Subgroups — #36711053

Order 22784.

664 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152037	C₂	1	2	yes	maximal, complemented
2	Q₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26031973	C₂	1	1	yes	maximal, characteristic
4	#26022505	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022361	C₂	1	2	yes	maximal, complemented
6	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022297	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022193	C₂	1	2	yes	maximal, complemented
9	#26022187	C₂	1	2	yes	maximal, complemented
10	#26022085	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668031	C₂²	1	2	yes	—
13	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
14	Q₈×D₃₅₆	—	2	1	no	—
15	#19152252	—	2	1	no	—
16	#19152037	—	2	8	no	—
17	C₂×Dic₇₁₂	C₂²	1	2	yes	—
18	C₂×Dic₇₁₂	C₂²	1	2	yes	—
19	C₂×#15667958	C₂²	1	2	yes	—
20	C₂×#15667956	C₂²	1	2	yes	—
21	C₂×#15667951	C₂²	1	2	yes	—
22	#19151978	C₂²	1	1	yes	characteristic
23	#19151978	C₂²	1	1	yes	characteristic
24	#19151840	C₂²	1	2	yes	—
25	#19151840	C₂²	1	2	yes	—
26	#19151808	—	2	1	no	—
27	#19151803	—	2	1	no	—
28	#19151761	—	2	2	no	—
29	#19151755	—	2	2	no	—
30	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
31	C₄⋊#13083549	C₂²	1	1	yes	characteristic
32	#19151401	C₂²	1	2	yes	—
33	#19151401	C₂²	1	2	yes	—
34	#19151377	C₂²	1	2	yes	—
35	C₂×SD₃₂×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
38	#367×C₈₉	C₂²	1	1	yes	characteristic
39	#358×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	1	yes	characteristic
41	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#13083573	—	2	2	no	—
43	C₂×Q₈×D₈₉	—	2	2	no	—
44	#15668031	—	2	4	no	—
45	#15668031	—	4	2	no	—
46	C₂×Dic₃₅₆	—	2	1	no	—
47	C₂×Dic₃₅₆	—	2	1	no	—
48	C₂×Dic₃₅₆	C₂³	1	2	yes	—
49	C₂×Dic₃₅₆	C₂³	1	2	yes	—
50	C₂×#13083564	—	2	2	no	—
51	Q₁₆×D₈₉	—	2	4	no	—
52	Q₁₆×D₈₉	—	4	2	no	—
53	C₂×#13083562	—	2	2	no	—
54	C₂×C₈×D₈₉	C₂³	1	2	yes	—
55	#15668012	—	2	1	no	—
56	#15668012	—	2	1	no	—
57	#15668012	—	4	1	no	—
58	#15667996	—	4	2	no	—
59	#15667992	—	2	2	no	—
60	#15667992	—	2	2	no	—
61	#15667992	—	4	2	no	—
62	#15667986	—	4	1	no	—
63	C₄×D₃₅₆	—	4	1	no	—
64	C₄×D₃₅₆	D₄	1	1	yes	characteristic
65	C₂×C₄⋊Dic₈₉	—	4	1	no	—
66	Dic₇₁₂	—	2	4	no	—
67	Dic₇₁₂	—	4	2	no	—
68	#15667958	—	2	4	no	—
69	#15667956	—	2	4	no	—
70	#15667951	—	2	4	no	—
71	C₂×#13083549	C₂³	1	2	yes	—
72	C₄⋊#10623494	D₄	1	1	yes	characteristic
73	#15667935	—	2	2	no	—
74	#15667935	—	2	2	no	—
75	#15667924	D₄	1	2	yes	—
76	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₈₉	—	2	1	no	—
79	C₄×D₄×C₈₉	—	2	1	no	—
80	SD₃₂×C₈₉	D₄	1	4	yes	complemented
81	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
83	#107×C₈₉	C₂³	1	1	yes	characteristic
84	#107×C₈₉	C₂³	1	1	yes	characteristic
85	#103×C₈₉	—	2	1	no	—
86	#102×C₈₉	—	2	1	no	—
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	#13083573	—	4	4	no	—
