Subgroups — #36470594

Order 22784.

438 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	C₂³×#13083549	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151423	C₂	1	2	yes	maximal, complemented
3	#26020281	C₂	1	1	yes	maximal, characteristic
4	#26018343	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	—	2	4	no	—
7	C₂²×#13083549	C₄	1	4	yes	complemented
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#15667946	C₂²	1	2	yes	—
11	#19151458	—	2	1	no	—
12	#19151423	—	2	4	no	—
13	#19151361	—	2	1	no	—
14	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₈₉	C₂²	1	2	yes	—
16	#380×C₈₉	—	2	1	no	—
17	#340×C₈₉	—	2	1	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
20	C₂×#13083549	—	2	8	no	—
21	C₂×#13083549	—	2	2	no	—
22	C₂×#13083549	C₂×C₄	1	2	yes	—
23	C₂×#13083549	C₂×C₄	1	2	yes	—
24	C₂×#13083549	D₄	1	4	yes	—
25	#15667946	—	2	2	no	—
26	#15667946	—	2	2	no	—
27	#15667919	—	4	1	no	—
28	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	—	2	2	no	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
37	#107×C₈₉	—	2	2	no	—
38	#107×C₈₉	—	2	2	no	—
39	#95×C₈₉	—	4	1	no	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₂²⋊Dic₈₉	—	8	2	no	—
44	#13083549	—	2	2	no	—
45	#13083549	C₂²⋊C₄	1	4	yes	complemented
46	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	4	2	no	—
50	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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	4	no	—
58	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	2	yes	—
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₂²⋊C₄×C₈₉	—	8	2	no	—
66	C₂³×C₈₉	—	2	2	no	—
67	C₂³×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	#107	1	2	yes	—
69	C₂³×C₈₉	D₁₆	1	2	yes	—
70	C₂³×C₈₉	Q₃₂	1	2	yes	—
71	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	2	4	no	—
76	C₂×C₄×C₈₉	—	8	2	no	—
77	C₂×C₄×C₈₉	—	8	2	no	—
78	C₂×C₄×C₈₉	—	8	2	no	—
79	C₂×C₄×C₈₉	#107	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#121	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#135	1	2	yes	—
84	C₂×Dic₈₉	—	8	2	no	—
85	C₂×Dic₈₉	—	8	2	no	—
86	C₂×Dic₈₉	—	8	2	no	—
87	C₈×C₈₉	—	2	2	no	—
88	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₈₉	#117	1	1	yes	characteristic
90	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
91	C₂²×C₈₉	—	2	2	no	—
92	C₂²×C₈₉	—	2	2	no	—
93	C₂²×C₈₉	—	2	4	no	—
94	C₂²×C₈₉	—	2	4	no	—
95	C₂²×C₈₉	#366	1	4	yes	complemented
96	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
97	C₂²×C₈₉	#466	1	2	yes	—
98	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
99	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
100	C₂²×C₈₉	#508	1	2	yes	—
101	C₄×C₈₉	—	2	2	no	—
102	C₄×C₈₉	—	16	2	no	—
103	C₄×C₈₉	#380	1	1	yes	characteristic
104	C₄×C₈₉	#427	1	1	yes	characteristic
105	C₄×C₈₉	#467	1	2	yes	—
106	Dic₈₉	—	16	2	no	—
107	#12834	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	—	2	4	no	—
109	C₂×C₈₉	#1808	1	1	yes	characteristic
110	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
111	C₂×C₈₉	#2158	1	1	yes	characteristic
112	C₂×C₈₉	#2215	1	2	yes	—
113	C₂×C₈₉	#2216	1	2	yes	—
114	C₂³×C₁₆	—	89	1	no	—
115	C₂×#366	—	89	2	no	—
116	#1808	—	89	1	no	—
117	#1808	D₈₉	1	1	yes	characteristic
118	#1380	—	89	2	no	—
119	C₈₉	#12834	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
121	C₂²×C₁₆	—	89	1	no	—
122	C₂²×C₁₆	—	89	4	no	—
123	C₂²×C₁₆	—	89	1	no	—
124	C₂²×C₁₆	—	178	4	no	—
125	C₂×#107	—	89	2	no	—
126	C₂×#107	C₂×D₈₉	1	2	yes	—
127	#380	—	2	1	no	—
128	#380	—	178	1	no	—
129	#366	—	178	4	no	—
130	#340	—	2	1	no	—
131	#340	—	178	1	no	—
132	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	Dic₁₇₈	1	2	yes	—
136	C₂²×C₈	C₄×D₈₉	1	2	yes	—
137	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	178	2	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	—	356	1	no	—
143	C₂×C₁₆	—	89	2	no	—
144	C₂×C₁₆	—	89	4	no	—
145	C₂×C₁₆	—	89	2	no	—
146	C₂×C₁₆	—	178	8	no	—
147	C₂×C₁₆	—	178	2	no	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	—	178	2	no	—
151	#107	—	178	2	no	—
152	#95	—	4	1	no	—
153	#95	—	356	1	no	—
154	C₂⁴	#13083559	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	356	2	no	—
159	C₂²×C₄	#13083553	1	2	yes	—
160	C₂²×C₄	#13083559	1	1	yes	characteristic
161	C₂²×C₄	#13083564	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	#13083553	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
169	C₂×C₈	D₄×D₈₉	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
171	C₂×C₈	#13083573	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	356	2	no	—
175	C₄⋊C₄	—	356	2	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₂²⋊C₄	—	712	2	no	—
178	C₁₆	—	89	4	no	—
179	C₁₆	—	178	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#15667945	1	2	yes	—
183	C₂³	#15667953	1	2	yes	—
184	C₂³	#15667958	1	2	yes	—
185	C₂³	C₂×#13083559	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	712	2	no	—
194	C₂×C₄	—	712	2	no	—
195	C₂×C₄	—	712	2	no	—
196	C₂×C₄	#15667945	1	2	yes	—
197	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
198	C₂×C₄	#15667984	1	2	yes	—
199	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
200	C₂×C₄	#15668032	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
203	C₈	#15667968	1	1	yes	characteristic
204	C₈	#15667989	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	#19151423	1	4	yes	complemented
210	C₂²	C₂×#15667945	1	1	yes	characteristic
211	C₂²	#19151832	1	2	yes	—
212	C₂²	C₂×#15667953	1	1	yes	characteristic
213	C₂²	C₂×#15667958	1	1	yes	characteristic
214	C₂²	#19152028	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	16	2	no	—
217	C₄	—	1424	2	no	—
218	C₄	#19151456	1	1	yes	characteristic
219	C₄	#19151634	1	1	yes	characteristic
220	C₄	#19151837	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	#26020279	1	1	yes	characteristic
223	C₂	C₂×#19151423	1	1	yes	characteristic
224	C₂	#26022034	1	1	yes	characteristic
225	C₂	#26022336	1	2	yes	—
226	C₂	#26022341	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1808	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1808	D₈₉
hypocenter	C₈₉	#12834
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#19151423	C₂	semidirect
4. #26018343	C₂	semidirect
7. C₂²×#13083549	C₄	semidirect
45. #13083549	C₂²⋊C₄	semidirect
95. C₂²×C₈₉	#366	semidirect
119. C₈₉	#12834	semidirect
209. C₂²	#19151423	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#13083549	2
C₂×#19151423	2
#26020281	2
#26018343	2
C₈₉×#1808	2
#12834	89