Subgroups — #36520149

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029219	C₂	1	2	yes	maximal
2	#26018555	C₂	1	2	yes	maximal, complemented
3	#26018549	C₂	1	2	yes	maximal, complemented
4	#26018539	C₂	1	2	yes	maximal, complemented
5	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#13083558	C₂	1	4	yes	maximal, complemented
7	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#131×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19151729	C₄	1	4	yes	—
10	#19151676	C₄	1	4	yes	—
11	#19151670	C₄	1	8	yes	—
12	#19151651	C₂²	1	4	yes	—
13	#19151651	C₂²	1	2	yes	—
14	C₄×#13083558	C₂²	1	4	yes	—
15	C₂×C₄×#10623494	C₂²	1	2	yes	—
16	#19151320	C₂²	1	4	yes	—
17	#19151320	C₂²	1	4	yes	—
18	#19151319	C₂²	1	2	yes	—
19	C₈×#10623494	C₂²	1	2	yes	—
20	C₈×#10623494	C₂²	1	1	yes	characteristic
21	C₈×#10623494	C₂²	1	1	yes	characteristic
22	C₈×#10623494	C₂²	1	2	yes	—
23	C₂×#131×C₈₉	C₄	1	2	yes	—
24	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
25	#405×C₈₉	C₄	1	4	yes	—
26	#405×C₈₉	C₄	1	2	yes	—
27	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
31	C₂×#13083558	C₂×C₄	1	8	yes	—
32	C₂²×#10623494	C₂×C₄	1	4	yes	—
33	C₄⋊#10623494	C₂×C₄	1	8	yes	—
34	C₄⋊#10623494	C₂×C₄	1	4	yes	—
35	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
36	C₂²⋊#10623494	C₂×C₄	1	8	yes	—
37	#15667923	C₂×C₄	1	8	yes	—
38	C₄×#10623494	C₂×C₄	1	4	yes	—
39	C₄×#10623494	C₂³	1	2	yes	—
40	C₄×#10623494	C₂³	1	4	yes	—
41	C₄×#10623494	C₂³	1	2	yes	—
42	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
43	#131×C₈₉	—	2	8	no	—
44	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
50	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
51	#117×C₈₉	C₂×C₄	1	2	yes	—
52	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
55	#97×C₈₉	C₂×C₄	1	2	yes	—
56	#97×C₈₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	#13083558	—	2	16	no	—
63	C₂×#10623494	—	2	8	no	—
64	C₂×#10623494	C₂²×C₄	1	4	yes	—
65	C₂×#10623494	C₂²×C₄	1	8	yes	—
66	C₂×#10623494	C₂²×C₄	1	4	yes	—
67	#41×C₈₉	—	2	4	no	—
68	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
69	C₂×D₄×C₈₉	C₄²	1	2	yes	—
70	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	M₁₆×C₈₉	—	2	4	no	—
76	M₁₆×C₈₉	—	2	8	no	—
77	C₂×C₈×C₈₉	—	2	8	no	—
78	C₂×C₈×C₈₉	M₁₆	1	8	yes	complemented
79	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₄⋊C₄×C₈₉	C₄²	1	4	yes	—
83	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
84	C₂²⋊C₄×C₈₉	C₄²	1	4	yes	—
85	C₂²⋊C₄×C₈₉	C₄²	1	2	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
92	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
93	Q₈×C₈₉	—	2	2	no	—
94	D₄×C₈₉	—	2	4	no	—
95	D₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	8	no	—
99	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₈₉	—	2	4	no	—
107	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
108	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
109	#10623494	—	2	4	no	—
110	#10623494	—	2	8	no	—
111	#10623494	#131	1	8	yes	complemented
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	2	4	no	—
115	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
116	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	4	no	—
120	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
121	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
122	C₄×C₈₉	C₂×#131	1	4	yes	—
123	#17335	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	2	no	—
125	C₂×C₈₉	—	4	2	no	—
126	C₂×C₈₉	C₂×C₄×M₁₆	1	1	yes	characteristic
127	C₂×C₈₉	C₄×#131	1	2	yes	—
128	#2965	—	89	2	no	—
129	C₄×#131	D₈₉	1	1	yes	characteristic
130	#1455	—	89	2	no	—
131	#1453	—	89	2	no	—
132	#1451	—	89	2	no	—
133	#1450	—	89	1	no	—
134	C₈×M₁₆	—	89	4	no	—
135	C₂×C₈²	—	89	1	no	—
136	C₈₉	#17335	1	1	yes	characteristic, complemented, Sylow
137	C₂×#131	Dic₈₉	1	2	yes	—
138	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
139	#446	—	89	4	no	—
140	#436	—	89	4	no	—
141	#435	—	89	8	no	—
142	#431	—	89	4	no	—
