Subgroups — #36470538

Order 22784.

400 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151714	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151418	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022002	C₂	1	1	yes	maximal, characteristic
4	#26021999	C₂	1	1	yes	maximal, characteristic
5	#26020422	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018355	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	1	yes	characteristic
9	C₂×#15668020	—	2	1	no	—
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	#19151714	—	2	1	no	—
12	#19151714	—	2	1	no	—
13	#19151714	C₄	1	2	yes	—
14	#19151714	C₄	1	2	yes	—
15	C₂×#15667948	C₂²	1	1	yes	characteristic
16	C₂×#15667947	—	2	1	no	—
17	#19151621	—	2	1	no	—
18	C₂×#15667938	—	2	1	no	—
19	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
20	#19151418	—	2	1	no	—
21	#19151418	—	2	1	no	—
22	#19151415	—	2	1	no	—
23	#19151413	—	2	1	no	—
24	#19151412	—	2	1	no	—
25	#19151410	—	2	1	no	—
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#13083574	—	2	1	no	—
30	#15668034	—	2	1	no	—
31	#15668034	—	2	1	no	—
32	#15668034	C₂×C₄	1	2	yes	—
33	#15668034	Q₈	1	2	yes	—
34	C₂×Dic₃₅₆	D₄	1	1	yes	characteristic
35	C₂×#13083563	—	2	1	no	—
36	C₂×D₇₁₂	D₄	1	1	yes	characteristic
37	#15668020	—	2	2	no	—
38	#15668020	—	2	2	no	—
39	#15668020	—	4	1	no	—
40	#15668020	—	4	1	no	—
41	C₂×#13083558	—	2	1	no	—
42	C₂×#13083557	—	4	1	no	—
43	C₂²×#10623494	D₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	—	4	1	no	—
45	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
46	#15667952	—	2	1	no	—
47	#15667952	—	2	1	no	—
48	C₂×#13083549	—	2	1	no	—
49	#15667948	—	2	1	no	—
50	#15667948	—	2	1	no	—
51	#15667947	—	4	1	no	—
52	#15667947	—	4	1	no	—
53	#15667941	—	4	2	no	—
54	#15667938	—	2	2	no	—
55	#15667938	—	2	2	no	—
56	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
57	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₈₉	D₄	1	1	yes	characteristic
59	C₈×Dic₈₉	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₈₉	—	2	1	no	—
61	C₂×M₁₆×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	M₃₂×C₈₉	—	2	1	no	—
64	M₃₂×C₈₉	—	2	1	no	—
65	C₂×C₁₆×C₈₉	—	2	1	no	—
66	#108×C₈₉	—	2	1	no	—
67	#108×C₈₉	—	2	1	no	—
68	#108×C₈₉	—	2	1	no	—
69	#108×C₈₉	C₂×C₄	1	2	yes	—
70	#13083574	—	2	2	no	—
71	#13083574	—	2	2	no	—
72	#13083574	—	4	1	no	—
73	#13083574	—	4	1	no	—
74	C₂×Dic₁₇₈	—	2	1	no	—
75	C₂×D₃₅₆	—	2	1	no	—
76	C₂×C₂²⋊D₈₉	—	4	1	no	—
77	C₂²×Dic₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	4	1	no	—
79	Dic₃₅₆	—	2	1	no	—
80	Dic₃₅₆	C₄⋊C₄	1	2	yes	—
81	#13083563	—	2	2	no	—
82	#13083563	—	2	2	no	—
83	D₇₁₂	—	2	1	no	—
84	D₇₁₂	C₄⋊C₄	1	2	yes	—
85	#13083558	—	4	1	no	—
86	#13083558	—	4	1	no	—
87	#13083557	—	4	2	no	—
88	#13083557	—	4	2	no	—
89	C₂×#10623494	—	4	1	no	—
90	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×#10623494	D₈	1	1	yes	characteristic
93	C₂×#10623494	D₈	1	1	yes	characteristic
94	C₂×#10623494	SD₁₆	1	1	yes	characteristic
95	C₂×#10623494	SD₁₆	1	1	yes	characteristic
96	C₈×D₈₉	—	4	2	no	—
97	C₈×D₈₉	—	4	2	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
