Subgroups — #36558847

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029054	C₂	1	1	yes	maximal, characteristic
2	#26028897	C₂	1	2	yes	maximal, complemented
3	#26021143	C₂	1	1	yes	maximal, characteristic
4	#26021138	C₂	1	1	yes	maximal, characteristic
5	#26021003	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020998	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020961	C₂	1	2	yes	maximal, complemented
8	#26020954	C₂	1	2	yes	maximal, complemented
9	#26020065	C₂	1	2	yes	maximal, complemented
10	#26019940	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1778	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	1	yes	characteristic
13	C₄×#13083574	C₂²	1	2	yes	—
14	#19151711	C₄	1	2	yes	complemented
15	#19151706	C₄	1	2	yes	complemented
16	#19151700	C₄	1	4	yes	complemented
17	#19151660	—	2	4	no	—
18	#19151649	—	2	4	no	—
19	#19151636	C₂²	1	1	yes	characteristic
20	#19151626	C₂²	1	2	yes	—
21	#19151626	C₂²	1	2	yes	—
22	C₂×#15667938	C₂²	1	2	yes	—
23	C₂×#15667938	C₂²	1	2	yes	—
24	#19151595	C₂²	1	2	yes	—
25	#19151566	C₂²	1	2	yes	—
26	#19151566	C₂²	1	2	yes	—
27	C₂×#15667922	C₂²	1	1	yes	characteristic
28	C₂×#15667922	C₂²	1	1	yes	characteristic
29	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	#19151444	C₂²	1	1	yes	characteristic
33	#19151359	C₂²	1	2	yes	—
34	#19151359	C₂²	1	2	yes	—
35	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
38	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
39	#341×C₈₉	C₂²	1	2	yes	—
40	C₂×#13083574	C₂³	1	2	yes	—
41	#15668034	—	2	2	no	—
42	#15668034	—	2	2	no	—
43	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
44	C₂×#13083563	C₂×C₄	1	2	yes	—
45	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂²×#10623494	C₂³	1	1	yes	characteristic
47	C₂²×#10623494	C₂³	1	1	yes	characteristic
48	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
49	C₄×D₃₅₆	C₂×C₄	1	4	yes	—
50	C₄×C₂²⋊D₈₉	—	2	4	no	—
51	#15667971	C₂³	1	2	yes	—
52	#15667969	—	2	2	no	—
53	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
55	C₄²×D₈₉	—	2	2	no	—
56	#15667946	C₂×C₄	1	2	yes	—
57	#15667944	C₂×C₄	1	2	yes	—
58	#15667941	—	2	4	no	—
59	#15667941	—	2	4	no	—
60	#15667938	—	2	4	no	—
61	#15667938	—	2	4	no	—
62	#15667937	C₂×C₄	1	4	yes	—
63	#15667934	C₂×C₄	1	4	yes	—
64	#15667924	—	2	4	no	—
65	#15667924	—	2	4	no	—
66	#15667922	—	2	2	no	—
67	#15667922	—	2	2	no	—
68	C₄×#10623494	D₄	1	4	yes	—
69	C₈×Dic₈₉	—	2	2	no	—
70	C₈×Dic₈₉	—	2	2	no	—
71	#15667919	—	2	2	no	—
72	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
73	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	#108×C₈₉	—	2	2	no	—
78	#108×C₈₉	—	2	2	no	—
79	#97×C₈₉	—	2	2	no	—
80	#97×C₈₉	C₂×C₄	1	2	yes	—
81	#97×C₈₉	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₈₉	—	2	2	no	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	#13083574	—	2	2	no	—
86	#13083574	—	2	2	no	—
87	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
88	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₈₉	—	2	2	no	—
90	C₂²×Dic₈₉	—	2	1	no	—
91	C₂²×Dic₈₉	—	2	1	no	—
92	C₂×C₄×D₈₉	—	2	4	no	—
93	C₂×C₄×D₈₉	—	2	2	no	—
94	Dic₃₅₆	—	2	2	no	—
95	#13083563	—	2	4	no	—
96	D₇₁₂	—	2	2	no	—
97	C₂×#10623494	—	2	1	no	—
98	C₂×#10623494	—	2	1	no	—
99	C₂×#10623494	—	2	1	no	—
100	C₂×#10623494	—	2	1	no	—
101	C₂×#10623494	C₂×D₄	1	2	yes	—
102	C₂×#10623494	C₂×D₄	1	2	yes	—
103	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
104	#13083553	—	2	4	no	—
105	C₂²⋊Dic₈₉	—	2	2	no	—
106	#13083551	—	2	4	no	—
107	C₄×Dic₈₉	—	2	4	no	—
108	C₄×Dic₈₉	—	2	2	no	—
109	C₄×Dic₈₉	—	2	1	no	—
110	C₄×Dic₈₉	—	2	1	no	—
