Subgroups — #36708028

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26022085	C₂	1	4	yes	maximal, complemented
3	#26022083	C₂	1	4	yes	maximal, complemented
4	#26022051	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021861	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₉	C₂²	1	2	yes	—
8	C₂×C₄×D₃₅₆	C₄	1	1	yes	characteristic
9	C₂×#15667951	C₂²	1	8	yes	complemented
10	C₈×D₃₅₆	C₄	1	2	yes	—
11	C₈×D₃₅₆	C₄	1	2	yes	—
12	C₈×D₃₅₆	C₂²	1	2	yes	—
13	C₈×D₃₅₆	C₂²	1	2	yes	—
14	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
15	C₂×#15667924	C₄	1	2	yes	—
16	C₄⋊#13083549	C₂²	1	2	yes	complemented
17	C₂²⋊#13083549	C₂²	1	2	yes	—
18	#19151377	C₂²	1	8	yes	complemented
19	C₄×#13083549	C₂²	1	2	yes	complemented
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
22	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
23	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
24	C₂²×D₃₅₆	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
26	C₂²×#10623494	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₈₉	—	4	4	no	—
28	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₈₉	C₂³	1	4	yes	—
30	C₄×D₃₅₆	C₈	1	2	yes	—
31	C₄×D₃₅₆	C₈	1	2	yes	—
32	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
33	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
35	C₂×#13083551	C₈	1	2	yes	—
36	#15667951	—	4	8	no	—
37	C₂×#13083549	C₂³	1	4	yes	—
38	C₄⋊#10623494	C₂×C₄	1	2	yes	—
39	C₄⋊#10623494	C₂×C₄	1	2	yes	—
40	#15667924	C₂×C₄	1	4	yes	—
41	#15667924	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
46	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂³×D₈₉	C₂×C₈	1	2	yes	—
51	C₂×D₃₅₆	—	2	4	no	—
52	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
53	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
54	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₈₉	—	4	2	no	—
56	C₂×C₄×D₈₉	—	4	2	no	—
57	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₈₉	C₂²×C₄	1	4	yes	—
59	C₂×#10623494	—	4	2	no	—
60	C₂×#10623494	C₂²×C₄	1	4	yes	—
61	C₈×D₈₉	—	2	8	no	—
62	C₈×D₈₉	—	4	4	no	—
63	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
64	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
65	#13083551	C₂×C₈	1	4	yes	—
66	#13083551	C₂×C₈	1	4	yes	—
67	#13083549	—	2	4	no	—
68	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	—	2	2	no	—
72	C₂×C₈×C₈₉	—	4	1	no	—
73	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
77	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₁₆×C₈₉	—	2	4	no	—
82	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
83	C₂²×D₈₉	—	2	4	no	—
84	C₂²×D₈₉	—	4	2	no	—
85	C₂²×D₈₉	—	4	2	no	—
86	C₂²×D₈₉	C₂²×C₈	1	4	yes	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	4	1	no	—
90	C₂×C₄×C₈₉	M₃₂	1	4	yes	complemented
91	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₈₉	—	4	2	no	—
98	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
99	D₃₅₆	—	2	4	no	—
100	D₃₅₆	—	2	4	no	—
101	C₄×D₈₉	—	2	4	no	—
102	C₄×D₈₉	—	2	4	no	—
103	C₄×D₈₉	—	4	4	no	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	—	2	1	no	—
106	C₈×C₈₉	—	2	1	no	—
107	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
108	#10623494	—	2	4	no	—
109	C₂²×C₈₉	—	2	1	no	—
110	C₂²×C₈₉	—	2	1	no	—
111	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
112	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
113	C₂×D₈₉	—	2	4	no	—
114	C₂×D₈₉	—	2	4	no	—
115	C₂×D₈₉	—	2	4	no	—
116	C₂×D₈₉	—	4	8	no	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
121	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
122	C₄×C₈₉	#518	1	1	yes	characteristic
123	C₄×C₈₉	#568	1	1	yes	characteristic
124	Dic₈₉	—	2	4	no	—
125	#33489	—	89	1	no	maximal, Sylow
126	C₂×C₈₉	—	2	2	no	—
127	C₂×C₈₉	#2972	1	1	yes	characteristic
128	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
129	C₂×C₈₉	#3407	1	1	yes	characteristic
130	D₈₉	—	4	4	no	—
131	C₂×C₈×D₄	—	89	1	no	—
132	#2169	—	89	4	no	—
133	#2169	—	89	4	no	—
134	#2162	—	89	1	no	—
135	#2162	D₈₉	1	1	yes	characteristic, complemented
136	#2112	—	89	4	no	—
