Subgroups — #36441320

Order 22784.

238 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	#26022051	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083549	—	2	1	no	—
5	C₄⋊#13083549	C₄	1	2	yes	—
6	C₂²⋊#13083549	—	2	1	no	—
7	C₄×#13083549	—	2	1	no	—
8	C₄×#13083549	—	2	1	no	—
9	C₄×#13083549	C₄	1	2	yes	—
10	#431×C₈₉	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#322×C₈₉	—	2	1	no	—
14	C₈²×C₈₉	—	2	1	no	—
15	C₂×#13083549	—	2	4	no	—
16	C₂×#13083549	—	2	2	no	—
17	C₂×#13083549	—	2	2	no	—
18	C₂×M₁₆×C₈₉	C₈	1	2	yes	—
19	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₈	1	2	yes	—
23	C₂²⋊C₈×C₈₉	—	2	1	no	—
24	#97×C₈₉	—	2	1	no	—
25	C₄×C₈×C₈₉	—	2	1	no	—
26	C₄×C₈×C₈₉	—	2	1	no	—
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	#13083549	—	2	4	no	—
32	#13083549	—	4	2	no	—
33	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
36	M₁₆×C₈₉	—	2	2	no	—
37	M₁₆×C₈₉	C₁₆	1	4	yes	complemented
38	C₂×C₈×C₈₉	—	2	2	no	—
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₄²	1	1	yes	characteristic
47	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
50	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	—	2	4	no	—
65	C₈×C₈₉	—	4	2	no	—
66	C₈×C₈₉	#104	1	2	yes	—
67	C₈×C₈₉	#108	1	2	yes	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	—	2	1	no	—
70	C₂²×C₈₉	#336	1	1	yes	characteristic
71	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	#339	1	1	yes	characteristic
75	C₄×C₈₉	#341	1	1	yes	characteristic
76	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
77	C₄×C₈₉	#346	1	1	yes	characteristic
78	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
79	C₄×C₈₉	#350	1	2	yes	—
80	#7163	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	2	no	—
82	C₂×C₈₉	#1278	1	1	yes	characteristic
83	C₂×C₈₉	#1374	1	1	yes	characteristic
84	C₂×C₈₉	#1379	1	1	yes	characteristic
85	#2162	—	89	1	no	—
86	C₂×C₄×C₁₆	—	89	1	no	—
87	C₈×M₁₆	D₈₉	1	1	yes	characteristic
88	C₈₉	#7163	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	178	1	no	—
90	#431	Dic₈₉	1	1	yes	characteristic
91	C₄×M₁₆	Dic₈₉	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
93	C₄⋊C₁₆	—	89	2	no	—
94	C₂²⋊C₁₆	—	178	1	no	—
95	C₄×C₁₆	—	89	2	no	—
96	C₄×C₁₆	—	178	1	no	—
97	C₄×C₁₆	—	178	1	no	—
98	#322	—	2	1	no	—
99	C₈²	—	2	1	no	—
100	C₂×M₁₆	#10623494	1	2	yes	—
101	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
102	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
104	C₂×C₁₆	—	178	2	no	—
105	C₂×C₁₆	—	178	4	no	—
106	C₂×C₁₆	—	178	2	no	—
107	C₄⋊C₈	#10623494	1	2	yes	—
108	C₂²⋊C₈	—	2	1	no	—
109	#97	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
113	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
117	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
118	C₂²×C₄	#13083558	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	#13083549	1	4	yes	complemented
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	#13083551	1	1	yes	characteristic
126	C₂×C₈	#13083551	1	1	yes	characteristic
127	C₂×C₈	#13083553	1	1	yes	characteristic
128	C₂×C₈	#13083553	1	1	yes	characteristic
129	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
130	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
131	C₄²	C₂×#10623494	1	1	yes	characteristic
132	C₄²	#13083558	1	1	yes	characteristic
133	C₁₆	—	356	2	no	—
134	C₁₆	—	178	4	no	—
135	C₂³	C₄⋊#10623494	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	#15667919	1	1	yes	characteristic
140	C₂×C₄	C₄×#10623494	1	1	yes	characteristic
141	C₂×C₄	#15667923	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
144	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
145	C₂×C₄	C₂×#13083549	1	2	yes	—
146	C₂×C₄	#15667952	1	2	yes	—
147	C₈	—	2	1	no	—
148	C₈	—	2	4	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	2	no	—
151	C₈	#15667938	1	2	yes	—
152	C₈	#15667947	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#19151353	1	1	yes	characteristic
156	C₂²	C₄⋊#13083549	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#19151359	1	1	yes	characteristic
160	C₄	#19151362	1	1	yes	characteristic
161	C₄	C₄×#13083549	1	1	yes	characteristic
162	C₄	#19151374	1	1	yes	characteristic
163	C₄	C₂²⋊#13083549	1	2	yes	—
164	C₄	#19151382	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#26018090	1	1	yes	characteristic
167	C₂	#26018330	1	1	yes	characteristic
168	C₂	#26018341	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#336
prime radical	C₂³×C₈₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#7163
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
37. M₁₆×C₈₉	C₁₆	semidirect
88. C₈₉	#7163	semidirect
120. M₁₆	#13083549	semidirect

Maximal subgroups

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

subgroup	index
#26022051	2
C₂×C₄×#13083549	2
C₈×M₁₆×C₈₉	2
#7163	89