Subgroups — #36453000

Order 22784.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#19151351	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	#19151351	—	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	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	—	2	1	no	—
34	C₄×C₈×C₈₉	—	2	1	no	—
35	C₄×C₈×C₈₉	—	2	1	no	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₄×C₈×C₈₉	Q₈	1	2	yes	—
38	C₄×C₈×C₈₉	Q₈	1	2	yes	—
39	C₂²×Dic₈₉	—	2	2	no	—
40	C₂²×Dic₈₉	—	2	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
48	C₄×Dic₈₉	—	4	2	no	—
49	C₄×Dic₈₉	—	4	2	no	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
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	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	1	no	—
59	C₂×C₈×C₈₉	—	2	1	no	—
60	C₂×C₈×C₈₉	—	2	2	no	—
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₈₉	#41	1	1	yes	characteristic
68	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₈₉	#131	1	2	yes	—
79	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₈₉	#137	1	2	yes	—
81	C₂×Dic₈₉	—	2	4	no	—
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₂×Dic₈₉	—	4	2	no	—
86	C₂×Dic₈₉	—	4	2	no	—
87	C₈×C₈₉	—	2	4	no	—
88	C₈×C₈₉	—	2	2	no	—
89	C₈×C₈₉	—	2	2	no	—
90	C₈×C₈₉	—	4	1	no	—
91	C₈×C₈₉	—	4	1	no	—
92	#10623494	—	8	2	no	—
93	C₂²×C₈₉	#389	1	1	yes	characteristic
94	C₂²×C₈₉	#435	1	2	yes	—
95	C₂²×C₈₉	#440	1	2	yes	—
96	C₂²×C₈₉	#443	1	1	yes	characteristic
97	C₂²×C₈₉	#443	1	1	yes	characteristic
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	#423	1	1	yes	characteristic
101	C₄×C₈₉	#424	1	1	yes	characteristic
102	C₄×C₈₉	#446	1	2	yes	—
103	C₄×C₈₉	#469	1	1	yes	characteristic
104	C₄×C₈₉	#470	1	1	yes	characteristic
105	C₄×C₈₉	#501	1	2	yes	—
106	Dic₈₉	—	4	4	no	—
107	Dic₈₉	—	8	2	no	—
108	#9600	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	#1585	1	2	yes	—
110	C₂×C₈₉	#1588	1	2	yes	—
111	C₂×C₈₉	#1940	1	1	yes	characteristic
112	C₂×C₈₉	#1944	1	1	yes	characteristic
113	C₂×C₈₉	#1945	1	1	yes	characteristic
114	#1939	—	89	2	no	—
115	C₄×#106	—	89	1	no	—
116	C₂×#335	—	89	1	no	—
117	C₂×#322	D₈₉	1	1	yes	characteristic
118	#1279	—	89	2	no	—
119	C₈₉	#9600	1	1	yes	characteristic, complemented, Sylow
120	C₂×#106	—	89	2	no	—
121	C₂×C₄⋊C₈	—	89	2	no	—
122	C₂×C₄⋊C₈	—	178	2	no	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	89	2	no	—
127	#340	—	89	1	no	—
128	#340	—	89	1	no	—
129	#336	—	178	2	no	—
130	#335	—	178	2	no	—
131	#322	—	2	2	no	—
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	178	2	no	—
134	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
136	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
137	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	89	2	no	—
139	C₂×C₄⋊C₄	—	89	2	no	—
140	C₂×C₄²	—	178	2	no	—
141	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
142	#106	—	178	4	no	—
143	C₄⋊C₈	—	356	4	no	—
144	C₄⋊C₈	—	178	2	no	—
145	C₄⋊C₈	—	178	2	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	Dic₁₇₈	1	2	yes	—
150	C₄×C₈	Dic₁₇₈	1	2	yes	—
151	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
152	#95	—	178	2	no	—
153	C₂²×C₄	—	178	2	no	—
154	C₂²×C₄	—	178	2	no	—
155	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
157	C₂²×C₄	#13083574	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	356	2	no	—
168	C₂×C₈	—	356	2	no	—
169	C₂×C₈	—	356	2	no	—
170	C₂×C₈	#13083553	1	2	yes	—
171	C₂×C₈	#13083553	1	2	yes	—
172	C₄⋊C₄	—	89	4	no	—
173	C₄⋊C₄	—	178	2	no	—
174	C₄⋊C₄	—	178	2	no	—
175	C₄⋊C₄	—	178	2	no	—
176	C₄²	—	356	2	no	—
177	C₄²	—	356	2	no	—
178	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
179	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
181	C₄²	#13083574	1	1	yes	characteristic
182	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	356	2	no	—
186	C₂×C₄	—	356	2	no	—
187	C₂×C₄	—	356	2	no	—
188	C₂×C₄	—	356	2	no	—
189	C₂×C₄	—	356	2	no	—
190	C₂×C₄	—	178	4	no	—
191	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
192	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
193	C₂×C₄	#15667986	1	1	yes	characteristic
194	C₂×C₄	#15667991	1	1	yes	characteristic
195	C₂×C₄	#15668004	1	1	yes	characteristic
196	C₂×C₄	#15668012	1	1	yes	characteristic
197	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
198	C₂×C₄	#15668020	1	2	yes	—
199	C₂×C₄	#15668040	1	2	yes	—
200	C₂×C₄	#15668045	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	712	2	no	—
207	C₂²	#19151496	1	1	yes	characteristic
208	C₂²	#19151671	1	2	yes	—
209	C₂²	#19151700	1	2	yes	—
210	C₂²	#19151716	1	1	yes	characteristic
211	C₂²	#19151716	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	356	4	no	—
215	C₄	—	712	2	no	—
216	C₄	#19151621	1	1	yes	characteristic
217	C₄	#19151626	1	1	yes	characteristic
218	C₄	#19151726	1	2	yes	—
219	C₄	#19151851	1	1	yes	characteristic
220	C₄	#19151856	1	1	yes	characteristic
221	C₄	#19152007	1	2	yes	—
222	C₂	#26019111	1	2	yes	—
223	C₂	#26019132	1	2	yes	—
224	C₂	#26020975	1	1	yes	characteristic
225	C₂	#26020995	1	1	yes	characteristic
226	C₂	#26021000	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	#9600
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
119. C₈₉	#9600	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×#19151351	2
#26018092	2
C₂×#322×C₈₉	2
#9600	89