Subgroups — #36524572

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030261	C₂	1	1	yes	maximal, characteristic
2	#26029346	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029275	C₂	1	1	yes	maximal, characteristic
4	#26019111	C₂	1	2	yes	maximal, complemented
5	#26019090	C₂	1	2	yes	maximal, complemented
6	#26019066	C₂	1	2	yes	maximal, complemented
7	C₈×#13083563	C₂	1	2	yes	maximal, complemented
8	#26019032	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018696	C₂	1	1	yes	maximal, characteristic
10	#26018674	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152155	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
14	#19151929	C₄	1	2	yes	—
15	#19151889	C₄	1	2	yes	—
16	#19151804	C₄	1	2	yes	—
17	#19151751	C₄	1	2	yes	—
18	#19151728	C₂²	1	2	yes	complemented
19	#19151727	—	2	2	no	—
20	C₈×Dic₁₇₈	C₂²	1	2	yes	complemented
21	C₄×#13083563	C₂²	1	2	yes	—
22	C₄×#13083563	C₂²	1	2	yes	—
23	C₈×D₃₅₆	C₂²	1	2	yes	—
24	C₈×D₃₅₆	C₂²	1	2	yes	—
25	#19151663	—	2	2	no	—
26	#19151656	—	2	2	no	—
27	#19151655	C₂²	1	1	yes	characteristic
28	#19151651	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×D₈₉	—	2	2	no	—
30	#19151351	C₂²	1	2	yes	complemented
31	#19151349	C₂²	1	2	yes	complemented
32	#19151330	C₂²	1	2	yes	complemented
33	#19151330	C₂²	1	2	yes	complemented
34	#19151327	C₂²	1	2	yes	—
35	#19151327	C₂²	1	2	yes	—
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	#431×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	#322×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	#322×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	#322×C₈₉	C₂²	1	1	yes	characteristic, complemented
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₈₉	—	2	1	no	—
45	C₂×#13083563	C₂×C₄	1	2	yes	—
46	C₂×#13083563	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₈₉	—	2	2	no	—
48	C₂×C₈×D₈₉	—	2	2	no	—
49	#15668012	C₂×C₄	1	1	yes	characteristic
50	#15668012	C₂×C₄	1	1	yes	characteristic
51	#15668004	—	2	1	no	—
52	#15667991	—	2	1	no	—
53	#15667991	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₇₈	—	2	1	no	—
55	C₄×Dic₁₇₈	C₂³	1	2	yes	—
56	C₄×D₃₅₆	—	2	1	no	—
57	C₄×D₃₅₆	—	2	1	no	—
58	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
59	C₄×D₃₅₆	C₂³	1	2	yes	—
60	#15667971	—	2	1	no	—
61	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×#13083551	—	2	1	no	—
63	#15667944	C₂×C₄	1	2	yes	—
64	#15667944	C₂×C₄	1	2	yes	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	C₄⋊#10623494	C₂³	1	2	yes	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	2	no	—
69	#15667937	C₂×C₄	1	2	yes	—
70	#15667937	C₂×C₄	1	2	yes	—
71	#15667934	C₂×C₄	1	2	yes	—
72	#15667934	C₂×C₄	1	2	yes	—
73	C₂²⋊#10623494	—	2	1	no	—
74	C₂²⋊#10623494	—	2	1	no	—
75	#15667924	—	2	2	no	—
76	#15667924	—	2	2	no	—
77	#15667923	—	2	1	no	—
78	#15667922	—	2	2	no	—
