Subgroups — #36453662

Order 22784.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151971	C₂	1	1	yes	maximal, characteristic
2	#26021204	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021179	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151345	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151341	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083563	—	2	1	no	—
9	C₂²×#13083563	—	2	1	no	—
10	C₂×#15668012	C₂²	1	1	yes	characteristic
11	C₂×#15668009	C₂²	1	1	yes	characteristic
12	#19151971	—	2	2	no	—
13	#19151971	C₄	1	4	yes	complemented
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×#15667932	—	2	2	no	—
17	#19151360	—	2	2	no	—
18	#19151349	—	2	2	no	—
19	#19151345	—	2	2	no	—
20	#19151341	—	2	2	no	—
21	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
25	#336×C₈₉	—	2	1	no	—
26	C₂²×Dic₁₇₈	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	1	no	—
28	C₂²×D₃₅₆	—	2	1	no	—
29	C₂²×D₃₅₆	—	2	1	no	—
30	C₂×#13083563	—	2	4	no	—
31	C₂×#13083563	—	2	4	no	—
32	C₂×#13083563	—	4	2	no	—
33	C₂×#13083563	—	4	2	no	—
34	C₂×#13083563	—	4	2	no	—
35	C₂×#13083563	—	4	2	no	—
36	C₂²×#10623494	—	2	1	no	—
37	C₂²×#10623494	—	2	1	no	—
38	#15668012	—	2	1	no	—
39	#15668012	—	2	1	no	—
40	#15668012	—	2	1	no	—
41	#15668012	C₂×C₄	1	2	yes	—
42	#15668009	—	2	1	no	—
43	#15668009	—	2	1	no	—
44	#15668009	—	2	1	no	—
45	#15668009	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	—	2	1	no	—
47	C₂×C₄⋊Dic₈₉	—	2	1	no	—
48	C₄⋊#10623494	—	2	1	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	—	2	1	no	—
52	#15667932	—	4	4	no	—
53	C₂²×C₈×C₈₉	—	2	1	no	—
54	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
57	C₂×C₄²×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
61	C₄×C₈×C₈₉	—	4	2	no	—
62	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₈₉	D₄	1	2	yes	—
64	#95×C₈₉	—	2	1	no	—
65	C₂³×D₈₉	—	4	1	no	—
66	C₂×Dic₁₇₈	—	2	2	no	—
67	C₂×Dic₁₇₈	—	4	1	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×Dic₁₇₈	—	4	1	no	—
70	C₂×Dic₁₇₈	—	4	1	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×D₃₅₆	—	2	2	no	—
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×D₃₅₆	—	2	2	no	—
75	C₂×D₃₅₆	—	4	1	no	—
76	C₂×D₃₅₆	—	4	1	no	—
77	C₂×D₃₅₆	—	4	2	no	—
78	C₂×D₃₅₆	—	4	2	no	—
79	C₂×D₃₅₆	—	4	1	no	—
80	C₂×D₃₅₆	—	4	1	no	—
81	C₂×D₃₅₆	—	4	1	no	—
82	C₂×D₃₅₆	—	4	1	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×D₃₅₆	—	8	2	no	—
85	C₂²×Dic₈₉	—	4	1	no	—
86	#13083563	—	4	8	no	—
87	#13083563	—	4	8	no	—
88	C₂×#10623494	—	4	1	no	—
89	C₂×#10623494	—	4	1	no	—
90	C₂×#10623494	—	4	2	no	—
91	C₂×#10623494	—	4	2	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	4	1	no	—
95	C₄⋊Dic₈₉	—	4	1	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	#41	1	1	yes	characteristic
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₈₉	D₈	1	2	yes	—
108	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	SD₁₆	1	4	yes	complemented
111	C₄²×C₈₉	—	4	1	no	—
112	C₄²×C₈₉	—	4	1	no	—
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₈₉	#41	1	1	yes	characteristic
117	C₂³×C₈₉	#121	1	1	yes	characteristic
118	C₂²×D₈₉	—	4	2	no	—
119	C₂²×D₈₉	—	8	1	no	—
120	C₂²×D₈₉	—	8	1	no	—
121	C₂²×D₈₉	—	8	2	no	—
122	C₂²×D₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	4	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₈₉	#135	1	2	yes	—
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	8	1	no	—
141	C₂×Dic₈₉	—	8	1	no	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	4	2	no	—
144	D₃₅₆	—	4	4	no	—
145	D₃₅₆	—	8	2	no	—
