Subgroups — #36573311

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
2	#26023210	C₂	1	2	yes	maximal
3	#26021023	C₂	1	2	yes	maximal
4	#26021020	C₂	1	2	yes	maximal
5	#26021009	C₂	1	2	yes	maximal
6	#26020419	C₂	1	2	yes	maximal
7	#26020416	C₂	1	1	yes	maximal, characteristic
8	#26020151	C₂	1	2	yes	maximal
9	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	#19151728	C₄	1	8	yes	complemented
12	C₂×#15667946	C₂²	1	1	yes	characteristic
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667945	C₂²	1	2	yes	—
15	C₂×#15667944	C₂²	1	2	yes	—
16	C₂×#15667943	C₂²	1	2	yes	—
17	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
18	C₂×#15667941	C₂²	1	2	yes	—
19	C₂×#15667922	C₂²	1	2	yes	—
20	#19151502	—	2	2	no	—
21	#19151482	—	2	2	no	—
22	#19151481	—	2	2	no	—
23	#19151467	C₂²	1	2	yes	—
24	#19151464	C₂²	1	2	yes	—
25	#19151463	C₂²	1	2	yes	—
26	C₄⋊C₄×Dic₈₉	C₂²	1	2	yes	—
27	C₄⋊C₄×Dic₈₉	C₂²	1	2	yes	—
28	C₄×#13083553	—	2	2	no	—
29	#19151360	C₂²	1	2	yes	—
30	#19151360	C₂²	1	2	yes	—
31	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
35	#340×C₈₉	C₂²	1	2	yes	—
36	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
37	C₂²×#10623494	C₂³	1	2	yes	—
38	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₇₈	Q₈	1	4	yes	—
40	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
43	C₂×#13083553	—	2	2	no	—
44	C₂×#13083553	—	2	2	no	—
45	C₂×#13083553	D₄	1	2	yes	—
46	C₂×C₄×Dic₈₉	—	2	2	no	—
47	C₂×C₄×Dic₈₉	—	2	2	no	—
48	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
49	#15667946	—	4	1	no	—
50	#15667946	—	4	1	no	—
51	#15667945	—	4	2	no	—
52	#15667944	—	2	4	no	—
53	#15667943	—	4	2	no	—
54	C₄⋊#10623494	C₂×C₄	1	4	yes	—
55	#15667941	C₂×C₄	1	8	yes	—
56	#15667922	C₂×C₄	1	8	yes	—
57	#15667919	—	2	2	no	—
58	#15667919	—	2	2	no	—
59	#15667919	—	2	2	no	—
60	#15667919	—	2	2	no	—
61	#15667919	—	2	2	no	—
62	#15667919	—	2	2	no	—
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
67	C₂×C₄²×C₈₉	—	2	2	no	—
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	#107×C₈₉	—	2	2	no	—
70	#107×C₈₉	Q₈	1	4	yes	complemented
71	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
72	#95×C₈₉	—	2	2	no	—
73	C₂×Dic₁₇₈	—	4	1	no	—
74	C₂×Dic₁₇₈	—	4	1	no	—
75	C₂×Dic₁₇₈	C₄⋊C₄	1	4	yes	—
76	C₂²×Dic₈₉	—	2	2	no	—
77	C₂²×Dic₈₉	—	2	2	no	—
78	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
79	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	C₂²×C₄	1	4	yes	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
88	#13083553	—	4	4	no	—
89	#13083553	—	8	2	no	—
90	#13083553	C₄⋊C₄	1	8	yes	—
91	C₄×Dic₈₉	—	4	4	no	—
92	C₄×Dic₈₉	—	8	2	no	—
93	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
94	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
95	C₂²×C₄×C₈₉	—	2	2	no	—
96	C₂²×C₄×C₈₉	—	2	2	no	—
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₈₉	#41	1	2	yes	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
110	C₄²×C₈₉	—	8	2	no	—
111	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
112	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
113	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	4	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×Dic₈₉	—	2	2	no	—
133	C₂×Dic₈₉	—	2	2	no	—
134	C₂×Dic₈₉	—	2	2	no	—
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
144	Dic₁₇₈	—	4	4	no	—
145	C₈×C₈₉	—	2	2	no	—
146	C₈×C₈₉	C₄×Q₈	1	4	yes	—
147	#10623494	—	2	4	no	—
148	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
150	C₂²×C₈₉	#550	1	1	yes	characteristic
151	C₂²×C₈₉	#575	1	1	yes	characteristic
152	C₂²×C₈₉	#575	1	1	yes	characteristic
153	C₂²×C₈₉	#576	1	1	yes	characteristic
154	C₂²×C₈₉	#578	1	1	yes	characteristic
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	4	4	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₄×C₈₉	#520	1	1	yes	characteristic
161	C₄×C₈₉	#557	1	1	yes	characteristic
162	C₄×C₈₉	Q₈²	1	1	yes	characteristic
163	Dic₈₉	—	2	4	no	—
164	Dic₈₉	—	2	4	no	—
165	Dic₈₉	—	8	2	no	—
166	Dic₈₉	—	8	2	no	—
167	#22731	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₈₉	#2911	1	1	yes	characteristic
170	C₂×C₈₉	#2913	1	1	yes	characteristic
171	C₂×C₈₉	#3389	1	1	yes	characteristic
172	C₂×C₈₉	#3390	1	1	yes	characteristic
173	C₂×C₈₉	#3402	1	1	yes	characteristic
174	C₂×C₈₉	#3403	1	1	yes	characteristic
175	C₂×#446	—	89	1	no	—
176	#2355	—	89	2	no	—
177	#1949	—	89	2	no	—
178	#1949	—	89	2	no	—
179	#1947	—	89	2	no	—
180	#1847	—	89	2	no	—
181	#1846	—	89	1	no	—
182	#1776	—	89	2	no	—
183	C₄×#107	D₈₉	1	1	yes	characteristic