111	C₄×Dic₈₉	#41	1	2	yes	—
112	C₄×Dic₈₉	#41	1	2	yes	—
113	C₂²×C₄×C₈₉	#41	1	2	yes	—
114	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
115	M₁₆×C₈₉	—	2	2	no	—
116	M₁₆×C₈₉	—	2	2	no	—
117	M₁₆×C₈₉	—	2	2	no	—
118	M₁₆×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	—	2	1	no	—
120	C₂×C₈×C₈₉	—	2	1	no	—
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	#41	1	2	yes	—
127	C₂×C₈×C₈₉	#41	1	2	yes	—
128	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
129	C₄²×C₈₉	C₂×D₄	1	2	yes	—
130	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
131	C₂²×D₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₂²⋊D₈₉	—	4	2	no	—
141	C₂²⋊D₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	2	4	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	4	2	no	—
147	C₂×Dic₈₉	—	4	2	no	—
148	D₃₅₆	—	2	2	no	—
149	D₃₅₆	—	2	2	no	—
150	C₄×D₈₉	—	4	8	no	—
151	C₄×D₈₉	—	4	2	no	—
152	C₄×D₈₉	—	4	2	no	—
153	Dic₁₇₈	—	2	2	no	—
154	Dic₁₇₈	—	2	2	no	—
155	C₈×C₈₉	—	2	2	no	—
156	C₈×C₈₉	—	4	2	no	—
157	C₈×C₈₉	—	4	2	no	—
158	C₈×C₈₉	#117	1	2	yes	—
159	C₈×C₈₉	#117	1	2	yes	—
160	#10623494	—	2	2	no	—
161	#10623494	—	2	2	no	—
162	C₂²×C₈₉	—	2	2	no	—
163	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂²×C₈₉	#529	1	1	yes	characteristic
165	C₂²×C₈₉	#529	1	1	yes	characteristic
166	C₂×D₈₉	—	4	4	no	—
167	C₂×D₈₉	—	4	2	no	—
168	C₄×C₈₉	—	2	4	no	—
169	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
170	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
171	C₄×C₈₉	#532	1	1	yes	characteristic
172	C₄×C₈₉	#532	1	1	yes	characteristic
173	Dic₈₉	—	4	2	no	—
174	Dic₈₉	—	4	2	no	—
175	Dic₈₉	—	4	4	no	—
176	Dic₈₉	—	4	2	no	—
177	#21296	—	89	1	no	maximal, Sylow
178	C₂×C₈₉	—	2	1	no	—
179	C₂×C₈₉	—	2	1	no	—
180	C₂×C₈₉	#2293	1	1	yes	characteristic
181	C₂×C₈₉	#2955	1	2	yes	—
182	D₈₉	—	8	2	no	—
183	#2943	—	89	1	no	—
184	#2922	—	89	2	no	—
185	#1975	—	89	1	no	—
186	#1974	—	89	1	no	—
187	#1946	—	89	1	no	—
188	#1945	—	89	1	no	—
189	#1938	—	89	2	no	—
190	#1937	—	89	2	no	—
191	#1778	D₈₉	1	1	yes	characteristic, complemented
192	#1759	—	89	2	no	—
193	#1726	—	89	1	no	—
194	C₈₉	#21296	1	1	yes	characteristic, complemented, Sylow
195	C₂×#135	—	89	1	no	—
196	C₄×#41	—	89	2	no	—
197	#442	—	89	2	no	—
198	#441	—	89	2	no	—
199	#440	—	89	4	no	—
200	#433	—	178	4	no	—
201	#431	—	178	4	no	—
202	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
203	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
204	#427	—	89	1	no	—
205	#424	—	89	2	no	—
206	#424	—	89	2	no	—
207	C₂×#104	—	89	2	no	—
208	C₂×#104	—	89	2	no	—
209	#416	—	89	2	no	—
210	#408	—	89	2	no	—
211	#408	—	89	2	no	—
212	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
213	C₂×#97	—	89	1	no	—
214	C₂×#97	—	89	1	no	—
215	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
216	C₂×C₄×C₈	—	89	1	no	—
217	C₂×C₄×C₈	—	89	1	no	—
218	C₂×C₄×C₈	—	89	1	no	—
219	#376	—	89	1	no	—
220	#341	C₂×D₈₉	1	2	yes	—
221	#339	—	89	2	no	—
222	#339	—	89	2	no	—
223	C₂×#41	—	89	2	no	—
224	#135	—	178	2	no	—
225	#135	—	178	2	no	—
226	C₂×Q₁₆	—	89	1	no	—
227	C₂×SD₁₆	—	89	2	no	—
228	C₂×D₈	—	89	1	no	—
229	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
230	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
231	C₂²×C₈	—	89	1	no	—
232	C₂²×C₈	—	89	1	no	—
233	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄×Q₈	—	89	4	no	—
236	C₄×D₄	—	89	4	no	—
237	C₄×D₄	—	178	4	no	—
238	#117	—	89	2	no	—
239	#117	—	178	2	no	—
240	C₂×C₄²	—	89	1	no	—
241	C₂×C₄²	—	89	1	no	—
242	C₂×C₄²	—	178	2	no	—
243	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
244	#108	—	2	2	no	—
245	#108	—	2	2	no	—
246	#107	—	89	2	no	—
