Subgroups — #36711263

Order 22784.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032645	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022438	C₂	1	2	yes	maximal, complemented
5	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022348	C₂	1	2	yes	maximal, complemented
7	#26022293	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022183	C₂	1	2	yes	maximal, complemented
9	C₄×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151400	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₁₂	C₂²	1	2	yes	—
13	#19152305	—	2	1	no	—
14	D₄×D₃₅₆	—	2	1	no	—
15	C₂×#15667957	C₂²	1	1	yes	characteristic
16	C₂×#15667957	C₂²	1	2	yes	complemented
17	#19152003	C₂²	1	1	yes	characteristic, complemented
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151934	—	2	2	no	—
20	#19151894	—	2	1	no	—
21	#19151839	C₂²	1	2	yes	complemented
22	#19151834	C₂²	1	2	yes	—
23	#19151794	—	2	1	no	—
24	#19151734	—	2	2	no	—
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic, complemented
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	C₂×#15667946	C₂²	1	2	yes	complemented
28	#19151426	C₂²	1	2	yes	complemented
29	#19151426	C₂²	1	1	yes	characteristic, complemented
30	#19151403	C₂²	1	1	yes	characteristic, complemented
31	#19151400	C₂²	1	2	yes	—
32	#19151400	C₂²	1	1	yes	characteristic
33	#19151400	C₂²	1	2	yes	complemented
34	#19151400	C₂²	1	2	yes	complemented
35	#19151400	C₂²	1	2	yes	complemented
36	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	complemented
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₅₆	—	2	2	no	—
42	C₂×D₄×D₈₉	—	4	1	no	—
43	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
44	C₂×D₇₁₂	—	2	4	no	—
45	C₂×D₇₁₂	—	2	2	no	—
46	C₂×D₇₁₂	—	2	1	no	—
47	C₂×D₇₁₂	—	4	2	no	—
48	C₂×D₇₁₂	C₂³	1	2	yes	—
49	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
50	#15668012	—	2	1	no	—
51	#15668009	—	2	1	no	—
52	#15668003	—	4	1	no	—
53	#15667991	—	2	2	no	—
54	#15667989	—	4	2	no	—
55	#15667987	—	4	1	no	—
56	#15667986	—	2	2	no	—
57	#15667982	—	4	2	no	—
58	C₄×Dic₁₇₈	—	2	1	no	—
59	C₄×D₃₅₆	—	2	1	no	—
60	D₄×Dic₈₉	—	4	1	no	—
61	C₂×C₄⋊Dic₈₉	—	2	2	no	—
62	#15667957	—	2	2	no	—
63	#15667957	—	4	2	no	—
64	#15667946	—	2	2	no	—
65	#15667946	—	2	1	no	—
66	#15667946	C₂³	1	1	yes	characteristic
67	#15667946	C₂³	1	2	yes	—
68	#15667946	C₂³	1	1	yes	characteristic
69	#15667937	—	2	2	no	—
70	#15667937	—	2	1	no	—
71	#15667934	—	2	1	no	—
72	#15667934	—	2	2	no	—
73	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
74	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₈₉	—	4	2	no	—
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
78	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×D₈₉	—	4	2	no	—
83	C₂×Dic₁₇₈	—	2	1	no	—
84	C₂×Dic₁₇₈	—	4	1	no	—
85	C₂×D₃₅₆	—	2	1	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	4	4	no	—