184	C₈₉	#22731	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	—	89	1	no	—
186	#446	—	89	8	no	—
187	C₂×#107	—	89	1	no	—
188	C₂×#107	—	89	1	no	—
189	C₂×#107	—	89	2	no	—
190	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
191	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
192	C₂×#106	—	89	2	no	—
193	C₂×#106	—	89	2	no	—
194	C₂×C₄⋊C₈	—	89	2	no	—
195	C₂×C₄⋊C₈	—	89	1	no	—
196	C₂×#97	—	89	2	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#391	—	178	2	no	—
199	#387	—	178	2	no	—
200	#387	—	178	2	no	—
201	#384	—	89	2	no	—
202	#382	—	89	2	no	—
203	#382	—	89	2	no	—
204	C₄×C₄⋊C₄	—	89	2	no	—
205	C₄×C₄⋊C₄	—	89	2	no	—
206	C₄×C₄⋊C₄	—	178	2	no	—
207	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
208	#340	—	89	2	no	—
209	#340	—	89	2	no	—
210	#340	C₂×D₈₉	1	2	yes	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂²×C₈	—	89	2	no	—
213	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄×Q₈	—	89	4	no	—
216	C₄×Q₈	—	89	4	no	—
217	C₂×C₄⋊C₄	—	89	2	no	—
218	C₂×C₄⋊C₄	—	89	1	no	—
219	C₂×C₄⋊C₄	—	89	2	no	—
220	C₂×C₄⋊C₄	—	89	2	no	—
221	C₂×C₄⋊C₄	—	178	2	no	—
222	C₂×C₄⋊C₄	—	178	2	no	—
223	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	—	89	2	no	—
227	C₂×C₄²	—	178	2	no	—
228	C₂×C₄²	—	178	2	no	—
229	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
230	#107	—	2	2	no	—
231	#107	—	356	2	no	—
232	#107	—	356	1	no	—
233	#107	—	356	1	no	—
234	#107	Dic₁₇₈	1	4	yes	complemented
235	#106	—	356	2	no	—
236	#106	—	178	4	no	—
237	C₄⋊C₈	—	89	8	no	—
238	C₄⋊C₈	—	89	4	no	—
239	#97	—	89	8	no	—
240	C₄×C₈	C₄×D₈₉	1	4	yes	—
241	#95	—	2	2	no	—
242	#95	—	178	2	no	—
243	#95	—	178	2	no	—
244	#95	—	178	2	no	—
245	#95	—	178	2	no	—
246	#95	—	178	2	no	—
247	#95	—	178	2	no	—
248	C₂×Q₈	—	89	4	no	—
249	C₂×Q₈	—	356	1	no	—
250	C₂×Q₈	—	356	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	89	2	no	—
254	C₂²×C₄	—	89	2	no	—
255	C₂²×C₄	—	178	2	no	—
256	C₂²×C₄	—	178	2	no	—
257	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	89	4	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
267	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
268	C₂×C₈	#13083574	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	89	8	no	—
273	C₄⋊C₄	—	89	4	no	—
274	C₄⋊C₄	—	356	4	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₄⋊C₄	—	178	2	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄⋊C₄	—	178	2	no	—
279	C₄⋊C₄	—	712	2	no	—
280	C₄⋊C₄	C₂×Dic₁₇₈	1	4	yes	—
281	C₄²	—	8	2	no	—
282	C₄²	—	89	4	no	—
283	C₄²	—	89	4	no	—
284	C₄²	—	356	4	no	—
285	C₄²	—	712	2	no	—
286	C₄²	C₂×C₄×D₈₉	1	2	yes	—
287	C₄²	Q₈×D₈₉	1	2	yes	—
288	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
289	Q₈	—	356	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	89	4	no	—
301	C₂×C₄	—	89	4	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	178	2	no	—
309	C₂×C₄	—	178	2	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
314	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
317	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
318	C₂×C₄	#15668058	1	1	yes	characteristic
319	C₂×C₄	#15668059	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	178	4	no	—
322	C₈	C₄×Dic₁₇₈	1	4	yes	—
323	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
324	C₂²	D₄×Dic₁₇₈	1	1	yes	characteristic
325	C₂²	#19152269	1	1	yes	characteristic
326	C₂²	#19152395	1	1	yes	characteristic
327	C₂²	#19152395	1	1	yes	characteristic
328	C₂²	#19152399	1	1	yes	characteristic
329	C₂²	#19152411	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	2	no	—
334	C₄	—	178	4	no	—
335	C₄	—	178	4	no	—
336	C₄	—	712	2	no	—
337	C₄	—	712	2	no	—
338	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
339	C₄	#19152078	1	1	yes	characteristic
340	C₄	#19152326	1	1	yes	characteristic
341	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
342	C₂	C₄⋊C₄×Dic₁₇₈	1	1	yes	characteristic
343	C₂	#26028840	1	1	yes	characteristic
344	C₂	#26028848	1	1	yes	characteristic
345	C₂	#26033208	1	1	yes	characteristic
346	C₂	#26033217	1	1	yes	characteristic
347	C₂	#26033329	1	1	yes	characteristic
348	C₂	#26033338	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#22731
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #19151728	C₄	semidirect
70. #107×C₈₉	Q₈	semidirect
184. C₈₉	#22731	semidirect
234. #107	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151728	2
#26023210	2
#26021023	2
#26021020	2
#26021009	2
#26020419	2
#26020416	2
#26020151	2
C₄×#107×C₈₉	2
#22731	89