101	#13083549	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	M₁₆×C₈₉	—	2	2	no	—
104	M₁₆×C₈₉	—	4	1	no	—
105	M₁₆×C₈₉	—	4	1	no	—
106	M₁₆×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
115	C₁₆×C₈₉	—	4	1	no	—
116	C₂³×C₈₉	#137	1	1	yes	characteristic
117	C₂²×D₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂²⋊D₈₉	—	4	2	no	—
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	D₃₅₆	—	2	2	no	—
131	D₃₅₆	—	4	1	no	—
132	C₄×D₈₉	—	4	2	no	—
133	C₄×D₈₉	—	4	2	no	—
134	Dic₁₇₈	—	2	2	no	—
135	Dic₁₇₈	—	4	1	no	—
136	C₈×C₈₉	—	4	2	no	—
137	C₈×C₈₉	—	8	1	no	—
138	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₈×C₈₉	#117	1	1	yes	characteristic
140	C₈×C₈₉	#121	1	1	yes	characteristic
141	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
142	#10623494	—	8	1	no	—
143	#10623494	#102	1	1	yes	characteristic
144	#10623494	#102	1	1	yes	characteristic
145	#10623494	#102	1	1	yes	characteristic
146	#10623494	#102	1	1	yes	characteristic
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	#417	1	1	yes	characteristic
149	C₂²×C₈₉	#461	1	1	yes	characteristic
150	C₂²×C₈₉	#474	1	1	yes	characteristic
151	C₂×D₈₉	—	4	2	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₄×C₈₉	#380	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
155	C₄×C₈₉	#459	1	1	yes	characteristic
156	C₄×C₈₉	#476	1	1	yes	characteristic
157	Dic₈₉	—	2	2	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	2	no	—
160	#12825	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	#1802	1	1	yes	characteristic
165	D₈₉	—	8	2	no	—
166	C₂×#443	—	89	1	no	—
167	C₂×#364	—	89	1	no	—
168	#2148	—	89	1	no	—
169	#2148	—	89	1	no	—
170	#1848	—	89	1	no	—
171	#1385	—	89	1	no	—
172	#1385	D₈₉	1	1	yes	characteristic, complemented
173	C₈₉	#12825	1	1	yes	characteristic, complemented, Sylow
174	C₂×#135	—	89	1	no	—
175	C₂×#131	—	178	1	no	—
176	C₂×M₃₂	—	89	1	no	—
177	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
178	#443	—	89	2	no	—
179	#443	—	89	2	no	—
180	#443	—	178	1	no	—
181	#443	—	178	1	no	—
182	C₂×#108	—	89	1	no	—
183	C₂×#108	—	178	1	no	—
184	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
185	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
186	#423	—	178	1	no	—
187	C₂×#104	—	178	1	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	#364	—	178	1	no	—
190	#364	—	178	1	no	—
191	#362	—	178	1	no	—
192	#362	—	178	1	no	—
193	#361	—	178	1	no	—
194	#361	—	178	1	no	—
195	C₂×#41	—	178	1	no	—
196	#135	—	89	2	no	—
197	#135	—	89	2	no	—
198	#135	—	178	1	no	—
199	#135	—	178	1	no	—
200	C₂×Q₁₆	—	89	1	no	—
201	C₂×SD₁₆	—	178	1	no	—
202	C₂×D₈	—	89	1	no	—
203	#131	—	356	1	no	—
204	#131	—	356	1	no	—
205	#131	—	178	2	no	—
206	#131	—	178	2	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	356	1	no	—
210	C₂×M₁₆	—	178	1	no	—
211	C₂²×C₈	—	89	1	no	—
212	C₂²×C₈	—	356	1	no	—
213	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₂×C₄²	—	89	1	no	—
215	M₃₂	—	2	1	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	178	1	no	—
218	M₃₂	—	178	1	no	—
219	C₂×C₁₆	—	2	1	no	—
220	C₂×C₁₆	—	178	1	no	—
221	#108	—	2	1	no	—
222	#108	—	2	1	no	—
223	#108	—	2	1	no	—
224	#108	—	356	1	no	—
225	#108	—	356	1	no	—
