Subgroups — #36452985

Order 22784.

378 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020970	C₂	1	2	yes	maximal
2	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
4	#26018092	C₂	1	2	yes	maximal
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667944	C₄	1	2	yes	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×#15667941	—	2	2	no	—
9	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	#19151361	C₄	1	1	yes	characteristic
11	#19151361	C₄	1	1	yes	characteristic
12	#19151352	—	2	2	no	—
13	#19151349	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#322×C₈₉	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₈₉	—	2	2	no	—
22	#15667944	—	2	4	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	#15667941	—	4	4	no	—
26	#15667919	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	1	no	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₈₉	—	2	1	no	—
33	C₄×C₈×C₈₉	—	2	1	no	—
34	C₄×C₈×C₈₉	—	2	1	no	—
35	C₄×C₈×C₈₉	D₄	1	2	yes	—
36	C₄×C₈×C₈₉	Q₈	1	2	yes	—
37	C₄×C₈×C₈₉	Q₈	1	2	yes	—
38	C₂²×Dic₈₉	—	2	2	no	—
39	C₂²×Dic₈₉	—	2	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
47	C₄×Dic₈₉	—	4	2	no	—
48	C₄×Dic₈₉	—	4	2	no	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	4	1	no	—
58	C₂×C₈×C₈₉	—	4	1	no	—
59	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
62	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂²⋊Q₈	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×M₁₆	1	2	yes	—
76	C₂×C₄×C₈₉	#131	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₈₉	#135	1	2	yes	—
79	C₂×Dic₈₉	—	2	4	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	—	4	2	no	—
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₈×C₈₉	—	2	2	no	—
86	C₈×C₈₉	—	2	2	no	—
87	C₈×C₈₉	—	2	4	no	—
88	C₈×C₈₉	—	4	2	no	—
89	#10623494	—	8	2	no	—
90	C₂²×C₈₉	#389	1	1	yes	characteristic
91	C₂²×C₈₉	#435	1	2	yes	—
92	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
93	C₂²×C₈₉	#444	1	1	yes	characteristic
94	C₂²×C₈₉	#444	1	1	yes	characteristic
95	C₄×C₈₉	—	2	1	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	#423	1	1	yes	characteristic
98	C₄×C₈₉	#424	1	1	yes	characteristic
99	C₄×C₈₉	#446	1	2	yes	—
100	C₄×C₈₉	#465	1	2	yes	—
101	C₄×C₈₉	#498	1	1	yes	characteristic
102	C₄×C₈₉	#499	1	1	yes	characteristic
103	Dic₈₉	—	4	4	no	—
104	Dic₈₉	—	8	2	no	—
105	#9597	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	#1582	1	2	yes	—
107	C₂×C₈₉	#1583	1	2	yes	—
108	C₂×C₈₉	#1940	1	1	yes	characteristic
109	C₂×C₈₉	#1941	1	1	yes	characteristic
110	C₂×C₈₉	#1946	1	1	yes	characteristic
111	#1939	—	89	2	no	—
112	C₄×#106	—	89	1	no	—
113	C₂×#334	—	89	1	no	—
114	C₂×#322	D₈₉	1	1	yes	characteristic
115	#1279	—	89	2	no	—
116	C₈₉	#9597	1	1	yes	characteristic, complemented, Sylow
117	C₂×#106	—	89	2	no	—
118	C₂×C₄⋊C₈	—	89	2	no	—
119	C₂×C₄⋊C₈	—	178	2	no	—
120	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
123	C₄×C₄⋊C₄	—	89	2	no	—
124	#340	—	89	1	no	—
125	#340	—	89	1	no	—
126	#336	—	178	2	no	—
127	#334	—	178	2	no	—
128	#322	—	2	2	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	178	2	no	—
131	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
133	C₂²×C₈	D₃₅₆	1	2	yes	—
134	C₂×C₄⋊C₄	—	89	2	no	—
135	C₂×C₄⋊C₄	—	89	2	no	—
136	C₂×C₄²	—	178	2	no	—
137	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
138	#106	—	178	4	no	—
139	C₄⋊C₈	—	356	4	no	—
140	C₄⋊C₈	—	178	2	no	—
141	C₄⋊C₈	—	178	2	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	Dic₁₇₈	1	2	yes	—
146	C₄×C₈	Dic₁₇₈	1	2	yes	—
147	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
148	#95	—	178	2	no	—
149	C₂²×C₄	—	178	2	no	—
150	C₂²×C₄	—	178	2	no	—
151	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
153	C₂²×C₄	#13083574	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	356	2	no	—
162	C₂×C₈	—	356	2	no	—
163	C₂×C₈	—	356	2	no	—
164	C₂×C₈	#13083553	1	2	yes	—
165	C₂×C₈	#13083553	1	2	yes	—
166	C₂×C₈	#13083563	1	4	yes	—
167	C₄⋊C₄	—	89	4	no	—
168	C₄⋊C₄	—	178	2	no	—
169	C₄⋊C₄	—	178	2	no	—
170	C₄⋊C₄	—	178	2	no	—
171	C₄²	—	356	2	no	—
172	C₄²	—	356	2	no	—
173	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
174	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
175	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
176	C₄²	#13083574	1	1	yes	characteristic
177	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	356	2	no	—
181	C₂×C₄	—	356	2	no	—
182	C₂×C₄	—	356	2	no	—
183	C₂×C₄	—	356	2	no	—
184	C₂×C₄	—	356	2	no	—
185	C₂×C₄	—	178	4	no	—
186	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
187	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
188	C₂×C₄	#15667986	1	1	yes	characteristic
189	C₂×C₄	#15667991	1	1	yes	characteristic
190	C₂×C₄	#15668004	1	1	yes	characteristic
191	C₂×C₄	#15668012	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
193	C₂×C₄	#15668020	1	2	yes	—
194	C₂×C₄	C₂×#13083563	1	2	yes	—
195	C₂×C₄	#15668034	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	712	2	no	—
201	C₂²	#19151496	1	1	yes	characteristic
202	C₂²	#19151671	1	2	yes	—
203	C₂²	C₄×#13083563	1	2	yes	—
204	C₂²	#19151721	1	1	yes	characteristic
205	C₂²	#19151721	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	356	4	no	—
209	C₄	—	712	2	no	—
210	C₄	#19151621	1	1	yes	characteristic
211	C₄	#19151626	1	1	yes	characteristic
212	C₄	#19151726	1	2	yes	—
213	C₄	#19151827	1	2	yes	—
214	C₄	#19151996	1	1	yes	characteristic
215	C₄	#19152000	1	1	yes	characteristic
216	C₂	#26019090	1	2	yes	—
217	C₂	#26019097	1	2	yes	—
218	C₂	#26020975	1	1	yes	characteristic
219	C₂	#26020980	1	1	yes	characteristic
220	C₂	#26021005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9597
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
116. C₈₉	#9597	semidirect

Maximal subgroups

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

subgroup	index
#26020970	2
C₄×#15667944	2
C₂×#19151349	2
#26018092	2
C₂×#322×C₈₉	2
#9597	89