Subgroups — #36441019

Order 22784.

292 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#26020369	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	1	no	—
5	C₂×C₄⋊#10623494	—	2	1	no	—
6	C₂×C₄⋊#10623494	—	2	1	no	—
7	C₂×C₄⋊#10623494	C₄	1	2	yes	—
8	#19151353	—	2	1	no	—
9	#19151353	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
11	C₄³×C₈₉	C₂²	1	1	yes	characteristic
12	#336×C₈₉	—	2	1	no	—
13	#336×C₈₉	C₄	1	2	yes	—
14	C₂²×#10623494	—	2	2	no	—
15	C₂²×#10623494	—	2	2	no	—
16	C₄⋊#10623494	—	4	2	no	—
17	C₄⋊#10623494	—	4	1	no	—
18	C₄⋊#10623494	—	4	1	no	—
19	C₄⋊#10623494	—	4	1	no	—
20	C₄⋊#10623494	—	4	1	no	—
21	C₄⋊#10623494	—	4	1	no	—
22	C₄⋊#10623494	—	4	1	no	—
23	C₂²×C₈×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	—	4	1	no	—
32	C₄⋊C₈×C₈₉	—	4	1	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	4	2	no	—
47	C₂×C₈×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₄²×C₈₉	—	2	1	no	—
50	C₄²×C₈₉	—	2	1	no	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₄²×C₈₉	—	4	1	no	—
53	C₄²×C₈₉	—	4	1	no	—
54	C₄²×C₈₉	—	4	1	no	—
55	C₄²×C₈₉	D₈	1	1	yes	characteristic
56	C₄²×C₈₉	D₈	1	1	yes	characteristic
57	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
58	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
59	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
60	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
61	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
62	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
63	C₂³×C₈₉	#95	1	1	yes	characteristic
64	C₂×C₄×C₈₉	—	2	1	no	—
65	C₂×C₄×C₈₉	—	2	1	no	—
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	1	no	—
77	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#108	1	2	yes	—
92	C₈×C₈₉	—	8	2	no	—
93	#10623494	—	8	2	no	—
94	#10623494	—	8	2	no	—
95	C₂²×C₈₉	H(3,ℤ/4)	1	1	yes	characteristic
96	C₂²×C₈₉	#340	1	1	yes	characteristic
97	C₂²×C₈₉	#340	1	1	yes	characteristic
98	C₂²×C₈₉	#341	1	1	yes	characteristic
99	C₂²×C₈₉	#343	1	1	yes	characteristic
100	C₂²×C₈₉	#344	1	1	yes	characteristic
101	C₂²×C₈₉	#344	1	1	yes	characteristic
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	—	2	1	no	—
105	C₄×C₈₉	—	2	1	no	—
106	C₄×C₈₉	—	2	2	no	—
107	C₄×C₈₉	—	2	1	no	—
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	—	4	1	no	—
112	C₄×C₈₉	#331	1	1	yes	characteristic
113	C₄×C₈₉	#332	1	1	yes	characteristic
114	C₄×C₈₉	#332	1	1	yes	characteristic
115	C₄×C₈₉	#333	1	1	yes	characteristic
116	#7040	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	#1286	1	1	yes	characteristic
118	C₂×C₈₉	#1286	1	1	yes	characteristic
119	C₂×C₈₉	#1290	1	1	yes	characteristic
120	C₂×C₈₉	#1296	1	1	yes	characteristic
121	C₂×C₈₉	#1301	1	1	yes	characteristic
122	C₂×C₈₉	#1301	1	1	yes	characteristic
123	C₂×C₈₉	#1307	1	1	yes	characteristic
124	#1842	—	89	1	no	—
125	#1831	—	89	1	no	—
126	#1831	D₈₉	1	1	yes	characteristic
127	C₈₉	#7040	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	—	89	2	no	—
130	C₂×C₄⋊C₈	—	178	1	no	—
131	C₂×C₄⋊C₈	—	178	1	no	—
132	C₂×C₄⋊C₈	—	178	1	no	—
133	C₄³	C₂×D₈₉	1	1	yes	characteristic
134	#336	—	2	1	no	—
135	#336	—	89	2	no	—
136	#336	—	178	1	no	—
