Subgroups — #36448178

Order 22784.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020404	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151350	C₂	1	2	yes	maximal
4	C₂×#19151348	C₂	1	2	yes	maximal
5	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	—	2	2	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×#15667923	C₄	1	1	yes	characteristic
9	C₂×#15667923	C₄	1	2	yes	—
10	C₂×#15667923	C₄	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	2	yes	—
12	#19151350	—	2	4	no	—
13	#19151348	—	2	4	no	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄³×C₈₉	C₂²	1	1	yes	characteristic
17	#336×C₈₉	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂²×#10623494	C₂×C₄	1	2	yes	—
20	C₂²×#10623494	C₂×C₄	1	2	yes	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	4	4	no	—
24	#15667923	—	2	4	no	—
25	#15667923	—	2	2	no	—
26	#15667923	C₈	1	4	yes	complemented
27	C₄×#10623494	D₄	1	4	yes	—
28	C₄×#10623494	Q₈	1	4	yes	—
29	C₂²×C₈×C₈₉	—	2	1	no	—
30	C₂²×C₈×C₈₉	—	2	1	no	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	D₄	1	2	yes	—
34	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₈₉	—	2	2	no	—
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	C₂×#10623494	—	2	2	no	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	4	no	—
42	C₂×#10623494	C₄⋊C₄	1	4	yes	—
43	C₂×#10623494	C₄⋊C₄	1	4	yes	—
44	C₂×#10623494	C₂×C₈	1	4	yes	—
45	C₂×#10623494	M₁₆	1	4	yes	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₈₉	—	4	1	no	—
52	C₂×C₈×C₈₉	—	4	1	no	—
53	C₂×C₈×C₈₉	—	4	2	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₄²×C₈₉	—	2	2	no	—
56	C₄²×C₈₉	—	2	2	no	—
57	C₄²×C₈₉	—	2	2	no	—
58	C₄²×C₈₉	—	4	2	no	—
59	C₄²×C₈₉	C₂×C₈	1	2	yes	—
60	C₄²×C₈₉	M₁₆	1	2	yes	—
61	C₄²×C₈₉	C₂×D₄	1	2	yes	—
62	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
63	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₈₉	#137	1	2	yes	—
80	C₈×C₈₉	—	8	1	no	—
81	C₈×C₈₉	—	8	1	no	—
82	#10623494	—	2	4	no	—
83	#10623494	—	8	2	no	—
84	#10623494	C₄⋊C₈	1	8	yes	complemented
85	C₂²×C₈₉	#384	1	1	yes	characteristic
86	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₈₉	#423	1	1	yes	characteristic
88	C₂²×C₈₉	#428	1	2	yes	—
89	C₂²×C₈₉	#430	1	2	yes	—
90	C₄×C₈₉	—	2	4	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	4	2	no	—
93	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₈₉	#423	1	2	yes	—
95	C₄×C₈₉	#496	1	1	yes	characteristic
96	C₄×C₈₉	#497	1	1	yes	characteristic
97	C₄×C₈₉	#501	1	2	yes	—
98	#8808	—	89	1	no	maximal, Sylow
99	C₂×C₈₉	#1565	1	2	yes	—
100	C₂×C₈₉	#1569	1	2	yes	—
101	C₂×C₈₉	#1842	1	1	yes	characteristic
102	C₂×C₈₉	#1847	1	1	yes	characteristic
103	C₂×C₈₉	#1849	1	1	yes	characteristic
104	#1842	—	89	1	no	—
105	#1831	D₈₉	1	1	yes	characteristic
106	C₄×#97	—	89	1	no	—
107	C₂×#335	—	89	2	no	—
108	C₂×#334	—	89	2	no	—
109	C₈₉	#8808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊C₈	—	89	2	no	—
111	C₂×C₄⋊C₈	—	178	2	no	—
112	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
113	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
114	C₂×#97	—	89	1	no	—
115	C₂×#97	—	89	2	no	—
116	C₂×#97	—	89	1	no	—
117	C₂×C₄×C₈	—	89	2	no	—
118	C₄³	C₂×D₈₉	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	#335	—	178	4	no	—
121	#334	—	178	4	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	89	2	no	—
125	C₂²×C₈	—	89	2	no	—
126	C₂²×C₈	—	178	2	no	—
127	C₂×C₄²	Dic₁₇₈	1	2	yes	—
128	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
129	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
131	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	356	4	no	—
135	C₄⋊C₈	—	178	2	no	—
136	C₄⋊C₈	—	178	2	no	—
137	#97	—	89	4	no	—
138	#97	—	178	4	no	—
139	#97	—	178	2	no	—
140	C₄×C₈	—	89	4	no	—
141	C₄×C₈	—	89	4	no	—
142	C₂²×C₄	#13083553	1	2	yes	—
143	C₂²×C₄	#13083553	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	89	4	no	—
152	C₂×C₈	—	89	4	no	—
153	C₂×C₈	—	89	4	no	—
154	C₂×C₈	—	89	4	no	—
155	C₂×C₈	—	356	2	no	—
156	C₂×C₈	—	356	4	no	—
157	C₂×C₈	—	178	2	no	—
158	C₂×C₈	—	178	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₈×D₈₉	1	2	yes	—
164	C₄²	#13083557	1	2	yes	—
165	C₄²	D₄×D₈₉	1	2	yes	—
166	C₄²	Q₈×D₈₉	1	2	yes	—
167	C₂³	C₂×#13083553	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#15667941	1	4	yes	—
175	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
176	C₂×C₄	#15668008	1	1	yes	characteristic
177	C₂×C₄	#15668010	1	1	yes	characteristic
178	C₂×C₄	#15668011	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
181	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
182	C₂×C₄	#15668037	1	2	yes	—
183	C₂×C₄	#15668042	1	2	yes	—
184	C₈	—	8	1	no	—
185	C₈	—	8	1	no	—
186	C₈	—	89	8	no	—
187	C₈	—	178	4	no	—
188	C₈	—	712	2	no	—
189	C₂²	#19151467	1	1	yes	characteristic
190	C₂²	C₂×#15667941	1	1	yes	characteristic
191	C₂²	#19151621	1	1	yes	characteristic
192	C₂²	#19151641	1	2	yes	—
193	C₂²	#19151646	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
198	C₄	#19151620	1	2	yes	—
199	C₄	#19151988	1	1	yes	characteristic
200	C₄	#19151993	1	1	yes	characteristic
201	C₄	#19152008	1	2	yes	—
202	C₂	#26019005	1	2	yes	—
203	C₂	#26019020	1	2	yes	—
204	C₂	#26020403	1	1	yes	characteristic
205	C₂	#26020421	1	1	yes	characteristic
206	C₂	#26020426	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#8808
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
26. #15667923	C₈	semidirect
84. #10623494	C₄⋊C₈	semidirect
109. C₈₉	#8808	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
C₄×#15667923	2
C₂×#19151350	2
C₂×#19151348	2
C₈₉×#1831	2
#8808	89