Subgroups — #36448078

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019000	C₂	1	2	yes	maximal
2	#26018097	C₂	1	2	yes	maximal
3	#26018081	C₂	1	2	yes	maximal
4	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	4	yes	—
6	#19151351	—	2	2	no	—
7	#19151349	—	2	2	no	—
8	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
9	C₂×#97×C₈₉	C₄	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	C₂²	1	1	yes	characteristic
15	C₄⋊#10623494	—	2	4	no	—
16	C₂²⋊#10623494	—	2	4	no	—
17	C₄×#10623494	—	2	4	no	—
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	C₂×C₄	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	C₂³	1	1	yes	characteristic
25	#97×C₈₉	—	2	2	no	—
26	#97×C₈₉	—	2	2	no	—
27	#97×C₈₉	C₈	1	4	yes	complemented
28	C₄×C₈×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	—	2	2	no	—
30	C₄×C₈×C₈₉	D₄	1	2	yes	—
31	C₄×C₈×C₈₉	Q₈	1	2	yes	—
32	#95×C₈₉	—	2	2	no	—
33	C₂×#10623494	—	4	4	no	—
34	C₂²×C₄×C₈₉	—	2	1	no	—
35	C₂²×C₄×C₈₉	—	2	1	no	—
36	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	#41	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
49	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
50	C₄²×C₈₉	—	2	1	no	—
51	C₄²×C₈₉	—	2	1	no	—
52	C₄²×C₈₉	C₂×C₈	1	2	yes	—
53	C₄²×C₈₉	M₁₆	1	2	yes	—
54	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₉	#41	1	2	yes	—
57	C₂³×C₈₉	#117	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	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#124	1	2	yes	—
66	C₂×C₄×C₈₉	#125	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²×C₈	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	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₈₉	—	2	2	no	—
72	C₈×C₈₉	—	2	2	no	—
73	C₈×C₈₉	—	2	2	no	—
74	C₈×C₈₉	C₄⋊C₈	1	4	yes	—
75	#10623494	—	8	4	no	—
76	C₂²×C₈₉	—	2	2	no	—
77	C₂²×C₈₉	#382	1	1	yes	characteristic
78	C₂²×C₈₉	#431	1	1	yes	characteristic
79	C₂²×C₈₉	#432	1	1	yes	characteristic
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	—	4	2	no	—
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₈₉	#423	1	1	yes	characteristic
86	C₄×C₈₉	#431	1	1	yes	characteristic
87	C₄×C₈₉	#432	1	1	yes	characteristic
88	#8781	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	—	2	1	no	—
90	C₂×C₈₉	—	2	1	no	—
91	C₂×C₈₉	#1831	1	1	yes	characteristic
92	C₂×C₈₉	#1833	1	2	yes	—
93	#1726	D₈₉	1	1	yes	characteristic
94	#1564	—	89	2	no	—
95	#1281	—	89	2	no	—
96	#1274	—	89	2	no	—
97	C₈₉	#8781	1	1	yes	characteristic, complemented, Sylow
98	#431	—	89	4	no	—
99	C₂×#97	Dic₈₉	1	1	yes	characteristic
100	C₂×#97	Dic₈₉	1	1	yes	characteristic
101	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
102	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
103	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	#376	C₂×D₈₉	1	1	yes	characteristic
106	#335	—	178	2	no	—
107	#334	—	178	2	no	—
108	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
114	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
115	C₄⋊C₈	—	178	4	no	—
116	C₂²⋊C₈	—	178	4	no	—
117	#97	—	2	2	no	—
118	#97	—	2	2	no	—
119	#97	#10623494	1	4	yes	complemented
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	178	4	no	—
123	C₄×C₈	Dic₁₇₈	1	2	yes	—
124	C₄×C₈	D₃₅₆	1	2	yes	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
129	C₂²×C₄	#13083574	1	2	yes	—
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	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	356	4	no	—
137	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
138	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
139	C₂×C₈	C₂×#10623494	1	2	yes	—
140	C₂×C₈	C₂×#10623494	1	2	yes	—
141	C₂×C₈	#13083558	1	2	yes	—
142	C₂×C₈	#13083558	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂×#10623494	1	2	yes	—
146	C₄²	#13083558	1	2	yes	—
147	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
148	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
149	C₄²	#13083574	1	2	yes	—
150	C₂³	#15667971	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
157	C₂×C₄	#15667971	1	1	yes	characteristic
158	C₂×C₄	#15668000	1	2	yes	—
159	C₂×C₄	#15668004	1	2	yes	—
160	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
161	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
162	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
163	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	712	4	no	—
168	C₈	C₄⋊#10623494	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	#19151464	1	1	yes	characteristic
171	C₂²	#19151651	1	1	yes	characteristic
172	C₂²	#19151655	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
178	C₄	#19151622	1	1	yes	characteristic
179	C₄	#19151651	1	1	yes	characteristic
180	C₄	#19151655	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#26020369	1	1	yes	characteristic
184	C₂	#26020376	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1726	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#8781
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
27. #97×C₈₉	C₈	semidirect
97. C₈₉	#8781	semidirect
119. #97	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
#26018097	2
#26018081	2
C₈₉×#1726	2
#8781	89