79	C₄×#10623494	—	2	1	no	—
80	C₈×Dic₈₉	—	2	2	no	—
81	C₂×M₁₆×C₈₉	—	2	1	no	—
82	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
85	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
87	#97×C₈₉	—	2	1	no	—
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂³×D₈₉	—	2	1	no	—
95	C₂×Dic₁₇₈	—	2	1	no	—
96	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
97	C₂×D₃₅₆	—	2	1	no	—
98	C₂×D₃₅₆	—	2	4	no	—
99	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
100	C₂²×Dic₈₉	—	2	1	no	—
101	C₂×C₄×D₈₉	—	2	2	no	—
102	C₂×C₄×D₈₉	—	2	2	no	—
103	C₂×C₄×D₈₉	—	8	2	no	—
104	#13083563	—	2	4	no	—
105	#13083563	—	2	4	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	2	no	—
108	C₈×D₈₉	—	8	2	no	—
109	C₈×D₈₉	—	8	2	no	—
110	C₄⋊Dic₈₉	—	2	1	no	—
111	C₄⋊Dic₈₉	—	2	1	no	—
112	C₄⋊Dic₈₉	—	2	1	no	—
113	C₄⋊Dic₈₉	—	2	1	no	—
114	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
116	#13083553	—	2	2	no	—
117	#13083553	—	2	2	no	—
118	C₂²⋊Dic₈₉	—	2	2	no	—
119	#13083551	—	2	2	no	—
120	#13083551	—	2	2	no	—
121	C₄×Dic₈₉	—	2	2	no	—
122	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
125	M₁₆×C₈₉	—	4	2	no	—
126	C₂×C₈×C₈₉	—	2	1	no	—
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
132	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	#41	1	1	yes	characteristic
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₈₉	—	2	2	no	—
138	C₂²×D₈₉	—	2	4	no	—
139	C₂²×D₈₉	—	8	1	no	—
140	C₂²×D₈₉	—	8	1	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	8	1	no	—
154	D₃₅₆	—	2	4	no	—
155	D₃₅₆	—	2	2	no	—
156	D₃₅₆	#131	1	4	yes	—
157	C₄×D₈₉	—	4	2	no	—
158	C₄×D₈₉	—	4	2	no	—
159	C₄×D₈₉	—	8	2	no	—
160	Dic₁₇₈	—	2	2	no	—
161	Dic₁₇₈	—	2	2	no	—
162	Dic₁₇₈	—	8	1	no	—
163	C₈×C₈₉	—	2	2	no	—
164	C₈×C₈₉	—	2	2	no	—
165	C₈×C₈₉	—	2	2	no	—
166	C₈×C₈₉	—	2	2	no	—
167	C₈×C₈₉	—	2	2	no	—
168	C₈×C₈₉	—	4	1	no	—
169	#10623494	—	4	2	no	—
170	#10623494	—	4	2	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	—	2	1	no	—
174	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	2	4	no	—
176	C₂×D₈₉	—	4	2	no	—
177	C₂×D₈₉	—	4	2	no	—
178	C₂×D₈₉	—	8	4	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	C₂×#131	1	2	yes	—
184	C₄×C₈₉	#577	1	2	yes	—
185	Dic₈₉	—	4	2	no	—
186	Dic₈₉	—	4	2	no	—
187	Dic₈₉	—	4	2	no	—
188	#17924	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	—	4	1	no	—
190	C₂×C₈₉	#2929	1	1	yes	characteristic
191	C₂×C₈₉	#2949	1	1	yes	characteristic
192	C₂×C₈₉	#2956	1	1	yes	characteristic
193	D₈₉	—	4	4	no	—
194	#3101	—	89	1	no	—
195	#2985	—	89	1	no	—
196	#2972	—	89	1	no	—
197	#1585	—	89	2	no	—
198	#1582	—	89	2	no	—
199	SD₁₆⋊C₈	—	89	2	no	—
200	C₈×SD₁₆	—	89	2	no	—
201	#1572	—	89	1	no	—
202	#1490	—	89	1	no	—
203	#1485	—	89	1	no	—
204	#1453	D₈₉	1	1	yes	characteristic, complemented
205	C₈₉	#17924	1	1	yes	characteristic, complemented, Sylow