88	C₂×D₃₅₆	—	4	2	no	—
89	C₂×D₃₅₆	—	4	1	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	—	8	2	no	—
92	C₂×C₂²⋊D₈₉	—	4	2	no	—
93	D₄×D₈₉	—	8	4	no	—
94	C₂²×Dic₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	—	4	1	no	—
96	Dic₃₅₆	—	2	1	no	—
97	Dic₃₅₆	—	2	1	no	—
98	D₇₁₂	—	2	2	no	—
99	D₇₁₂	—	2	1	no	—
100	D₇₁₂	—	2	2	no	—
101	D₇₁₂	—	4	4	no	—
102	D₇₁₂	—	8	1	no	—
103	D₇₁₂	#41	1	2	yes	—
104	C₄⋊Dic₈₉	—	2	1	no	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	4	1	no	—
108	C₄⋊Dic₈₉	—	4	2	no	—
109	#13083553	—	4	1	no	—
110	#13083553	—	4	2	no	—
111	C₂²⋊Dic₈₉	—	4	2	no	—
112	#13083551	—	4	1	no	—
113	#13083551	—	4	2	no	—
114	C₄×Dic₈₉	—	4	1	no	—
115	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₈₉	—	2	1	no	—
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	2	no	—
128	C₂³×C₈₉	C₂×D₈	1	2	yes	—
129	C₂²×D₈₉	—	4	1	no	—
130	C₂²×D₈₉	—	4	2	no	—
131	C₂²×D₈₉	—	8	4	no	—
132	C₂²×D₈₉	—	8	4	no	—
133	C₂²×D₈₉	—	16	2	no	—
134	D₄×C₈₉	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	8	4	no	—
142	C₂×Dic₈₉	—	4	1	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	C₂×Dic₈₉	—	16	2	no	—
147	D₃₅₆	—	2	2	no	—
148	D₃₅₆	—	4	2	no	—
149	D₃₅₆	—	4	1	no	—
150	D₃₅₆	—	4	2	no	—
151	D₃₅₆	—	8	4	no	—
152	D₃₅₆	—	8	2	no	—
153	D₃₅₆	—	16	1	no	—
154	C₄×D₈₉	—	8	2	no	—
155	Dic₁₇₈	—	4	1	no	—
156	Dic₁₇₈	—	4	1	no	—
157	Dic₁₇₈	—	16	1	no	—
158	C₈×C₈₉	—	2	2	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₈₉	—	2	2	no	—
163	C₂²×C₈₉	—	2	2	no	—
164	C₂²×C₈₉	C₂×SD₃₂	1	4	yes	—
165	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₈₉	—	4	2	no	—
167	C₂×D₈₉	—	8	1	no	—
168	C₂×D₈₉	—	8	1	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₂×D₈₉	—	8	2	no	—
172	C₂×D₈₉	—	16	8	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	2	no	—
175	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
176	C₄×C₈₉	#546	1	1	yes	characteristic
177	C₄×C₈₉	#577	1	1	yes	characteristic
178	Dic₈₉	—	8	1	no	—
179	Dic₈₉	—	8	2	no	—
180	Dic₈₉	—	8	1	no	—
181	Dic₈₉	—	8	1	no	—
182	#33862	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	4	no	—
184	C₂×C₈₉	#3294	1	1	yes	characteristic
185	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
186	C₂×C₈₉	#3415	1	1	yes	characteristic
187	D₈₉	—	8	2	no	—
188	D₈₉	—	16	2	no	—
189	#3338	—	89	1	no	—
190	D₄×D₈	—	89	1	no	—
191	#2240	—	89	1	no	—
192	#2233	—	89	2	no	—
193	#2226	—	89	1	no	—
194	#2217	—	89	2	no	—
195	#2207	—	89	1	no	—
196	#2187	—	89	2	no	—
197	C₄×SD₃₂	—	89	1	no	—
198	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
199	C₂×#357	—	89	2	no	—
200	C₈₉	#33862	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₈	—	89	2	no	—
202	#554	—	178	1	no	—
203	D₄²	—	178	1	no	—
204	C₂×SD₃₂	—	89	2	no	—
205	C₂×SD₃₂	—	89	1	no	—
206	C₂²×C₁₆	C₂×D₈₉	1	2	yes	complemented
207	#500	—	89	1	no	—
208	#493	—	89	1	no	—
209	#484	—	178	2	no	—
210	#477	—	178	1	no	—