146	D₃₅₆	—	8	2	no	—
147	D₃₅₆	—	8	1	no	—
148	D₃₅₆	—	8	1	no	—
149	D₃₅₆	—	8	1	no	—
150	D₃₅₆	—	8	1	no	—
151	Dic₁₇₈	—	4	2	no	—
152	Dic₁₇₈	—	4	2	no	—
153	Dic₁₇₈	—	8	1	no	—
154	Dic₁₇₈	—	8	1	no	—
155	C₈×C₈₉	—	2	2	no	—
156	C₈×C₈₉	—	4	2	no	—
157	C₈×C₈₉	#102	1	4	yes	complemented
158	#10623494	—	8	1	no	—
159	#10623494	—	8	1	no	—
160	C₂²×C₈₉	#390	1	1	yes	characteristic
161	C₂²×C₈₉	#459	1	1	yes	characteristic
162	C₂²×C₈₉	#460	1	1	yes	characteristic
163	C₂²×C₈₉	#465	1	2	yes	—
164	C₂²×C₈₉	#471	1	2	yes	—
165	C₂×D₈₉	—	8	2	no	—
166	C₂×D₈₉	—	8	2	no	—
167	C₂×D₈₉	—	8	2	no	—
168	C₂×D₈₉	—	16	2	no	—
169	C₂×D₈₉	—	16	2	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	4	4	no	—
173	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
174	C₄×C₈₉	#418	1	1	yes	characteristic
175	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
176	C₄×C₈₉	#459	1	1	yes	characteristic
177	C₄×C₈₉	#460	1	1	yes	characteristic
178	C₄×C₈₉	#492	1	1	yes	characteristic
179	C₄×C₈₉	#495	1	1	yes	characteristic
180	Dic₈₉	—	8	2	no	—
181	#9732	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	#1665	1	2	yes	—
183	C₂×C₈₉	#1666	1	2	yes	—
184	C₂×C₈₉	#1968	1	1	yes	characteristic
185	C₂×C₈₉	#1977	1	1	yes	characteristic
186	C₂×C₈₉	#1982	1	1	yes	characteristic
187	D₈₉	—	16	2	no	—
188	C₂×#492	—	89	1	no	—
189	#1988	—	89	1	no	—
190	#1982	—	89	1	no	—
191	C₈×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
192	C₂×#334	—	89	1	no	—
193	C₂×#332	—	89	1	no	—
194	C₂×#331	—	89	1	no	—
195	C₈₉	#9732	1	1	yes	characteristic, complemented, Sylow
196	C₂²×SD₁₆	—	178	1	no	—
197	C₂²×SD₁₆	—	178	1	no	—
198	C₂×C₄⋊Q₈	—	89	1	no	—
199	C₂×#127	—	89	1	no	—
200	#492	—	89	4	no	—
201	#492	—	178	2	no	—
202	C₂×C₄⋊C₈	—	89	1	no	—
203	C₂×C₄⋊C₈	—	89	1	no	—
204	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#102	—	178	2	no	—
206	C₂×C₄×C₈	—	2	1	no	—
207	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
208	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
209	#340	—	178	2	no	—
210	#336	—	2	1	no	—
211	#334	—	178	2	no	—
212	#332	—	178	2	no	—
213	#331	—	178	2	no	—
214	C₂²×Q₈	—	178	1	no	—
215	C₂²×D₄	—	178	1	no	—
216	C₂²×D₄	—	178	1	no	—
217	C₂²×D₄	—	178	1	no	—
218	C₂×SD₁₆	—	356	2	no	—
219	C₂×SD₁₆	—	356	2	no	—
220	C₂×SD₁₆	—	356	2	no	—
221	C₂×SD₁₆	—	356	2	no	—
222	C₂×SD₁₆	—	178	4	no	—
223	C₂×SD₁₆	—	178	4	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	178	1	no	—
226	C₂²×C₈	—	178	1	no	—
227	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	89	2	no	—
230	C₄⋊Q₈	—	178	1	no	—
231	C₄⋊Q₈	—	178	1	no	—
232	C₄⋊Q₈	—	178	1	no	—
233	#127	—	89	2	no	—
234	#127	—	178	1	no	—
235	#127	—	178	1	no	—
236	#127	—	178	1	no	—
237	C₂×C₄⋊C₄	—	178	1	no	—
238	C₂×C₄⋊C₄	—	178	1	no	—
239	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	178	1	no	—
244	C₄⋊C₈	—	178	1	no	—
245	C₄⋊C₈	—	178	1	no	—
246	C₄⋊C₈	—	178	1	no	—
247	C₄⋊C₈	D₃₅₆	1	2	yes	—
248	#102	—	356	4	no	—
249	C₄×C₈	—	4	2	no	—
250	C₄×C₈	C₄×D₈₉	1	2	yes	—
251	C₄×C₈	D₃₅₆	1	2	yes	—
252	#95	—	2	1	no	—
253	C₂⁴	—	356	1	no	—
254	C₂×Q₈	—	356	1	no	—
255	C₂×Q₈	—	356	1	no	—
256	C₂×Q₈	—	356	1	no	—
257	C₂×Q₈	—	356	1	no	—
258	C₂×Q₈	—	356	1	no	—
259	C₂×Q₈	—	178	2	no	—
260	C₂×D₄	—	356	1	no	—
261	C₂×D₄	—	356	1	no	—
262	C₂×D₄	—	356	1	no	—
263	C₂×D₄	—	356	1	no	—
264	C₂×D₄	—	356	1	no	—
265	C₂×D₄	—	356	2	no	—
266	C₂×D₄	—	356	2	no	—
267	C₂×D₄	—	356	1	no	—
268	C₂×D₄	—	356	1	no	—
269	C₂×D₄	—	178	2	no	—
270	C₂×D₄	—	178	2	no	—
271	C₂×D₄	—	178	2	no	—
272	C₂×D₄	—	712	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	356	1	no	—