89	#13083573	—	8	2	no	—
90	C₂×Dic₁₇₈	—	2	2	no	—
91	C₂×Dic₁₇₈	—	2	2	no	—
92	C₂×Dic₁₇₈	—	4	2	no	—
93	Q₈×D₈₉	—	4	4	no	—
94	Q₈×D₈₉	—	8	2	no	—
95	C₂×D₃₅₆	D₈	1	1	yes	characteristic
96	C₂×C₂²⋊D₈₉	—	4	2	no	—
97	C₂²×Dic₈₉	—	4	2	no	—
98	C₂×C₄×D₈₉	—	4	2	no	—
99	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
100	Dic₃₅₆	—	2	2	no	—
101	Dic₃₅₆	—	2	2	no	—
102	Dic₃₅₆	—	2	2	no	—
103	Dic₃₅₆	—	2	2	no	—
104	Dic₃₅₆	—	8	1	no	—
105	Dic₃₅₆	—	8	1	no	—
106	#13083564	—	4	4	no	—
107	#13083562	—	4	4	no	—
108	C₂×#10623494	C₂×D₄	1	2	yes	—
109	C₈×D₈₉	—	2	2	no	—
110	C₈×D₈₉	—	2	2	no	—
111	C₄⋊Dic₈₉	—	4	1	no	—
112	C₄⋊Dic₈₉	—	4	1	no	—
113	C₄⋊Dic₈₉	—	4	1	no	—
114	C₄⋊Dic₈₉	—	4	1	no	—
115	C₄⋊Dic₈₉	—	4	2	no	—
116	C₄⋊Dic₈₉	—	4	2	no	—
117	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
118	#13083551	—	4	2	no	—
119	#13083551	—	4	2	no	—
120	#13083551	D₈	1	2	yes	—
121	#13083549	—	2	2	no	—
122	C₂×Q₈×C₈₉	—	2	1	no	—
123	C₂×D₄×C₈₉	—	2	1	no	—
124	C₂²×C₄×C₈₉	—	4	1	no	—
125	Q₁₆×C₈₉	—	2	1	no	—
126	Q₁₆×C₈₉	C₂×D₄	1	2	yes	—
127	D₈×C₈₉	—	2	1	no	—
128	D₈×C₈₉	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₈₉	—	2	1	no	—
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	4	1	no	—
134	C₂²⋊C₄×C₈₉	—	4	1	no	—
135	C₄²×C₈₉	—	4	1	no	—
136	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₁₆×C₈₉	—	2	1	no	—
138	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
139	C₂³×C₈₉	—	4	1	no	—
140	C₂²×D₈₉	C₂×D₈	1	2	yes	—
141	Q₈×C₈₉	—	2	2	no	—
142	Q₈×C₈₉	—	4	1	no	—
143	D₄×C₈₉	—	2	2	no	—
144	D₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	8	2	no	—
149	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂²⋊D₈₉	—	8	4	no	—
153	C₂×Dic₈₉	—	4	2	no	—
154	C₂×Dic₈₉	—	4	2	no	—
155	C₂×Dic₈₉	—	8	4	no	—
156	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
157	D₃₅₆	—	2	2	no	—
158	C₄×D₈₉	—	2	2	no	—
159	C₄×D₈₉	—	2	2	no	—
160	C₄×D₈₉	—	8	4	no	—
161	Dic₁₇₈	—	4	2	no	—
162	Dic₁₇₈	—	4	2	no	—
163	Dic₁₇₈	—	4	2	no	—
164	Dic₁₇₈	—	4	2	no	—
165	Dic₁₇₈	—	8	4	no	—
166	C₈×C₈₉	—	2	1	no	—
167	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
168	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
169	#10623494	—	2	2	no	—
170	C₂²×C₈₉	—	4	2	no	—
171	C₂²×C₈₉	—	8	1	no	—
172	C₂²×C₈₉	—	8	1	no	—
173	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₈₉	—	2	2	no	—
175	C₂×D₈₉	—	2	2	no	—
176	C₂×D₈₉	—	2	2	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	—	8	1	no	—
180	C₄×C₈₉	—	8	1	no	—
181	C₄×C₈₉	#510	1	2	yes	complemented
182	C₄×C₈₉	D₄²	1	1	yes	characteristic
183	C₄×C₈₉	#576	1	1	yes	characteristic
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	8	2	no	—
186	Dic₈₉	—	8	2	no	—
187	#33842	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	8	2	no	—
189	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
190	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
191	C₂×C₈₉	#3417	1	1	yes	characteristic
192	D₈₉	—	4	2	no	—