226	#108	—	178	1	no	—
227	#108	—	178	1	no	—
228	#108	C₄×D₈₉	1	2	yes	—
229	C₄⋊C₈	—	356	2	no	—
230	#104	—	178	2	no	—
231	#104	—	178	2	no	—
232	C₄×C₈	—	89	1	no	—
233	C₄×C₈	—	89	1	no	—
234	C₄×C₈	—	89	1	no	—
235	C₄×C₈	—	89	1	no	—
236	#41	—	356	1	no	—
237	#41	—	356	1	no	—
238	#41	—	178	2	no	—
239	#41	—	178	2	no	—
240	C₂×Q₈	—	178	1	no	—
241	C₂×D₄	—	356	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂²×C₄	—	356	1	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
246	Q₁₆	—	89	2	no	—
247	Q₁₆	—	178	1	no	—
248	SD₁₆	—	178	2	no	—
249	SD₁₆	—	178	2	no	—
250	D₈	—	89	2	no	—
251	D₈	—	178	1	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	356	2	no	—
257	M₁₆	—	356	2	no	—
258	M₁₆	—	356	1	no	—
259	M₁₆	—	356	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	89	1	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	356	2	no	—
269	C₂×C₈	—	356	2	no	—
270	C₂×C₈	—	356	1	no	—
271	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
273	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
275	C₂×C₈	#13083574	1	1	yes	characteristic
276	C₂×C₈	#13083574	1	1	yes	characteristic
277	C₄²	—	89	1	no	—
278	C₄²	—	89	1	no	—
279	C₄²	—	178	1	no	—
280	C₁₆	—	4	1	no	—
281	C₁₆	—	356	1	no	—
282	C₂³	—	356	1	no	—
283	C₂³	#15668044	1	1	yes	characteristic
284	Q₈	—	356	1	no	—
285	Q₈	—	178	2	no	—
286	D₄	—	356	2	no	—
287	D₄	—	356	2	no	—
288	D₄	—	356	1	no	—
289	D₄	—	178	2	no	—
290	C₂×C₄	—	356	1	no	—
291	C₂×C₄	—	356	2	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₄	#15667985	1	1	yes	characteristic
298	C₂×C₄	#15667990	1	1	yes	characteristic
299	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
301	C₂×C₄	#15668039	1	1	yes	characteristic
302	C₈	—	4	2	no	—
303	C₈	—	8	1	no	—
304	C₈	—	89	1	no	—
305	C₈	—	89	1	no	—
306	C₈	—	89	1	no	—
307	C₈	—	89	1	no	—
308	C₈	—	712	1	no	—
309	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
310	C₈	#15667969	1	1	yes	characteristic
311	C₈	#15667985	1	1	yes	characteristic
312	C₈	#15667990	1	1	yes	characteristic
313	C₂²	—	4	1	no	—
314	C₂²	—	356	2	no	—
315	C₂²	—	712	2	no	—
316	C₂²	#19151598	1	1	yes	characteristic
317	C₂²	#19151802	1	1	yes	characteristic
318	C₂²	#19151877	1	1	yes	characteristic
319	C₄	—	356	1	no	—
320	C₄	—	356	2	no	—
321	C₄	—	178	2	no	—
322	C₄	#19151457	1	1	yes	characteristic
323	C₄	#102×D₈₉	1	1	yes	characteristic
324	C₄	#19151792	1	1	yes	characteristic
325	C₄	#19151887	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	712	2	no	—
330	C₂	#26020251	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151887
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1385	C₂
socle	C₂×C₈₉	#1802
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1385	D₈₉
hypocenter	C₈₉	#12825
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

Complements

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

N	H ≅ G/N	type
2. C₂×#19151418	C₂	semidirect
5. #26020422	C₂	semidirect
6. #26018355	C₂	semidirect
7. C₈₉×#1385	C₂	semidirect
172. #1385	D₈₉	semidirect
173. C₈₉	#12825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151714	2
C₂×#19151418	2
#26022002	2
#26021999	2
#26020422	2
#26018355	2
C₈₉×#1385	2
#12825	89