Subgroups — #36709786

Order 22784.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	#26022497	C₂	1	4	yes	maximal, complemented
3	#26022479	C₂	1	4	yes	maximal, complemented
4	#26021972	C₂	1	4	yes	maximal
5	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
6	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19152155	—	2	1	no	—
8	C₂×#15667952	C₂²	1	2	yes	—
9	#19152003	C₂²	1	2	yes	—
10	#19152000	—	2	1	no	—
11	#19152000	C₂²	1	2	yes	—
12	#19151996	—	2	1	no	—
13	#19151636	—	2	1	no	—
14	#19151636	C₂²	1	2	yes	—
15	#19151402	C₂²	1	16	yes	—
16	#19151374	C₂²	1	2	yes	—
17	C₄×#13083549	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₈₉	—	2	1	no	—
19	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
21	#494×C₈₉	—	2	1	no	—
22	#494×C₈₉	—	2	1	no	—
23	#494×C₈₉	C₂²	1	2	yes	—
24	#494×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#15668012	—	2	2	no	—
27	#15668004	—	2	2	no	—
28	#15667991	—	4	2	no	—
29	C₄×Dic₁₇₈	—	4	2	no	—
30	#15667971	—	2	2	no	—
31	#15667952	—	2	4	no	—
32	C₂×#13083549	C₂³	1	4	yes	—
33	#15667946	—	2	2	no	—
34	#15667946	C₂³	1	4	yes	—
35	#15667944	—	2	2	no	—
36	#15667944	—	2	2	no	—
37	C₂²×Q₈×C₈₉	—	2	2	no	—
38	C₂×Q₁₆×C₈₉	—	2	2	no	—
39	C₂×Q₁₆×C₈₉	—	2	2	no	—
40	C₂×Q₁₆×C₈₉	—	2	2	no	—
41	C₂×Q₁₆×C₈₉	—	2	8	no	—
42	C₂×Q₁₆×C₈₉	—	4	4	no	—
43	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₈₉	—	2	2	no	—
47	C₄⋊Q₈×C₈₉	—	2	2	no	—
48	C₄⋊Q₈×C₈₉	—	4	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₇₈	—	4	2	no	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	#13083553	—	4	4	no	—
59	C₂²⋊Dic₈₉	—	4	2	no	—
60	C₄×Dic₈₉	—	4	2	no	—
61	#13083549	—	2	4	no	—
62	C₂×Q₈×C₈₉	—	2	4	no	—
63	C₂×Q₈×C₈₉	—	4	2	no	—
64	C₂×Q₈×C₈₉	—	4	4	no	—
65	C₂×Q₈×C₈₉	—	4	4	no	—
66	C₂²×C₄×C₈₉	—	4	2	no	—
67	C₂²×C₄×C₈₉	D₈	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	Q₁₆×C₈₉	—	2	8	no	—
70	Q₁₆×C₈₉	—	4	4	no	—
71	Q₁₆×C₈₉	—	4	4	no	—
72	Q₁₆×C₈₉	—	4	4	no	—
73	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
74	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	#41	1	4	yes	—
78	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₈₉	—	4	4	no	—
80	C₄⋊C₄×C₈₉	—	4	4	no	—
81	C₄²×C₈₉	D₈	1	2	yes	—
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₈₉	—	4	4	no	—
86	Q₈×C₈₉	—	4	4	no	—
87	Q₈×C₈₉	—	8	4	no	—
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	4	4	no	—
90	C₂×C₄×C₈₉	—	8	4	no	—
91	C₂×C₄×C₈₉	#124	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₈₉	—	4	4	no	—
98	Dic₁₇₈	—	8	4	no	—
99	C₈×C₈₉	#124	1	2	yes	—
100	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
101	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
102	C₈×C₈₉	C₂×#41	1	2	yes	—
103	C₂²×C₈₉	—	2	2	no	—
104	C₂²×C₈₉	#486	1	2	yes	—
105	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₈₉	—	2	2	no	—
107	C₄×C₈₉	—	8	4	no	—
108	C₄×C₈₉	#486	1	2	yes	—
109	C₄×C₈₉	#510	1	4	yes	—
110	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
111	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
112	Dic₈₉	—	8	4	no	—
113	#33705	—	89	1	no	maximal, Sylow
114	C₂×C₈₉	—	2	2	no	—