275	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
277	C₂²×C₄	#13083574	1	1	yes	characteristic
278	SD₁₆	—	356	8	no	—
279	SD₁₆	—	356	8	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	356	1	no	—
286	C₂×C₈	—	356	1	no	—
287	C₂×C₈	—	356	2	no	—
288	C₂×C₈	—	356	2	no	—
289	C₂×C₈	#13083551	1	2	yes	—
290	C₂×C₈	#13083551	1	2	yes	—
291	C₂×C₈	#13083559	1	2	yes	—
292	C₂×C₈	#13083561	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	356	1	no	—
295	C₄⋊C₄	—	356	1	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	#13083563	1	4	yes	complemented
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
302	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
303	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
304	C₄²	#13083574	1	1	yes	characteristic
305	C₂³	—	356	2	no	—
306	C₂³	—	712	1	no	—
307	C₂³	—	712	1	no	—
308	C₂³	—	712	2	no	—
309	C₂³	—	712	2	no	—
310	C₂³	#15667986	1	1	yes	characteristic
311	Q₈	—	356	2	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	712	1	no	—
314	Q₈	—	712	1	no	—
315	D₄	—	356	4	no	—
316	D₄	—	356	2	no	—
317	D₄	—	356	2	no	—
318	D₄	—	712	2	no	—
319	D₄	—	712	2	no	—
320	D₄	—	712	1	no	—
321	D₄	—	712	1	no	—
322	D₄	—	712	1	no	—
323	D₄	—	712	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	712	1	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
332	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
333	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
334	C₂×C₄	#15667986	1	1	yes	characteristic
335	C₂×C₄	#15668000	1	1	yes	characteristic
336	C₂×C₄	#15668009	1	1	yes	characteristic
337	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
338	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
339	C₂×C₄	C₂×#13083563	1	2	yes	—
340	C₂×C₄	#15668034	1	2	yes	—
341	C₂×C₄	#15668037	1	1	yes	characteristic
342	C₂×C₄	#15668037	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	712	1	no	—
346	C₈	—	712	1	no	—
347	C₈	#15667932	1	4	yes	complemented
348	C₂²	—	1424	2	no	—
349	C₂²	—	1424	2	no	—
350	C₂²	—	712	2	no	—
351	C₂²	—	712	2	no	—
352	C₂²	—	712	2	no	—
353	C₂²	#19151500	1	1	yes	characteristic
354	C₂²	#19151793	1	1	yes	characteristic
355	C₂²	#19151798	1	1	yes	characteristic
356	C₂²	#19151827	1	2	yes	—
357	C₂²	#19151861	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	4	no	—
361	C₄	—	712	2	no	—
362	C₄	C₂×#15667932	1	1	yes	characteristic
363	C₄	#19151602	1	1	yes	characteristic
364	C₄	C₄×#13083563	1	2	yes	—
365	C₄	#19151793	1	1	yes	characteristic
366	C₄	#19151798	1	1	yes	characteristic
367	C₄	#19151971	1	1	yes	characteristic
368	C₄	#19151982	1	1	yes	characteristic
369	C₂	—	1424	2	no	—
370	C₂	#26019593	1	2	yes	—
371	C₂	#26019600	1	2	yes	—
372	C₂	#26021106	1	1	yes	characteristic
373	C₂	#26021155	1	1	yes	characteristic
374	C₂	#26021179	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×C₄⋊C₄×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂×#15668009	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#9732
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #26021204	C₂	semidirect
4. C₂×#19151349	C₂	semidirect
5. C₂×#19151345	C₂	semidirect
7. C₈×C₄⋊C₄×C₈₉	C₂	semidirect
13. #19151971	C₄	semidirect
110. C₄⋊C₄×C₈₉	SD₁₆	semidirect
157. C₈×C₈₉	#102	semidirect
191. C₈×C₄⋊C₄	D₈₉	semidirect
195. C₈₉	#9732	semidirect
298. C₄⋊C₄	#13083563	semidirect
347. C₈	#15667932	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151971	2
#26021204	2
#26021179	2
C₂×#19151349	2
C₂×#19151345	2
C₂×#19151341	2
C₈×C₄⋊C₄×C₈₉	2
#9732	89