211	#467	—	89	2	no	—
212	#466	—	89	2	no	—
213	#459	—	178	1	no	—
214	#447	—	178	2	no	—
215	C₄×Q₁₆	—	89	1	no	—
216	C₄×D₈	—	89	1	no	—
217	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
218	C₂×#107	—	89	2	no	—
219	#367	—	89	1	no	—
220	#367	—	89	2	no	—
221	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
222	#358	—	89	1	no	—
223	#357	—	89	1	no	—
224	#357	—	89	2	no	—
225	#357	—	89	2	no	—
226	#357	—	89	2	no	—
227	#357	—	89	2	no	—
228	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
229	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
230	C₂²×D₄	—	356	1	no	—
231	C₂²×D₄	—	178	2	no	—
232	C₂×Q₁₆	—	89	1	no	—
233	C₂×D₈	—	89	1	no	—
234	C₂×D₈	—	89	2	no	—
235	C₂×D₈	—	356	2	no	—
236	C₂×D₈	—	178	4	no	—
237	C₂×D₈	—	178	2	no	—
238	C₂×D₈	—	178	1	no	—
239	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
240	C₄⋊Q₈	—	178	1	no	—
241	#127	—	178	1	no	—
242	#125	—	356	1	no	—
243	C₂²⋊Q₈	—	356	2	no	—
244	C₂²⋊Q₈	—	178	2	no	—
245	#121	—	356	1	no	—
246	#121	—	178	2	no	—
247	#120	—	356	2	no	—
248	C₄×Q₈	—	178	1	no	—
249	C₄×D₄	—	356	1	no	—
250	C₄×D₄	—	178	1	no	—
251	C₄×D₄	D₃₅₆	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	178	2	no	—
253	SD₃₂	—	356	2	no	—
254	SD₃₂	—	178	2	no	—
255	C₂×C₁₆	—	4	2	no	—
256	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
259	#107	—	89	1	no	—
260	#107	—	89	2	no	—
261	#107	—	89	1	no	—
262	#107	—	178	2	no	—
263	#107	—	178	1	no	—
264	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
265	#103	—	178	1	no	—
266	#103	—	178	2	no	—
267	#102	—	178	1	no	—
268	#102	—	178	2	no	—
269	C₂²⋊C₈	D₃₅₆	1	2	yes	—
270	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₂⁴	—	356	2	no	—
272	C₂×Q₈	—	356	1	no	—
273	C₂×Q₈	—	178	1	no	—
274	C₂×D₄	—	356	4	no	—
275	C₂×D₄	—	356	2	no	—
276	C₂×D₄	—	356	1	no	—
277	C₂×D₄	—	356	2	no	—
278	C₂×D₄	—	356	1	no	—
279	C₂×D₄	—	178	2	no	—
280	C₂×D₄	—	178	1	no	—
281	C₂×D₄	—	712	2	no	—
282	C₂×D₄	—	712	4	no	—
283	C₂×D₄	D₇₁₂	1	1	yes	characteristic
284	C₂²×C₄	—	356	2	no	—
285	C₂²×C₄	—	356	1	no	—
286	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
287	Q₁₆	—	178	1	no	—
288	Q₁₆	—	178	1	no	—
289	D₈	—	89	2	no	—
290	D₈	—	356	4	no	—
291	D₈	—	178	2	no	—
292	D₈	—	178	2	no	—
293	D₈	—	178	1	no	—
294	D₈	—	712	1	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
298	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
299	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	356	2	no	—
301	C₄⋊C₄	—	356	1	no	—
302	C₄⋊C₄	—	356	1	no	—
303	C₄⋊C₄	—	356	2	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
308	C₂²⋊C₄	—	356	2	no	—
309	C₂²⋊C₄	—	356	1	no	—
310	C₂²⋊C₄	—	356	2	no	—
311	C₂²⋊C₄	D₇₁₂	1	2	yes	—
312	C₄²	—	356	1	no	—
313	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	2	2	no	—
317	C₂³	—	356	1	no	—
318	C₂³	—	356	2	no	—
319	C₂³	—	1424	2	no	—
320	C₂³	—	712	4	no	—
321	C₂³	—	712	4	no	—