193	C₂×#510	—	89	2	no	—
194	D₄×Q₁₆	—	89	1	no	—
195	#3283	—	89	1	no	—
196	#2247	—	89	1	no	—
197	#2219	—	89	2	no	—
198	#2209	—	89	1	no	—
199	#2208	—	89	1	no	—
200	#2189	—	89	2	no	—
201	#2188	—	89	2	no	—
202	C₄×SD₃₂	D₈₉	1	1	yes	characteristic, complemented
203	#2169	—	89	1	no	—
204	C₈₉	#33842	1	1	yes	characteristic, complemented, Sylow
205	C₂×#135	—	89	2	no	—
206	C₂²×Q₁₆	—	89	2	no	—
207	D₄×Q₈	—	178	1	no	—
208	#547	—	178	1	no	—
209	#510	—	178	8	no	—
210	C₂×Q₃₂	—	89	2	no	—
211	C₂×Q₃₂	—	89	2	no	—
212	C₂×Q₃₂	—	89	2	no	—
213	C₂×SD₃₂	—	89	2	no	—
214	C₂×SD₃₂	C₂×D₈₉	1	1	yes	characteristic
215	C₂×M₃₂	—	89	2	no	—
216	#494	—	89	1	no	—
217	#494	—	89	1	no	—
218	#467	—	89	2	no	—
219	#467	—	89	2	no	—
220	#462	—	178	1	no	—
221	#461	—	178	1	no	—
222	#452	—	178	2	no	—
223	#451	—	178	2	no	—
224	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
225	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
226	C₈×D₄	—	89	1	no	—
227	#367	C₂×D₈₉	1	1	yes	characteristic
228	C₄⋊C₁₆	—	89	1	no	—
229	#358	—	89	2	no	—
230	#358	—	89	2	no	—
231	#358	C₂×D₈₉	1	1	yes	characteristic
232	#357	C₂×D₈₉	1	1	yes	characteristic
233	C₂²⋊C₁₆	—	89	2	no	—
234	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
235	C₂×#41	—	178	2	no	—
236	C₂²×Q₈	—	178	2	no	—
237	#135	—	356	2	no	—
238	#135	—	178	4	no	—
239	C₂×Q₁₆	—	89	2	no	—
240	C₂×Q₁₆	—	89	2	no	—
241	C₂×Q₁₆	—	356	2	no	—
242	C₂×Q₁₆	—	178	2	no	—
243	C₂×Q₁₆	—	178	4	no	—
244	C₂×Q₁₆	—	178	1	no	—
245	C₂×Q₁₆	—	178	1	no	—
246	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×SD₁₆	—	178	2	no	—
248	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂²×C₈	—	89	2	no	—
250	C₄⋊Q₈	—	356	1	no	—
251	C₄⋊Q₈	—	178	1	no	—
252	C₄⋊Q₈	—	178	1	no	—
253	#123	—	356	2	no	—
254	C₂²⋊Q₈	—	356	2	no	—
255	C₂²⋊Q₈	—	178	2	no	—
256	C₂²⋊Q₈	—	178	2	no	—
257	#121	—	356	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×D₄	—	2	1	no	—
260	C₄×D₄	—	89	1	no	—
261	C₄×D₄	—	356	1	no	—
262	C₂×C₄⋊C₄	—	356	1	no	—
263	Q₃₂	—	356	2	no	—
264	Q₃₂	—	178	4	no	—
265	Q₃₂	—	178	4	no	—
266	SD₃₂	—	178	4	no	—
267	SD₃₂	D₃₅₆	1	4	yes	complemented
268	M₃₂	—	178	4	no	—
269	C₂×C₁₆	—	89	2	no	—
270	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
271	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
272	#107	C₂²×D₈₉	1	1	yes	characteristic
273	#107	C₂²×D₈₉	1	1	yes	characteristic
274	C₄⋊C₈	—	89	1	no	—
275	#103	—	2	1	no	—
276	#103	—	178	2	no	—
277	#103	—	178	2	no	—
278	#102	—	2	1	no	—
279	C₂²⋊C₈	—	89	2	no	—
280	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	#41	—	356	4	no	—
282	#41	—	712	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	356	2	no	—
285	C₂×Q₈	—	356	4	no	—
286	C₂×Q₈	—	178	2	no	—
287	C₂×Q₈	—	178	2	no	—
288	C₂×Q₈	—	712	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	89	1	no	—
291	C₂×D₄	—	356	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	89	2	no	—
