Subgroups — #36469728

Order 22784.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151721	C₂	1	1	yes	maximal, characteristic
2	#26022002	C₂	1	1	yes	maximal, characteristic
3	#26021981	C₂	1	1	yes	maximal, characteristic
4	#26020158	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018364	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018356	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#347×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	1	yes	characteristic
9	C₂×#15668018	—	2	1	no	—
10	#19151721	—	2	2	no	—
11	#19151721	C₄	1	4	yes	—
12	C₂×#15667948	C₂²	1	1	yes	characteristic
13	C₂×#15667948	C₂²	1	1	yes	characteristic
14	C₂×#15667948	C₂²	1	1	yes	characteristic
15	C₂×#15667940	—	2	1	no	—
16	C₄×#13083558	—	2	1	no	—
17	#19151415	C₄	1	2	yes	complemented
18	#19151412	C₄	1	2	yes	complemented
19	#19151406	C₄	1	4	yes	complemented
20	#19151363	—	2	1	no	—
21	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
24	#347×C₈₉	—	2	2	no	—
25	C₂×#13083574	—	2	1	no	—
26	#15668034	—	2	1	no	—
27	#15668034	—	2	1	no	—
28	#15668034	C₂×C₄	1	2	yes	—
29	#15668034	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083563	—	2	1	no	—
32	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
33	#15668018	—	2	4	no	—
34	#15668018	—	4	2	no	—
35	C₂×#13083558	—	2	1	no	—
36	C₂×#13083558	—	2	1	no	—
37	C₂×#13083558	—	2	1	no	—
38	C₂×#13083557	—	4	1	no	—
39	C₂×C₈×D₈₉	—	4	1	no	—
40	#15667948	—	2	1	no	—
41	#15667948	—	2	1	no	—
42	#15667948	—	2	1	no	—
43	#15667948	C₂×C₄	1	2	yes	—
44	#15667948	C₂×C₄	1	2	yes	—
45	#15667948	C₂×C₄	1	2	yes	—
46	#15667940	—	2	4	no	—
47	#15667923	—	4	1	no	—
48	C₄×#10623494	—	4	1	no	—
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	—	2	1	no	—
54	M₃₂×C₈₉	—	2	1	no	—
55	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
58	#97×C₈₉	C₂×C₄	1	2	yes	—
59	#97×C₈₉	D₄	1	2	yes	—
60	#13083574	—	2	2	no	—
61	#13083574	—	2	2	no	—
62	#13083574	—	4	1	no	—
63	#13083574	—	4	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	4	1	no	—
67	C₂×C₄×D₈₉	—	4	1	no	—
68	Dic₃₅₆	—	2	1	no	—
69	Dic₃₅₆	C₄²	1	2	yes	—
70	#13083563	—	2	2	no	—
71	#13083563	—	2	2	no	—
72	D₇₁₂	—	2	1	no	—
73	D₇₁₂	C₄²	1	2	yes	—
74	#13083558	—	2	2	no	—
75	#13083558	—	2	2	no	—
76	#13083558	—	2	2	no	—
77	#13083558	—	4	1	no	—
78	#13083558	—	4	1	no	—
79	#13083558	—	4	1	no	—
80	#13083557	—	4	4	no	—
81	C₂×#10623494	—	4	1	no	—
82	C₂×#10623494	—	4	1	no	—
83	C₂×#10623494	—	4	1	no	—
84	C₈×D₈₉	—	4	4	no	—
85	C₂²×C₄×C₈₉	—	2	1	no	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
98	C₁₆×C₈₉	—	2	2	no	—
99	C₁₆×C₈₉	—	2	2	no	—
100	C₂³×C₈₉	#137	1	1	yes	characteristic
101	C₂²×D₈₉	—	4	1	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	8	1	no	—
104	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂²⋊D₈₉	—	4	2	no	—
111	C₂²⋊D₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	4	1	no	—
113	D₃₅₆	—	2	2	no	—
114	D₃₅₆	—	4	1	no	—
115	C₄×D₈₉	—	4	2	no	—
116	C₄×D₈₉	—	4	2	no	—
117	Dic₁₇₈	—	2	2	no	—
118	Dic₁₇₈	—	4	1	no	—
119	C₈×C₈₉	—	2	2	no	—
120	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
121	C₈×C₈₉	#117	1	1	yes	characteristic
122	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
124	#10623494	—	4	2	no	—
125	#10623494	—	4	2	no	—
126	#10623494	—	4	2	no	—
127	C₂²×C₈₉	—	4	1	no	—
128	C₂²×C₈₉	#417	1	1	yes	characteristic
129	C₂²×C₈₉	#440	1	1	yes	characteristic
130	C₂²×C₈₉	#440	1	1	yes	characteristic
131	C₂×D₈₉	—	4	2	no	—
132	C₂×D₈₉	—	8	2	no	—
133	C₄×C₈₉	—	4	2	no	—
134	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₈₉	#418	1	1	yes	characteristic
136	C₄×C₈₉	#442	1	1	yes	characteristic
137	C₄×C₈₉	#442	1	1	yes	characteristic
138	Dic₈₉	—	4	2	no	—
139	#12651	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	#1762	1	1	yes	characteristic
144	D₈₉	—	8	2	no	—
145	C₂×#444	—	89	1	no	—
146	#2148	—	89	1	no	—
147	#2143	—	89	1	no	—
148	C₂×#347	D₈₉	1	1	yes	characteristic, complemented
149	#1778	—	89	1	no	—
150	#1388	—	89	1	no	—
151	#1385	—	89	1	no	—
152	C₈₉	#12651	1	1	yes	characteristic, complemented, Sylow
153	C₂×#135	—	89	1	no	—
154	C₂×#131	—	178	1	no	—