322	C₂³	C₂×D₇₁₂	1	2	yes	—
323	Q₈	—	356	1	no	—
324	Q₈	—	356	1	no	—
325	Q₈	—	1424	1	no	—
326	D₄	—	356	1	no	—
327	D₄	—	356	2	no	—
328	D₄	—	356	2	no	—
329	D₄	—	1424	1	no	—
330	D₄	—	178	2	no	—
331	D₄	—	712	4	no	—
332	D₄	—	712	4	no	—
333	D₄	—	712	2	no	—
334	D₄	#15667957	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	356	1	no	—
339	C₂×C₄	—	356	1	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	—	1424	2	no	—
342	C₂×C₄	—	712	2	no	—
343	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
344	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
345	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
346	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	2	no	—
349	C₈	#41×D₈₉	1	1	yes	characteristic
350	C₈	#15668057	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	356	2	no	—
354	C₂²	—	1424	8	no	—
355	C₂²	—	712	2	no	—
356	C₂²	—	712	2	no	—
357	C₂²	—	712	2	no	—
358	C₂²	—	712	1	no	—
359	C₂²	—	712	1	no	—
360	C₂²	C₂×#15667957	1	4	yes	—
361	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	1	no	—
364	C₄	—	712	1	no	—
365	C₄	—	712	1	no	—
366	C₄	—	712	1	no	—
367	C₄	—	712	2	no	—
368	C₄	C₂×#15667957	1	2	yes	—
369	C₄	#19152243	1	1	yes	characteristic
370	C₄	#19152410	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	1424	2	no	—
373	C₂	—	712	2	no	—
374	C₂	#26032066	1	1	yes	characteristic
375	C₂	C₂²×#15667957	1	1	yes	characteristic
376	C₂	#26033393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	D₄×D₇₁₂	C₂
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33862
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #26032645	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26022438	C₂	semidirect
5. #26022400	C₂	semidirect
6. #26022348	C₂	semidirect
7. #26022293	C₂	semidirect
8. #26022183	C₂	semidirect
9. C₄×#15667957	C₂	semidirect
10. C₂×#19151400	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
16. C₂×#15667957	C₂²	semidirect
17. #19152003	C₂²	semidirect
21. #19151839	C₂²	semidirect
25. C₄×Dic₃₅₆	C₂²	semidirect
27. C₂×#15667946	C₂²	semidirect
28. #19151426	C₂²	semidirect
29. #19151426	C₂²	semidirect
30. #19151403	C₂²	semidirect
33. #19151400	C₂²	semidirect
34. #19151400	C₂²	semidirect
35. #19151400	C₂²	semidirect
36. C₂²×C₁₆×C₈₉	C₂²	semidirect
38. C₄⋊C₁₆×C₈₉	C₂²	semidirect
39. C₂²⋊C₁₆×C₈₉	C₂²	semidirect
40. C₄×C₁₆×C₈₉	C₂²	semidirect
134. D₄×C₈₉	SD₃₂	semidirect
198. D₄×C₁₆	D₈₉	semidirect
200. C₈₉	#33862	semidirect
206. C₂²×C₁₆	C₂×D₈₉	semidirect
221. C₄⋊C₁₆	C₂×D₈₉	semidirect
228. C₂²⋊C₁₆	C₂×D₈₉	semidirect
229. C₄×C₁₆	C₂×D₈₉	semidirect
334. D₄	#15667957	semidirect

Maximal subgroups

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

subgroup	index
#26032645	2
D₄×D₇₁₂	2
#26022476	2
#26022438	2
#26022400	2
#26022348	2
#26022293	2
#26022183	2
C₄×#15667957	2
C₂×#19151400	2
D₄×C₁₆×C₈₉	2
#33862	89