Subgroups — #36465982

Order 22784.

234 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	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	#26018317	C₂	1	4	yes	maximal, complemented
4	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₄	1	1	yes	characteristic
6	C₂²×#13083549	C₄	1	1	yes	characteristic
7	#19151651	—	2	1	no	—
8	#19151374	C₄	1	2	yes	—
9	#19151374	C₂²	1	2	yes	—
10	#19151351	—	2	1	no	—
11	#19151349	C₂²	1	2	yes	—
12	#431×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#335×C₈₉	—	2	1	no	—
15	#334×C₈₉	C₂²	1	2	yes	—
16	C₂×#13083549	—	2	2	no	—
17	C₂×#13083549	C₈	1	4	yes	complemented
18	C₂×#13083549	C₂×C₄	1	2	yes	—
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₂²⋊#10623494	—	4	1	no	—
22	C₄×#10623494	—	4	1	no	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	—	2	2	no	—
27	C₂²⋊C₈×C₈₉	—	4	1	no	—
28	C₄×C₈×C₈₉	—	4	1	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×#10623494	—	4	2	no	—
34	#13083549	—	2	2	no	—
35	#13083549	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
49	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
50	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₈₉	#106	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	4	1	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#108	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	—	8	2	no	—
65	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
66	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
68	#10623494	—	8	2	no	—
69	C₂²×C₈₉	—	2	1	no	—
70	C₂²×C₈₉	—	2	1	no	—
71	C₂²×C₈₉	#334	1	2	yes	—
72	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	#334	1	2	yes	—
78	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
80	#11999	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	2	no	—
82	C₂×C₈₉	C₂×#334	1	1	yes	characteristic
83	C₂×C₈₉	#2155	1	1	yes	characteristic
84	C₂×C₈₉	#2155	1	1	yes	characteristic
85	C₂×#346	—	89	1	no	—
86	#1564	—	89	1	no	—
87	#1564	D₈₉	1	1	yes	characteristic
88	#1368	—	89	4	no	—
89	C₈₉	#11999	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	—	89	1	no	—
91	C₂²×C₁₆	—	89	1	no	—
92	#431	—	2	1	no	—
93	#431	—	178	1	no	—
94	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
95	#346	—	89	2	no	—
96	#346	—	89	2	no	—
97	#335	—	2	1	no	—
98	#335	—	178	1	no	—
99	#334	—	89	2	no	—
100	#334	C₂×D₈₉	1	2	yes	—
101	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
102	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂×C₁₆	—	89	4	no	—
105	C₂×C₁₆	—	89	2	no	—
106	C₂×C₁₆	—	89	2	no	—
107	C₂×C₁₆	—	178	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	178	2	no	—
110	C₂²⋊C₈	—	4	1	no	—
111	C₂²⋊C₈	—	356	1	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	—	356	1	no	—
114	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
115	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₂²×C₄	#13083553	1	1	yes	characteristic
119	C₂²×C₄	#13083561	1	1	yes	characteristic
120	C₂²×C₄	#13083562	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	356	2	no	—
124	C₂×C₈	#13083553	1	1	yes	characteristic
125	C₂×C₈	#13083553	1	1	yes	characteristic
126	C₂×C₈	C₈×D₈₉	1	2	yes	—
127	C₂×C₈	#13083557	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
130	C₄²	#13083553	1	1	yes	characteristic
131	C₄²	#13083561	1	1	yes	characteristic
132	C₄²	#13083562	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
134	C₁₆	—	89	4	no	—
135	C₁₆	—	178	2	no	—
136	C₂³	#15667943	1	1	yes	characteristic
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#15667941	1	2	yes	—
140	C₂×C₄	#15667943	1	1	yes	characteristic
141	C₂×C₄	#15667943	1	1	yes	characteristic
142	C₂×C₄	#15667943	1	1	yes	characteristic
143	C₂×C₄	#15667947	1	2	yes	—
144	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
145	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
146	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	8	2	no	—
150	C₈	—	712	2	no	—
151	C₈	#15667941	1	2	yes	—
152	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
153	C₈	C₂×#13083557	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#19151348	1	2	yes	—
157	C₂²	C₂×#15667943	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	#19151348	1	2	yes	—
163	C₄	C₂×#15667941	1	1	yes	characteristic
164	C₄	C₂×#15667947	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	C₂×#19151348	1	1	yes	characteristic
167	C₂	#26022024	1	1	yes	characteristic
168	C₂	#26022026	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667943
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×C₈₉	#106
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#11999
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #26018317	C₂	semidirect
17. C₂×#13083549	C₈	semidirect
89. C₈₉	#11999	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151374	2
#26019000	2
#26018317	2
C₈₉×#1564	2
#11999	89