206	#533	—	89	1	no	—
207	C₂×C₄×D₄	—	89	1	no	—
208	#483	—	89	2	no	—
209	#476	—	89	2	no	—
210	#461	—	89	2	no	—
211	#450	—	89	2	no	—
212	#446	—	89	2	no	—
213	#446	—	178	2	no	—
214	C₈×Q₈	—	89	2	no	—
215	C₄×SD₁₆	—	89	2	no	—
216	C₄×SD₁₆	—	89	2	no	—
217	C₈×D₄	—	89	2	no	—
218	C₈×D₄	—	89	2	no	—
219	C₈×D₄	—	178	2	no	—
220	#433	—	178	2	no	—
221	#432	—	89	1	no	—
222	#431	—	89	1	no	—
223	#431	C₂×D₈₉	1	1	yes	characteristic
224	#431	C₂×D₈₉	1	1	yes	characteristic
225	C₂×C₂²⋊C₈	—	178	2	no	—
226	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
227	#335	—	89	2	no	—
228	#334	—	89	2	no	—
229	Q₈⋊C₈	—	89	2	no	—
230	Q₈⋊C₈	—	89	2	no	—
231	D₄⋊C₈	—	89	2	no	—
232	D₄⋊C₈	—	89	2	no	—
233	#322	C₂×D₈₉	1	1	yes	characteristic, complemented
234	#322	C₂×D₈₉	1	1	yes	characteristic, complemented
235	#322	C₂×D₈₉	1	1	yes	characteristic, complemented
236	C₈²	C₂×D₈₉	1	1	yes	characteristic, complemented
237	C₂²×D₄	—	89	1	no	—
238	C₂³×C₄	—	178	1	no	—
239	C₂×SD₁₆	—	89	2	no	—
240	C₂×SD₁₆	—	89	2	no	—
241	C₂×M₁₆	—	2	1	no	—
242	C₂²×C₈	—	178	2	no	—
243	C₂²×C₈	—	178	2	no	—
244	C₄⋊Q₈	—	89	1	no	—
245	C₄⋊Q₈	—	89	1	no	—
246	#125	—	178	1	no	—
247	C₂²⋊Q₈	—	89	2	no	—
248	C₂²⋊Q₈	—	178	1	no	—
249	C₄×Q₈	—	89	2	no	—
250	C₄×Q₈	—	178	1	no	—
251	C₄×D₄	—	89	2	no	—
252	C₄×D₄	—	89	2	no	—
253	C₄×D₄	—	178	1	no	—
254	C₄×D₄	—	178	1	no	—
255	#117	—	178	1	no	—
256	C₂×C₄⋊C₄	—	89	1	no	—
257	C₂×C₂²⋊C₄	—	178	1	no	—
258	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	#106	—	89	2	no	—
260	#106	—	89	2	no	—
261	C₄⋊C₈	—	89	2	no	—
262	C₄⋊C₈	—	89	2	no	—
263	C₄⋊C₈	—	178	2	no	—
264	C₄⋊C₈	—	178	2	no	—
265	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
266	C₄⋊C₈	D₃₅₆	1	2	yes	—
267	#103	—	89	2	no	—
268	#103	—	89	2	no	—
269	#102	—	89	2	no	—
270	#102	—	89	2	no	—
271	C₂²⋊C₈	—	178	1	no	—
272	C₂²⋊C₈	—	178	1	no	—
273	C₂²⋊C₈	—	178	2	no	—
274	C₂²⋊C₈	—	178	2	no	—
275	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
276	C₂²⋊C₈	D₃₅₆	1	2	yes	—
277	#97	—	2	1	no	—
278	#97	—	178	2	no	—
279	#97	—	178	1	no	—
280	C₄×C₈	—	178	2	no	—
281	C₄×C₈	—	178	1	no	—
282	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₂⁴	—	178	1	no	—
289	C₂×Q₈	—	89	2	no	—
290	C₂×Q₈	—	178	1	no	—
291	C₂×D₄	—	89	2	no	—
292	C₂×D₄	—	178	4	no	—
293	C₂×D₄	—	178	1	no	—
294	C₂²×C₄	—	178	2	no	—
295	C₂²×C₄	—	178	1	no	—
296	C₂²×C₄	—	178	2	no	—
297	C₂²×C₄	—	712	2	no	—
298	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
300	C₂²×C₄	#13083574	1	1	yes	characteristic
301	SD₁₆	—	178	4	no	—
302	SD₁₆	—	178	4	no	—
303	M₁₆	—	4	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	—	712	2	no	—
309	C₂×C₈	—	712	2	no	—
310	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
311	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
312	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
313	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