115	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
116	C₂×C₈₉	C₂×#510	1	2	yes	—
117	#3160	—	89	1	no	—
118	C₂×#494	D₈₉	1	1	yes	characteristic
119	#2245	—	89	4	no	—
120	#2241	—	89	4	no	—
121	#2141	—	89	4	no	—
122	C₄×M₃₂	—	89	1	no	—
123	C₈₉	#33705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	—	2	1	no	—
125	C₂²×Q₁₆	C₂×D₈₉	1	2	yes	—
126	#533	—	178	1	no	—
127	C₂×C₄⋊Q₈	—	2	1	no	—
128	C₂×M₃₂	—	89	2	no	—
129	#500	—	89	2	no	—
130	#499	—	89	2	no	—
131	#499	—	178	1	no	—
132	#498	—	178	1	no	—
133	#494	—	2	1	no	—
134	#494	—	2	1	no	—
135	#494	C₂×D₈₉	1	2	yes	—
136	#494	C₂×D₈₉	1	2	yes	—
137	#427	—	89	2	no	—
138	#427	—	178	1	no	—
139	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
140	#358	—	89	16	no	—
141	#346	—	89	2	no	—
142	C₄×C₁₆	—	89	2	no	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	4	4	no	—
149	C₂×Q₁₆	C₂²×D₈₉	1	4	yes	—
150	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	178	2	no	—
156	#125	—	178	2	no	—
157	C₂²⋊Q₈	—	356	2	no	—
158	C₄×Q₈	—	356	2	no	—
159	#117	—	178	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
162	M₃₂	—	178	4	no	—
163	C₂×C₁₆	—	89	4	no	—
164	#107	—	89	4	no	—
165	#107	—	178	2	no	—
166	#106	—	178	2	no	—
167	#106	—	178	2	no	—
168	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	356	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#13083559	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	Q₁₆	—	4	4	no	—
184	C₂×C₈	#13083562	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
188	C₂×C₈	#13083573	1	4	yes	—
189	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	356	4	no	—
193	C₄⋊C₄	—	356	2	no	—
194	C₄⋊C₄	—	178	4	no	—
195	C₂²⋊C₄	—	356	2	no	—
196	C₄²	—	356	2	no	—
197	C₄²	#13083559	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
200	C₁₆	—	178	4	no	—
201	C₂³	C₂×#13083559	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	712	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	356	4	no	—
210	C₂×C₄	#15667999	1	2	yes	—
211	C₂×C₄	C₂×#13083559	1	2	yes	—
212	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
213	C₂×C₄	C₂×#13083559	1	2	yes	—
214	C₂×C₄	C₂×#13083562	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
216	C₈	#15667999	1	2	yes	—
217	C₈	C₂×#13083562	1	2	yes	—
218	C₈	C₂×#13083562	1	2	yes	—
219	C₈	C₂×#13083573	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#19151944	1	2	yes	—
222	C₂²	C₂²×#13083559	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	712	4	no	—
226	C₄	#19151944	1	2	yes	—
227	C₄	#19152039	1	4	yes	—
228	C₄	C₂×#15667999	1	1	yes	characteristic
229	C₄	C₂²×#13083562	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#19151944	1	1	yes	characteristic
232	C₂	C₂×#19152039	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#494×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#494	D₈₉
hypocenter	C₈₉	#33705
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #26022497	C₂	semidirect
3. #26022479	C₂	semidirect
123. C₈₉	#33705	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26022497	2
#26022479	2
#26021972	2
C₄×#15667952	2
C₂×#494×C₈₉	2
#33705	89