₂×#13083549	C₂×C₄	1	4	yes	—
19	#15667948	—	2	4	no	—
20	#15667948	—	4	2	no	—
21	C₄⋊#10623494	—	4	2	no	—
22	C₂×M₁₆×C₈₉	—	2	2	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	#108×C₈₉	—	2	4	no	—
27	#108×C₈₉	—	4	2	no	—
28	C₄⋊C₈×C₈₉	—	4	2	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
33	#13083558	—	4	2	no	—
34	#13083558	—	4	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	#13083549	—	2	4	no	—
37	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₈₉	—	4	2	no	—
40	M₁₆×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	—	4	2	no	—
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₂×C₈×C₈₉	#41	1	2	yes	—
50	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₉	D₈	1	2	yes	—
53	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	4	no	—
55	C₂×C₄×C₈₉	#107	1	2	yes	—
56	C₂×C₄×C₈₉	#107	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₈₉	—	8	2	no	—
64	C₈×C₈₉	#106	1	2	yes	—
65	C₈×C₈₉	#106	1	2	yes	—
66	C₈×C₈₉	#117	1	2	yes	—
67	C₈×C₈₉	#125	1	2	yes	—
68	#10623494	—	8	2	no	—
69	C₂²×C₈₉	—	2	2	no	—
70	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
71	C₂²×C₈₉	#487	1	1	yes	characteristic
72	C₂²×C₈₉	#487	1	1	yes	characteristic
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	#368	1	4	yes	—
75	C₄×C₈₉	#382	1	1	yes	characteristic
76	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
77	C₄×C₈₉	#486	1	1	yes	characteristic
78	C₄×C₈₉	#486	1	1	yes	characteristic
79	#12891	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	#1834	1	1	yes	characteristic
83	C₂×C₈₉	C₂×#368	1	2	yes	—
84	C₂×C₄×C₁₆	—	89	1	no	—
85	#1848	—	89	1	no	—
86	#1848	D₈₉	1	1	yes	characteristic
87	#1381	—	89	4	no	—
88	C₈₉	#12891	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	89	2	no	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	—	89	2	no	—
92	C₂×#108	—	178	1	no	—
93	C₂×#108	C₂×D₈₉	1	2	yes	—
94	#423	—	2	1	no	—
95	#423	—	178	1	no	—
96	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
97	C₄×C₁₆	—	89	2	no	—
98	C₄×C₁₆	—	89	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	178	2	no	—
101	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂×C₁₆	—	89	4	no	—
105	C₂×C₁₆	—	178	2	no	—
106	C₂×C₁₆	—	178	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	356	2	no	—
110	#108	—	178	4	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	356	2	no	—
113	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
114	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
115	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
116	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
117	C₂²×C₄	#13083564	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	356	2	no	—
122	M₁₆	—	356	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	356	2	no	—
125	C₂×C₈	#13083553	1	1	yes	characteristic
126	C₂×C₈	#13083553	1	1	yes	characteristic
127	C₂×C₈	#13083561	1	2	yes	—
128	C₂×C₈	#13083562	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
131	C₂×C₈	#13083573	1	2	yes	—
132	C₂×C₈	#13083575	1	2	yes	—
133	C₄²	#13083553	1	1	yes	characteristic
134	C₄²	#13083553	1	1	yes	characteristic
135	C₄²	#13083559	1	2	yes	—
136	C₁₆	—	178	4	no	—
137	C₂³	C₂×#13083564	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#15667945	1	2	yes	—
140	C₂×C₄	#15667945	1	2	yes	—
141	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
142	C₂×C₄	#15667999	1	1	yes	characteristic
143	C₂×C₄	#15668001	1	1	yes	characteristic
144	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
145	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
146	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	712	2	no	—
149	C₈	#15667943	1	2	yes	—
150	C₈	#15667943	1	2	yes	—
151	C₈	#15667970	1	2	yes	—
152	C₈	#15668003	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#15667945	1	1	yes	characteristic
155	C₂²	#19151948	1	1	yes	characteristic
156	C₂²	#19151950	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#19151427	1	4	yes	—
159	C₄	#19151463	1	1	yes	characteristic
160	C₄	C₂×#15667943	1	1	yes	characteristic
161	C₄	#19151944	1	1	yes	characteristic
162	C₄	#19151946	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#26020379	1	1	yes	characteristic
166	C₂	C₂×#19151427	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#12891
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
88. C₈₉	#12891	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083549	2
#26020424	2
#26018345	2
C₈₉×#1848	2
#12891	89