314	C₄⋊C₄	—	89	2	no	—
315	C₄⋊C₄	—	89	2	no	—
316	C₄⋊C₄	—	178	1	no	—
317	C₄⋊C₄	—	178	1	no	—
318	C₄⋊C₄	—	178	1	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	—	178	1	no	—
321	C₄⋊C₄	—	178	2	no	—
322	C₂²⋊C₄	—	178	2	no	—
323	C₂²⋊C₄	—	178	2	no	—
324	C₂²⋊C₄	—	178	2	no	—
325	C₄²	—	178	2	no	—
326	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
327	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
328	C₄²	#13083574	1	1	yes	characteristic
329	C₄²	C₂³×D₈₉	1	1	yes	characteristic
330	C₂³	—	178	4	no	—
331	C₂³	—	178	2	no	—
332	C₂³	—	712	1	no	—
333	C₂³	—	712	1	no	—
334	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
335	Q₈	—	178	2	no	—
336	Q₈	—	178	2	no	—
337	Q₈	—	712	1	no	—
338	D₄	—	89	4	no	—
339	D₄	—	178	4	no	—
340	D₄	—	178	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	356	2	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	178	2	no	—
347	C₂×C₄	—	178	2	no	—
348	C₂×C₄	—	178	2	no	—
349	C₂×C₄	—	712	2	no	—
350	C₂×C₄	—	712	1	no	—
351	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
352	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
353	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
356	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	1	no	—
363	C₈	—	356	2	no	—
364	C₈	—	356	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	356	2	no	—
369	C₂²	—	356	2	no	—
370	C₂²	—	178	4	no	—
371	C₂²	—	712	4	no	—
372	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	356	2	no	—
378	C₄	—	356	2	no	—
379	C₄	—	356	2	no	—
380	C₄	#131×D₈₉	1	2	yes	—
381	C₄	#19152410	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	356	4	no	—
384	C₂	#26028948	1	1	yes	characteristic
385	C₂	#26029113	1	1	yes	characteristic
386	C₂	#26029163	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1453	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1453	D₈₉
hypocenter	C₈₉	#17924
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
2. #26029346	C₂	semidirect
4. #26019111	C₂	semidirect
5. #26019090	C₂	semidirect
6. #26019066	C₂	semidirect
7. C₈×#13083563	C₂	semidirect
8. #26019032	C₂	semidirect
10. #26018674	C₂	semidirect
11. C₈₉×#1453	C₂	semidirect
18. #19151728	C₂²	semidirect
20. C₈×Dic₁₇₈	C₂²	semidirect
30. #19151351	C₂²	semidirect
31. #19151349	C₂²	semidirect
32. #19151330	C₂²	semidirect
33. #19151330	C₂²	semidirect
39. #322×C₈₉	C₂²	semidirect
40. #322×C₈₉	C₂²	semidirect
41. #322×C₈₉	C₂²	semidirect
42. C₈²×C₈₉	C₂²	semidirect
204. #1453	D₈₉	semidirect
205. C₈₉	#17924	semidirect
233. #322	C₂×D₈₉	semidirect
234. #322	C₂×D₈₉	semidirect
235. #322	C₂×D₈₉	semidirect
236. C₈²	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030261	2
#26029346	2
#26029275	2
#26019111	2
#26019090	2
#26019066	2
C₈×#13083563	2
#26019032	2
#26018696	2
#26018674	2
C₈₉×#1453	2
#17924	89