Subgroups — #36453170

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020424	C₂	1	1	yes	maximal, characteristic
2	#26020119	C₂	1	1	yes	maximal, characteristic
3	#26019023	C₂	1	2	yes	maximal
4	#26018154	C₂	1	2	yes	maximal
5	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	—	2	1	no	—
7	C₂×#15667948	—	2	1	no	—
8	#19151622	C₂²	1	2	yes	—
9	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₂×#15667922	—	2	1	no	—
11	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
12	#19151352	—	2	1	no	—
13	#19151349	—	2	4	no	—
14	#431×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#335×C₈₉	C₄	1	2	yes	complemented
17	#334×C₈₉	C₄	1	2	yes	complemented
18	C₂×#13083558	—	2	2	no	—
19	C₂²×#10623494	—	2	1	no	—
20	C₂×C₄⋊Dic₈₉	—	2	1	no	—
21	C₂×C₄×Dic₈₉	—	2	1	no	—
22	#15667948	—	4	2	no	—
23	#15667948	—	4	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	—	2	1	no	—
26	C₄⋊#10623494	—	2	2	no	—
27	C₄⋊#10623494	D₄	1	2	yes	—
28	#15667922	—	4	2	no	—
29	#15667919	Q₈	1	2	yes	—
30	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
40	C₂²×Dic₈₉	—	2	2	no	—
41	#13083558	—	4	2	no	—
42	#13083558	—	4	2	no	—
43	C₂×#10623494	—	2	2	no	—
44	C₂×#10623494	—	4	1	no	—
45	C₂×#10623494	—	4	1	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	C₄×Dic₈₉	—	4	1	no	—
49	C₄×Dic₈₉	—	4	1	no	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₉	—	2	1	no	—
54	C₂×C₈×C₈₉	—	2	1	no	—
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
61	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
62	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	4	1	no	—
69	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	—	4	2	no	—
82	C₈×C₈₉	—	2	1	no	—
83	C₈×C₈₉	—	2	1	no	—
84	C₈×C₈₉	—	4	4	no	—
85	C₈×C₈₉	#106	1	2	yes	—
86	C₈×C₈₉	#106	1	2	yes	—
87	#10623494	—	4	2	no	—
88	#10623494	—	8	2	no	—
89	C₂²×C₈₉	—	2	1	no	—
90	C₂²×C₈₉	—	2	1	no	—
91	C₂²×C₈₉	#389	1	1	yes	characteristic
92	C₂²×C₈₉	#475	1	1	yes	characteristic
93	C₂²×C₈₉	#476	1	1	yes	characteristic
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	—	2	1	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
98	C₄×C₈₉	#428	1	1	yes	characteristic
99	C₄×C₈₉	#444	1	2	yes	—
100	C₄×C₈₉	#460	1	1	yes	characteristic
101	C₄×C₈₉	#461	1	1	yes	characteristic
102	Dic₈₉	—	8	2	no	—
103	#9631	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	—	2	1	no	—
105	C₂×C₈₉	—	2	1	no	—
106	C₂×C₈₉	#1946	1	1	yes	characteristic
107	C₂×C₈₉	#1948	1	1	yes	characteristic
108	C₂×C₈₉	#1953	1	1	yes	characteristic
109	#1848	—	89	1	no	—
110	#1770	—	89	1	no	—
111	#1570	—	89	2	no	—
112	#1564	D₈₉	1	1	yes	characteristic
113	#1303	—	89	2	no	—
114	C₈₉	#9631	1	1	yes	characteristic, complemented, Sylow
115	#431	C₂×D₈₉	1	2	yes	—
116	C₂×#108	—	178	1	no	—
117	C₂×#108	—	178	1	no	—
118	#423	—	89	2	no	—
119	C₂×C₄⋊C₈	—	89	1	no	—
120	C₂×#97	—	178	1	no	—
121	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	89	1	no	—
123	#336	—	178	1	no	—
124	#335	Dic₈₉	1	2	yes	complemented
125	#334	—	178	4	no	—
126	#334	Dic₈₉	1	2	yes	complemented
127	C₂×M₁₆	—	178	2	no	—
128	C₂²×C₈	—	178	1	no	—
129	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
130	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	178	1	no	—
132	C₂×C₄²	—	178	1	no	—
133	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
134	#108	—	356	2	no	—
135	#108	—	356	2	no	—
136	C₄⋊C₈	—	89	2	no	—
137	C₄⋊C₈	—	178	2	no	—
138	C₄⋊C₈	—	178	2	no	—
139	C₄⋊C₈	—	178	1	no	—
140	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	356	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
145	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
148	#95	—	89	2	no	—
149	C₂²×C₄	—	178	2	no	—
150	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
151	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
152	C₂²×C₄	#13083573	1	1	yes	characteristic
153	M₁₆	—	356	2	no	—
154	M₁₆	—	356	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	356	1	no	—
159	C₂×C₈	—	356	1	no	—
160	C₂×C₈	—	356	2	no	—
161	C₂×C₈	—	178	2	no	—
162	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
163	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
164	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
166	C₂×C₈	#13083563	1	2	yes	—
167	C₂×C₈	#13083563	1	2	yes	—
168	C₄⋊C₄	—	356	2	no	—
169	C₄²	—	356	1	no	—
170	C₄²	—	356	1	no	—
171	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
172	C₄²	D₄×D₈₉	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
174	C₄²	#13083575	1	1	yes	characteristic
175	C₂³	#15667992	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	356	2	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	356	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
184	C₂×C₄	#15667987	1	1	yes	characteristic
185	C₂×C₄	#15668003	1	1	yes	characteristic
186	C₂×C₄	#15668011	1	1	yes	characteristic
187	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
188	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
189	C₂×C₄	#15668040	1	1	yes	characteristic
190	C₂×C₄	#15668045	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	356	2	no	—
195	C₈	—	712	2	no	—
196	C₈	#15667944	1	2	yes	—
197	C₈	#15667944	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#19151497	1	1	yes	characteristic
201	C₂²	#19151884	1	1	yes	characteristic
202	C₂²	#19151889	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	712	2	no	—
207	C₄	C₂×#15667944	1	1	yes	characteristic
208	C₄	#19151640	1	1	yes	characteristic
209	C₄	#19151720	1	2	yes	—
210	C₄	#19151799	1	1	yes	characteristic
211	C₄	#19151804	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#26021006	1	1	yes	characteristic
215	C₂	#26021016	1	1	yes	characteristic
216	C₂	#26021043	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	#389
prime radical	C₂³×C₈₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#9631
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. #335×C₈₉	C₄	semidirect
17. #334×C₈₉	C₄	semidirect
114. C₈₉	#9631	semidirect
124. #335	Dic₈₉	semidirect
126. #334	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020424	2
#26020119	2
#26019023	2
#26018154	2
C₈₉×#1564	2
#9631	89