294	C₂²×C₄	—	356	2	no	—
295	C₂²×C₄	—	356	2	no	—
296	Q₁₆	—	2	1	no	—
297	Q₁₆	—	356	4	no	—
298	Q₁₆	—	178	2	no	—
299	Q₁₆	—	178	2	no	—
300	Q₁₆	—	178	2	no	—
301	Q₁₆	—	178	2	no	—
302	Q₁₆	—	712	1	no	—
303	Q₁₆	—	712	1	no	—
304	Q₁₆	C₂×D₃₅₆	1	2	yes	—
305	SD₁₆	—	356	4	no	—
306	D₈	—	2	1	no	—
307	D₈	C₂×D₃₅₆	1	2	yes	—
308	C₂×C₈	—	89	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	2	no	—
311	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	89	1	no	—
317	C₄⋊C₄	—	356	2	no	—
318	C₄⋊C₄	—	356	2	no	—
319	C₄⋊C₄	—	356	1	no	—
320	C₄⋊C₄	—	356	1	no	—
321	C₄⋊C₄	—	356	1	no	—
322	C₄⋊C₄	—	356	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₂²⋊C₄	—	89	2	no	—
325	C₂²⋊C₄	—	356	2	no	—
326	C₂²⋊C₄	—	356	2	no	—
327	C₄²	—	4	1	no	—
328	C₄²	D₄×D₈₉	1	1	yes	characteristic
329	C₁₆	—	2	1	no	—
330	C₁₆	—	178	2	no	—
331	C₁₆	C₂×D₃₅₆	1	2	yes	—
332	C₂³	—	4	1	no	—
333	C₂³	—	89	2	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	356	2	no	—
338	Q₈	—	356	2	no	—
339	Q₈	—	356	2	no	—
340	Q₈	—	712	4	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	1	no	—
343	D₄	—	178	2	no	—
344	D₄	—	712	4	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	89	2	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	178	2	no	—
353	C₂×C₄	—	178	2	no	—
354	C₂×C₄	—	712	4	no	—
355	C₂×C₄	—	712	4	no	—
356	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
357	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	178	2	no	—
361	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
362	C₈	#15668058	1	1	yes	characteristic
363	C₂²	—	4	2	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	178	2	no	—
367	C₂²	—	178	2	no	—
368	C₂²	—	178	2	no	—
369	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	178	2	no	—
375	C₄	—	712	2	no	—
376	C₄	—	712	2	no	—
377	C₄	#19152037	1	2	yes	complemented
378	C₄	D₄×D₃₅₆	1	1	yes	characteristic
379	C₄	#19152400	1	1	yes	characteristic
380	C₂	—	8	2	no	—
381	C₂	—	356	2	no	—
382	C₂	D₈×D₃₅₆	1	1	yes	characteristic
383	C₂	C₂×#19152037	1	1	yes	characteristic
384	C₂	#26033402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×SD₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26031973	C₂
hypercenter	C₄×SD₃₂	D₈₉
hypocenter	C₈₉	#33842
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#19152037	C₂	semidirect
2. Q₁₆×D₃₅₆	C₂	semidirect
4. #26022505	C₂	semidirect
5. #26022361	C₂	semidirect
6. #26022302	C₂	semidirect
7. #26022297	C₂	semidirect
8. #26022193	C₂	semidirect
9. #26022187	C₂	semidirect
10. #26022085	C₂	semidirect
11. C₄×SD₃₂×C₈₉	C₂	semidirect
80. SD₃₂×C₈₉	D₄	semidirect
181. C₄×C₈₉	#510	semidirect
202. C₄×SD₃₂	D₈₉	semidirect
204. C₈₉	#33842	semidirect
267. SD₃₂	D₃₅₆	semidirect
377. C₄	#19152037	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152037	2
Q₁₆×D₃₅₆	2
#26031973	2
#26022505	2
#26022361	2
#26022302	2
#26022297	2
#26022193	2
#26022187	2
#26022085	2
C₄×SD₃₂×C₈₉	2
#33842	89