137	C₈₉	#33489	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	—	89	2	no	—
139	C₂×C₄×D₄	—	89	1	no	—
140	C₂×M₃₂	—	89	8	no	—
141	C₈×D₄	—	89	2	no	—
142	C₈×D₄	—	89	2	no	—
143	C₈×D₄	—	89	2	no	—
144	C₈×D₄	—	89	2	no	—
145	C₂×C₄⋊C₈	—	89	1	no	—
146	C₂×C₂²⋊C₈	—	89	2	no	—
147	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	C₄⋊C₁₆	—	89	2	no	—
149	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
150	C₂²⋊C₁₆	—	89	2	no	—
151	C₂²⋊C₁₆	—	89	8	no	—
152	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
153	C₄×C₁₆	—	89	2	no	—
154	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
155	C₂²×D₄	—	89	1	no	—
156	C₂³×C₄	—	89	2	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₂²×C₈	—	89	4	no	—
159	C₂²×C₈	—	89	4	no	—
160	C₂²×C₈	—	356	4	no	—
161	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	C₄×D₄	—	89	2	no	—
164	C₄×D₄	—	89	2	no	—
165	C₄×D₄	—	89	2	no	—
166	C₄×D₄	—	89	2	no	—
167	C₂×C₄⋊C₄	—	89	1	no	—
168	C₂×C₂²⋊C₄	—	89	2	no	—
169	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
170	M₃₂	—	356	8	no	—
171	C₂×C₁₆	—	89	4	no	—
172	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
173	C₄⋊C₈	—	89	2	no	—
174	C₄⋊C₈	—	89	2	no	—
175	C₂²⋊C₈	—	89	4	no	—
176	C₂²⋊C₈	—	89	4	no	—
177	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
178	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
181	C₂⁴	—	89	2	no	—
182	C₂×D₄	—	89	2	no	—
183	C₂×D₄	—	89	2	no	—
184	C₂×D₄	—	178	4	no	—
185	C₂²×C₄	—	89	2	no	—
186	C₂²×C₄	—	89	4	no	—
187	C₂²×C₄	—	89	4	no	—
188	C₂²×C₄	—	356	2	no	—
189	C₂²×C₄	—	356	2	no	—
190	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
191	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	89	4	no	—
196	C₂×C₈	—	356	4	no	—
197	C₂×C₈	—	356	2	no	—
198	C₂×C₈	—	178	8	no	—
199	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
201	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
203	C₄⋊C₄	—	89	2	no	—
204	C₄⋊C₄	—	89	2	no	—
205	C₂²⋊C₄	—	89	4	no	—
206	C₂²⋊C₄	—	89	4	no	—
207	C₄²	C₈×D₈₉	1	1	yes	characteristic
208	C₄²	C₈×D₈₉	1	1	yes	characteristic
209	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
210	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₁₆	—	178	4	no	—
213	C₂³	—	89	4	no	—
214	C₂³	—	356	2	no	—
215	C₂³	—	356	2	no	—
216	C₂³	—	178	4	no	—
217	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
218	D₄	—	178	4	no	—
219	D₄	—	178	4	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	89	4	no	—
224	C₂×C₄	—	356	4	no	—
225	C₂×C₄	—	356	2	no	—
226	C₂×C₄	—	178	4	no	—
227	C₂×C₄	—	178	4	no	—
228	C₂×C₄	#15667951	1	4	yes	complemented
229	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
230	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	178	4	no	—
239	C₈	#15668058	1	2	yes	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	356	8	no	—
243	C₂²	—	178	4	no	—
244	C₂²	—	178	4	no	—
245	C₂²	—	178	4	no	—
246	C₂²	C₂×#15667951	1	2	yes	—
247	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	178	4	no	—
252	C₄	C₂×#15667951	1	2	yes	—
253	C₄	C₂×#15667951	1	2	yes	—
254	C₄	#19152072	1	1	yes	characteristic
255	C₄	#19152364	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	356	4	no	—
258	C₂	#26029275	1	1	yes	characteristic
259	C₂	C₂²×#15667951	1	1	yes	characteristic
260	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2162	C₂
prime radical	C₂²×D₃₅₆	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2162	D₈₉
hypocenter	C₈₉	#33489
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #26022085	C₂	semidirect
3. #26022083	C₂	semidirect
4. #26022051	C₂	semidirect
5. #26021861	C₂	semidirect
6. C₈₉×#2162	C₂	semidirect
9. C₂×#15667951	C₂²	semidirect
16. C₄⋊#13083549	C₂²	semidirect
18. #19151377	C₂²	semidirect
19. C₄×#13083549	C₂²	semidirect
21. C₄⋊C₁₆×C₈₉	C₂²	semidirect
23. C₄×C₁₆×C₈₉	C₂²	semidirect
90. C₂×C₄×C₈₉	M₃₂	semidirect
135. #2162	D₈₉	semidirect
137. C₈₉	#33489	semidirect
149. C₄⋊C₁₆	C₂×D₈₉	semidirect
154. C₄×C₁₆	C₂×D₈₉	semidirect
228. C₂×C₄	#15667951	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₅₆	2
#26022085	2
#26022083	2
#26022051	2
#26021861	2
C₈₉×#2162	2
#33489	89