247	#106	—	89	2	no	—
248	C₄⋊C₈	—	178	4	no	—
249	C₄⋊C₈	—	178	4	no	—
250	#104	—	178	4	no	—
251	#104	—	178	4	no	—
252	#103	—	89	4	no	—
253	#102	—	89	4	no	—
254	C₂²⋊C₈	—	178	4	no	—
255	C₂²⋊C₈	—	178	4	no	—
256	#97	—	2	2	no	—
257	#97	—	178	2	no	—
258	#97	—	178	2	no	—
259	#97	C₄×D₈₉	1	2	yes	—
260	#97	C₄×D₈₉	1	2	yes	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	89	4	no	—
263	C₄×C₈	—	178	2	no	—
264	C₄×C₈	—	178	2	no	—
265	#95	—	178	2	no	—
266	#41	—	178	2	no	—
267	#41	—	178	2	no	—
268	#41	—	178	2	no	—
269	#41	—	178	2	no	—
270	C₂×Q₈	—	89	2	no	—
271	C₂×D₄	—	89	2	no	—
272	C₂×D₄	—	178	2	no	—
273	C₂²×C₄	—	178	1	no	—
274	C₂²×C₄	—	178	1	no	—
275	C₂²×C₄	—	178	4	no	—
276	C₂²×C₄	—	178	2	no	—
277	C₂²×C₄	#13083575	1	2	yes	—
278	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
279	Q₁₆	—	178	2	no	—
280	SD₁₆	—	178	4	no	—
281	D₈	—	178	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	89	2	no	—
291	C₂×C₈	—	89	2	no	—
292	C₂×C₈	—	178	1	no	—
293	C₂×C₈	—	178	1	no	—
294	C₂×C₈	—	178	1	no	—
295	C₂×C₈	—	178	1	no	—
296	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
297	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₂×C₈	#13083574	1	2	yes	—
300	C₂×C₈	#13083574	1	2	yes	—
301	C₄⋊C₄	—	89	4	no	—
302	C₄⋊C₄	—	178	4	no	—
303	C₂²⋊C₄	—	178	2	no	—
304	C₂²⋊C₄	—	178	4	no	—
305	C₄²	—	89	2	no	—
306	C₄²	—	89	2	no	—
307	C₄²	—	178	4	no	—
308	C₄²	—	178	2	no	—
309	C₄²	—	178	1	no	—
310	C₄²	—	178	1	no	—
311	C₄²	C₂×C₄×D₈₉	1	2	yes	—
312	C₄²	D₄×D₈₉	1	2	yes	—
313	C₂³	—	178	2	no	—
314	C₂³	C₂×#13083575	1	1	yes	characteristic
315	Q₈	—	178	2	no	—
316	Q₈	—	178	2	no	—
317	D₄	—	356	2	no	—
318	D₄	—	356	2	no	—
319	D₄	—	178	2	no	—
320	D₄	—	178	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	356	8	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	4	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	#15667974	1	4	yes	—
332	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
334	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
335	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
337	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	178	2	no	—
342	C₈	—	178	2	no	—
343	C₈	#15667969	1	2	yes	—
344	C₈	#15667969	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	356	2	no	—
347	C₂²	—	356	4	no	—
348	C₂²	C₂×#15667974	1	1	yes	characteristic
349	C₂²	#19152135	1	1	yes	characteristic
350	C₂²	#19152135	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	356	2	no	—
353	C₄	—	356	2	no	—
354	C₄	—	356	2	no	—
355	C₄	—	356	4	no	—
356	C₄	C₂×#15667969	1	1	yes	characteristic
357	C₄	#19152066	1	1	yes	characteristic
358	C₄	#19152145	1	1	yes	characteristic
359	C₄	#19152145	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	712	2	no	—
363	C₂	#26022713	1	1	yes	characteristic
364	C₂	#26029154	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1778	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1778	D₈₉
hypocenter	C₈₉	#21296
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
2. #26028897	C₂	semidirect
5. #26021003	C₂	semidirect
6. #26020998	C₂	semidirect
7. #26020961	C₂	semidirect
8. #26020954	C₂	semidirect
9. #26020065	C₂	semidirect
10. #26019940	C₂	semidirect
11. C₈₉×#1778	C₂	semidirect
14. #19151711	C₄	semidirect
15. #19151706	C₄	semidirect
16. #19151700	C₄	semidirect
191. #1778	D₈₉	semidirect
194. C₈₉	#21296	semidirect

Maximal subgroups

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

subgroup	index
#26029054	2
#26028897	2
#26021143	2
#26021138	2
#26021003	2
#26020998	2
#26020961	2
#26020954	2
#26020065	2
#26019940	2
C₈₉×#1778	2
#21296	89