Subgroups — #36472579

Order 22784.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022024	C₂	1	2	yes	maximal
2	C₂×#19151419	C₂	1	1	yes	maximal, characteristic
3	#26020424	C₂	1	1	yes	maximal, characteristic
4	#26018363	C₂	1	2	yes	maximal
5	C₈₉×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂×#15667948	—	2	1	no	—
8	C₂×#15667948	C₂²	1	2	yes	—
9	#19151622	—	2	1	no	—
10	#19151420	—	2	4	no	—
11	#19151419	C₄	1	2	yes	complemented
12	#19151419	C₄	1	2	yes	complemented
13	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
14	#427×C₈₉	C₂²	1	1	yes	characteristic
15	#405×C₈₉	—	2	1	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#339×C₈₉	—	2	1	no	—
18	C₂×#13083558	—	2	2	no	—
19	#15667952	—	2	2	no	—
20	#15667952	C₂×C₄	1	4	yes	—
21	C₂×#13083549	—	2	2	no	—
22	#15667948	—	2	2	no	—
23	#15667948	—	2	2	no	—
24	#15667948	—	4	1	no	—
25	#15667948	—	4	1	no	—
26	C₄⋊#10623494	—	4	2	no	—
27	C₂×M₁₆×C₈₉	—	2	1	no	—
28	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	#117×C₈₉	—	2	1	no	—
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	#108×C₈₉	—	2	1	no	—
33	#108×C₈₉	Q₈	1	2	yes	—
34	#107×C₈₉	—	2	1	no	—
35	#106×C₈₉	D₄	1	2	yes	—
36	#97×C₈₉	—	4	1	no	—
37	C₄×C₈×C₈₉	—	4	1	no	—
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
42	#13083558	—	4	2	no	—
43	#13083558	—	4	2	no	—
44	C₂×#10623494	—	4	2	no	—
45	#13083549	—	2	4	no	—
46	C₂²×C₄×C₈₉	—	2	1	no	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₈₉	—	2	2	no	—
49	M₁₆×C₈₉	—	4	1	no	—
50	C₂×C₈×C₈₉	—	4	1	no	—
51	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₂²⋊C₄×C₈₉	—	4	1	no	—
65	C₄²×C₈₉	—	2	1	no	—
66	C₄²×C₈₉	—	4	1	no	—
67	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₈₉	#137	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₈×C₈₉	—	4	2	no	—
81	C₈×C₈₉	#106	1	1	yes	characteristic
82	C₈×C₈₉	#106	1	1	yes	characteristic
83	C₈×C₈₉	#106	1	1	yes	characteristic
84	C₈×C₈₉	#106	1	1	yes	characteristic
85	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₈₉	#125	1	1	yes	characteristic
88	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
89	#10623494	—	8	2	no	—
90	C₂²×C₈₉	—	4	1	no	—
91	C₂²×C₈₉	#428	1	1	yes	characteristic
92	C₂²×C₈₉	#461	1	1	yes	characteristic
93	C₂²×C₈₉	#475	1	1	yes	characteristic
94	C₄×C₈₉	—	2	2	no	—
95	C₄×C₈₉	—	4	1	no	—
96	C₄×C₈₉	—	8	2	no	—
97	C₄×C₈₉	#389	1	1	yes	characteristic
98	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
99	C₄×C₈₉	#460	1	1	yes	characteristic
100	C₄×C₈₉	#476	1	1	yes	characteristic
101	#13188	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	—	2	1	no	—
103	C₂×C₈₉	—	2	1	no	—
104	C₂×C₈₉	—	2	1	no	—
105	C₂×C₈₉	#1948	1	1	yes	characteristic
106	#2155	—	89	2	no	—
107	C₂×#364	—	89	1	no	—
108	#1848	—	89	1	no	—
109	#1772	D₈₉	1	1	yes	characteristic
110	#1388	—	89	2	no	—
111	C₈₉	#13188	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₃₂	—	89	2	no	—
113	C₂×#108	—	89	2	no	—
114	C₂×#108	—	178	1	no	—
115	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
116	#427	C₂×D₈₉	1	1	yes	characteristic
117	#423	—	178	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
120	#365	—	178	4	no	—
121	#364	—	89	2	no	—
122	#364	—	89	2	no	—
123	#339	—	2	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	178	2	no	—
126	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
130	M₃₂	—	89	4	no	—
131	M₃₂	—	178	2	no	—
132	C₂×C₁₆	—	178	2	no	—
133	#108	—	2	1	no	—
134	#108	—	356	1	no	—
135	#108	—	356	1	no	—
136	#108	—	178	2	no	—
137	#108	—	178	2	no	—
138	#108	Dic₁₇₈	1	2	yes	—
139	#107	—	2	1	no	—
140	#106	D₃₅₆	1	2	yes	—
141	C₄⋊C₈	—	356	2	no	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
145	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
146	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
147	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	356	2	no	—
153	M₁₆	—	356	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	356	2	no	—
156	C₂×C₈	#13083553	1	1	yes	characteristic
157	C₂×C₈	#13083553	1	1	yes	characteristic
158	C₂×C₈	#13083561	1	1	yes	characteristic
159	C₂×C₈	#13083561	1	1	yes	characteristic
160	C₂×C₈	#13083562	1	1	yes	characteristic
161	C₂×C₈	#13083562	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
163	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
166	C₂×C₈	#13083574	1	1	yes	characteristic
167	C₂×C₈	#13083574	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	#13083553	1	1	yes	characteristic
173	C₄²	#13083553	1	1	yes	characteristic
174	C₁₆	—	178	4	no	—
175	C₂³	#15668042	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
181	C₂×C₄	#15667986	1	1	yes	characteristic
182	C₂×C₄	#15667991	1	1	yes	characteristic
183	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
184	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
185	C₂×C₄	#15668037	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	712	2	no	—
188	C₈	#15667943	1	1	yes	characteristic
189	C₈	#15667943	1	1	yes	characteristic
190	C₈	#15667943	1	1	yes	characteristic
191	C₈	#15667943	1	1	yes	characteristic
192	C₈	C₄×D₃₅₆	1	1	yes	characteristic
193	C₈	C₄×Dic₁₇₈	1	1	yes	characteristic
194	C₈	#15668004	1	1	yes	characteristic
195	C₈	#15668012	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#19151641	1	1	yes	characteristic
198	C₂²	#19151803	1	1	yes	characteristic
199	C₂²	#19151883	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	#19151496	1	1	yes	characteristic
204	C₄	C₂×#15667943	1	1	yes	characteristic
205	C₄	#19151798	1	1	yes	characteristic
206	C₄	#19151888	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#26021015	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151888
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1772	C₂
socle	C₂×C₈₉	#1948
prime radical	C₂³×C₈₉	#137
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1772	D₈₉
hypocenter	C₈₉	#13188
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
11. #19151419	C₄	semidirect
12. #19151419	C₄	semidirect
111. C₈₉	#13188	semidirect

Maximal subgroups

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

subgroup	index
#26022024	2
C₂×#19151419	2
#26020424	2
#26018363	2
C₈₉×#1772	2
#13188	89