Subgroups — #36469754

Order 22784.

334 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022008	C₂	1	2	yes	maximal
2	C₂×#19151376	C₂	1	2	yes	maximal
3	#26018332	C₂	1	2	yes	maximal
4	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₂²	1	4	yes	—
6	C₄⋊#13083549	C₄	1	8	yes	complemented
7	#19151376	—	2	4	no	—
8	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
9	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#376×C₈₉	C₄	1	1	yes	characteristic
15	#15667952	—	2	4	no	—
16	#15667952	—	2	4	no	—
17	C₂×#13083549	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	#97×C₈₉	—	2	2	no	—
26	#97×C₈₉	—	2	2	no	—
27	#97×C₈₉	D₄	1	2	yes	—
28	#97×C₈₉	Q₈	1	2	yes	—
29	C₄×C₈×C₈₉	—	2	2	no	—
30	C₄×C₈×C₈₉	—	2	2	no	—
31	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
32	#95×C₈₉	—	2	2	no	—
33	#13083549	—	2	8	no	—
34	C₂²×C₄×C₈₉	—	2	1	no	—
35	C₂²×C₄×C₈₉	—	2	1	no	—
36	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₂×C₈×C₈₉	—	2	1	no	—
40	C₂×C₈×C₈₉	—	2	1	no	—
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	C₄²	1	2	yes	—
45	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₈₉	#41	1	2	yes	—
51	C₄²×C₈₉	—	2	1	no	—
52	C₄²×C₈₉	—	2	1	no	—
53	C₄²×C₈₉	C₄²	1	2	yes	—
54	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₈₉	M₁₆	1	2	yes	—
56	C₄²×C₈₉	M₁₆	1	2	yes	—
57	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	4	2	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	—	4	2	no	—
63	C₂×C₄×C₈₉	#97	1	4	yes	—
64	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₈₉	—	2	2	no	—
71	C₈×C₈₉	—	2	2	no	—
72	C₈×C₈₉	—	2	2	no	—
73	C₈×C₈₉	C₄×D₄	1	2	yes	—
74	C₈×C₈₉	C₄×Q₈	1	2	yes	—
75	C₂²×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
77	C₂²×C₈₉	#423	1	1	yes	characteristic
78	C₂²×C₈₉	#432	1	1	yes	characteristic
79	C₄×C₈₉	—	2	4	no	—
80	C₄×C₈₉	—	4	2	no	—
81	C₄×C₈₉	—	4	2	no	—
82	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
84	C₄×C₈₉	#423	1	1	yes	characteristic
85	C₄×C₈₉	#432	1	1	yes	characteristic
86	#12657	—	89	1	no	maximal, Sylow
87	C₂×C₈₉	—	2	1	no	—
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	#1767	1	1	yes	characteristic
90	C₂×C₈₉	#2110	1	2	yes	—
91	#2150	—	89	2	no	—
92	C₂×#347	—	89	2	no	—
93	#1726	D₈₉	1	1	yes	characteristic
94	#1375	—	89	2	no	—
95	C₈₉	#12657	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₃₂	—	89	4	no	—
97	C₂×#97	Dic₈₉	1	1	yes	characteristic
98	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
99	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
100	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
101	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
103	#376	Dic₈₉	1	1	yes	characteristic
104	C₄⋊C₁₆	—	89	8	no	—
105	#347	—	178	4	no	—
106	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
113	M₃₂	—	178	4	no	—
114	M₃₂	—	178	4	no	—
115	C₂×C₁₆	—	89	8	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	Dic₁₇₈	1	2	yes	—
119	#97	D₃₅₆	1	2	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₈₉	1	4	yes	—
123	#95	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	#13083558	1	2	yes	—
127	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
135	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
136	C₂×C₈	#13083558	1	2	yes	—
137	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
138	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
140	C₂×C₈	#13083574	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄×Dic₈₉	1	2	yes	—
144	C₄²	C₄⋊Dic₈₉	1	2	yes	—
145	C₄²	#13083558	1	2	yes	—
146	C₄²	#13083558	1	2	yes	—
147	C₁₆	—	178	8	no	—
148	C₂³	C₂×#13083558	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#15667923	1	4	yes	—
155	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
157	C₂×C₄	#15667971	1	1	yes	characteristic
158	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
159	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
160	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	C₄×D₃₅₆	1	2	yes	—
165	C₈	C₄×Dic₁₇₈	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₂×#15667923	1	1	yes	characteristic
168	C₂²	#19151622	1	1	yes	characteristic
169	C₂²	#19151655	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
174	C₄	C₄×#13083558	1	1	yes	characteristic
175	C₄	#19151622	1	1	yes	characteristic
176	C₄	#19151655	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#26020105	1	1	yes	characteristic
180	C₂	#26021857	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1726	C₂
socle	C₂²×C₈₉	C₂×#97
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#12657
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. C₄⋊#13083549	C₄	semidirect
95. C₈₉	#12657	semidirect

Maximal subgroups

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

subgroup	index
#26022008	2
C₂×#19151376	2
#26018332	2
C₈₉×#1726	2
#12657	89