143	#431	—	89	2	no	—
144	#405	Dic₈₉	1	4	yes	—
145	#405	Dic₈₉	1	2	yes	—
146	C₄×M₁₆	—	89	4	no	—
147	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
148	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
149	C₂×C₄×C₈	—	89	2	no	—
150	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
152	#322	—	89	4	no	—
153	#322	—	89	4	no	—
154	#322	—	89	2	no	—
155	C₈²	—	89	1	no	—
156	C₈²	—	89	1	no	—
157	C₈²	—	89	2	no	—
158	C₈²	—	89	2	no	—
159	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
160	#131	—	2	8	no	—
161	C₂×M₁₆	—	89	8	no	—
162	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
163	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
164	C₂²×C₈	—	89	4	no	—
165	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
166	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
167	C₄×Q₈	C₄×D₈₉	1	2	yes	—
168	C₄×D₄	C₄×D₈₉	1	4	yes	—
169	C₄×D₄	C₄×D₈₉	1	2	yes	—
170	#117	C₂×Dic₈₉	1	2	yes	—
171	#117	C₂×Dic₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
174	C₄⋊C₈	—	89	8	no	—
175	C₄⋊C₈	—	89	4	no	—
176	C₂²⋊C₈	—	89	4	no	—
177	C₂²⋊C₈	—	89	8	no	—
178	#97	—	89	8	no	—
179	#97	C₂×Dic₈₉	1	2	yes	—
180	#97	C₂×Dic₈₉	1	4	yes	—
181	C₄×C₈	—	89	4	no	—
182	C₄×C₈	—	89	2	no	—
183	C₄×C₈	—	89	2	no	—
184	C₄×C₈	—	89	4	no	—
185	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
186	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
187	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₈₉	1	2	yes	—
189	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
190	#41	—	2	4	no	—
191	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
192	C₂×D₄	C₄×Dic₈₉	1	2	yes	—
193	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
196	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	8	no	—
200	M₁₆	—	178	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	89	4	no	—
203	C₂×C₈	—	89	4	no	—
204	C₂×C₈	—	89	8	no	—
205	C₂×C₈	—	178	8	no	—
206	C₂×C₈	#13083558	1	8	yes	complemented
207	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
208	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
209	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
210	C₄⋊C₄	C₄×Dic₈₉	1	4	yes	—
211	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
212	C₂²⋊C₄	C₄×Dic₈₉	1	4	yes	—
213	C₂²⋊C₄	C₄×Dic₈₉	1	2	yes	—
214	C₄²	C₂×C₄×D₈₉	1	2	yes	—
215	C₄²	C₂×C₄×D₈₉	1	4	yes	—
216	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
217	C₄²	C₂²×Dic₈₉	1	2	yes	—
218	C₄²	C₂³×D₈₉	1	1	yes	characteristic
219	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
220	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
230	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂×#13083558	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
233	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	—	89	8	no	—
236	C₈	—	178	4	no	—
237	C₈	—	178	8	no	—
238	C₈	C₂×#13083558	1	4	yes	—
239	C₈	C₂×#13083558	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₄×#13083558	1	2	yes	—
244	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	C₄×#13083558	1	2	yes	—
249	C₄	C₂²×#13083558	1	2	yes	—
250	C₄	#131×D₈₉	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×#13083558	1	1	yes	characteristic
254	C₂	#131×Dic₈₉	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₈₉
Fitting	C₄×#131×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₄×#131	D₈₉
hypocenter	C₈₉	#17335
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #26018555	C₂	semidirect
3. #26018549	C₂	semidirect
4. #26018539	C₂	semidirect
5. #26018535	C₂	semidirect
6. C₈×#13083558	C₂	semidirect
7. C₂×C₈×#10623494	C₂	semidirect
78. C₂×C₈×C₈₉	M₁₆	semidirect
111. #10623494	#131	semidirect
136. C₈₉	#17335	semidirect
206. C₂×C₈	#13083558	semidirect

Maximal subgroups

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

subgroup	index
#26029219	2
#26018555	2
#26018549	2
#26018539	2
#26018535	2
C₈×#13083558	2
C₂×C₈×#10623494	2
C₄×#131×C₈₉	2
#17335	89