155	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
156	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
157	#444	—	89	4	no	—
158	#444	—	178	2	no	—
159	C₂×#108	—	89	1	no	—
160	C₂×#108	—	89	1	no	—
161	C₂×#108	—	89	1	no	—
162	C₂×#104	—	178	1	no	—
163	C₄×M₁₆	—	178	1	no	—
164	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
165	#362	—	89	2	no	—
166	#361	—	89	2	no	—
167	#359	—	89	4	no	—
168	#347	—	2	2	no	—
169	#341	—	178	1	no	—
170	C₂×#41	—	178	1	no	—
171	#135	—	89	2	no	—
172	#135	—	89	2	no	—
173	#135	—	178	1	no	—
174	#135	—	178	1	no	—
175	C₂×Q₁₆	—	89	1	no	—
176	C₂×SD₁₆	—	178	1	no	—
177	C₂×D₈	—	89	1	no	—
178	#131	—	356	2	no	—
179	#131	—	178	4	no	—
180	C₂×M₁₆	—	356	1	no	—
181	C₂×M₁₆	—	178	1	no	—
182	C₂×M₁₆	—	178	1	no	—
183	C₂×M₁₆	—	178	1	no	—
184	C₂²×C₈	—	356	1	no	—
185	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
187	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	—	2	1	no	—
191	M₃₂	C₄×D₈₉	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
194	#108	—	89	2	no	—
195	#108	—	89	2	no	—
196	#108	—	89	2	no	—
197	#108	—	178	1	no	—
198	#108	—	178	1	no	—
199	#108	—	178	1	no	—
200	#104	—	178	4	no	—
201	#97	—	356	1	no	—
202	#97	C₄×D₈₉	1	2	yes	—
203	#97	C₂²⋊D₈₉	1	2	yes	—
204	C₄×C₈	—	356	1	no	—
205	#41	—	356	1	no	—
206	#41	—	356	1	no	—
207	#41	—	178	2	no	—
208	#41	—	178	2	no	—
209	C₂×Q₈	—	178	1	no	—
210	C₂×D₄	—	356	1	no	—
211	C₂×D₄	—	178	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	356	1	no	—
214	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
215	Q₁₆	—	89	2	no	—
216	Q₁₆	—	178	1	no	—
217	SD₁₆	—	178	2	no	—
218	SD₁₆	—	178	2	no	—
219	D₈	—	89	2	no	—
220	D₈	—	178	1	no	—
221	M₁₆	—	356	4	no	—
222	M₁₆	—	356	1	no	—
223	M₁₆	—	356	1	no	—
224	M₁₆	—	356	1	no	—
225	M₁₆	—	178	2	no	—
226	M₁₆	—	178	2	no	—
227	M₁₆	—	178	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	356	4	no	—
231	C₂×C₈	—	356	1	no	—
232	C₂×C₈	—	356	1	no	—
233	C₂×C₈	—	356	1	no	—
234	C₂×C₈	#13083551	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
239	C₂×C₈	#13083574	1	1	yes	characteristic
240	C₂×C₈	#13083574	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	#13083551	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₂³	—	356	1	no	—
246	C₂³	#15668045	1	1	yes	characteristic
247	Q₈	—	356	1	no	—
248	Q₈	—	178	2	no	—
249	D₄	—	356	2	no	—
250	D₄	—	356	2	no	—
251	D₄	—	356	1	no	—
252	D₄	—	178	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	356	1	no	—
256	C₂×C₄	—	356	2	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
259	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
261	C₂×C₄	#15668040	1	1	yes	characteristic
262	C₂×C₄	#15668040	1	1	yes	characteristic
263	C₂×C₄	#15668040	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	356	2	no	—
266	C₈	—	356	2	no	—
267	C₈	—	356	2	no	—
268	C₈	C₄²×D₈₉	1	1	yes	characteristic
269	C₈	#15667969	1	1	yes	characteristic
270	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
271	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	356	2	no	—
274	C₂²	—	712	2	no	—
275	C₂²	#19151600	1	1	yes	characteristic
276	C₂²	#19151700	1	1	yes	characteristic
277	C₂²	#19151700	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	356	2	no	—
280	C₄	C₄×#13083551	1	1	yes	characteristic
281	C₄	#19151604	1	1	yes	characteristic
282	C₄	#19151711	1	1	yes	characteristic
283	C₄	#19151711	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	712	2	no	—
288	C₂	#26020078	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151711
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#347×C₈₉	C₂
socle	C₂×C₈₉	#1762
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#347	D₈₉
hypocenter	C₈₉	#12651
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #26020158	C₂	semidirect
5. #26018364	C₂	semidirect
6. #26018356	C₂	semidirect
7. C₂×#347×C₈₉	C₂	semidirect
17. #19151415	C₄	semidirect
18. #19151412	C₄	semidirect
19. #19151406	C₄	semidirect
148. C₂×#347	D₈₉	semidirect
152. C₈₉	#12651	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151721	2
#26022002	2
#26021981	2
#26020158	2
#26018364	2
#26018356	2
C₂×#347×C₈₉	2
#12651	89