137	#336	Dic₈₉	1	2	yes	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	178	2	no	—
140	C₂²×C₈	—	178	2	no	—
141	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
144	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	356	2	no	—
151	C₄⋊C₈	—	356	1	no	—
152	C₄⋊C₈	—	356	1	no	—
153	C₄⋊C₈	—	356	1	no	—
154	C₄⋊C₈	—	356	1	no	—
155	C₄⋊C₈	—	356	1	no	—
156	C₄⋊C₈	—	356	1	no	—
157	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
158	C₂²×C₄	#13083551	1	1	yes	characteristic
159	C₂²×C₄	#13083551	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
161	C₂²×C₄	#13083553	1	1	yes	characteristic
162	C₂²×C₄	#13083553	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	356	2	no	—
168	C₂×C₈	—	356	2	no	—
169	C₂×C₈	—	356	2	no	—
170	C₂×C₈	—	356	2	no	—
171	C₂×C₈	—	356	2	no	—
172	C₂×C₈	—	356	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	1	no	—
177	C₄²	—	4	1	no	—
178	C₄²	—	4	1	no	—
179	C₄²	#13083559	1	1	yes	characteristic
180	C₄²	D₇₁₂	1	1	yes	characteristic
181	C₄²	#13083561	1	1	yes	characteristic
182	C₄²	#13083562	1	1	yes	characteristic
183	C₄²	#13083563	1	1	yes	characteristic
184	C₄²	#13083563	1	1	yes	characteristic
185	C₄²	#13083564	1	1	yes	characteristic
186	C₄²	Dic₃₅₆	1	1	yes	characteristic
187	C₂³	#15667919	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	#15667928	1	1	yes	characteristic
202	C₂×C₄	#15667930	1	1	yes	characteristic
203	C₂×C₄	#15667932	1	1	yes	characteristic
204	C₂×C₄	#15667933	1	1	yes	characteristic
205	C₂×C₄	#15667934	1	1	yes	characteristic
206	C₂×C₄	#15667934	1	1	yes	characteristic
207	C₂×C₄	#15667935	1	1	yes	characteristic
208	C₂×C₄	#15667936	1	1	yes	characteristic
209	C₂×C₄	#15667937	1	1	yes	characteristic
210	C₂×C₄	#15667937	1	1	yes	characteristic
211	C₂×C₄	#15667943	1	1	yes	characteristic
212	C₂×C₄	#15667944	1	1	yes	characteristic
213	C₂×C₄	#15667945	1	1	yes	characteristic
214	C₂×C₄	#15667946	1	1	yes	characteristic
215	C₂×C₄	#15667947	1	2	yes	—
216	C₈	—	8	2	no	—
217	C₈	—	712	2	no	—
218	C₈	—	712	2	no	—
219	C₂²	#19151354	1	1	yes	characteristic
220	C₂²	#19151360	1	1	yes	characteristic
221	C₂²	#19151361	1	1	yes	characteristic
222	C₂²	#19151362	1	1	yes	characteristic
223	C₂²	#19151366	1	1	yes	characteristic
224	C₂²	#19151368	1	1	yes	characteristic
225	C₂²	#19151369	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	1	no	—
236	C₄	#19151341	1	1	yes	characteristic
237	C₄	#19151343	1	1	yes	characteristic
238	C₄	#19151345	1	1	yes	characteristic
239	C₄	#19151346	1	1	yes	characteristic
240	C₂	#26018110	1	1	yes	characteristic
241	C₂	#26018111	1	1	yes	characteristic
242	C₂	#26018121	1	1	yes	characteristic
243	C₂	#26018136	1	1	yes	characteristic
244	C₂	#26018147	1	1	yes	characteristic
245	C₂	#26018148	1	1	yes	characteristic
246	C₂	#26018162	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₄³	C₂×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	#95
cosocle	C₄³×C₈₉	C₂²
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#7040
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
127. C₈₉	#7040	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
#26020369	2
C₈₉×#1831	2
#7040	89