Subgroups — #36711262

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	#26022475	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022435	C₂	1	2	yes	maximal, complemented
4	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022346	C₂	1	2	yes	maximal, complemented
6	#26022289	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022181	C₂	1	2	yes	maximal, complemented
8	C₄×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151399	C₂	1	2	yes	maximal, complemented
11	D₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152305	—	2	1	no	—
13	C₂×#15667956	C₂²	1	1	yes	characteristic
14	C₂×#15667956	C₂²	1	2	yes	complemented
15	C₂²×#13083549	C₂²	1	2	yes	complemented
16	#19152003	C₂²	1	1	yes	characteristic, complemented
17	#19151934	—	2	2	no	—
18	#19151894	—	2	1	no	—
19	#19151839	C₂²	1	2	yes	complemented
20	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×#15667946	C₂²	1	2	yes	complemented
22	#19151425	C₂²	1	1	yes	characteristic, complemented
23	#19151425	C₂²	1	2	yes	complemented
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic, complemented
25	#19151401	C₂²	1	1	yes	characteristic, complemented
26	#19151399	C₂²	1	2	yes	—
27	#19151399	C₂²	1	1	yes	characteristic
28	#19151399	C₂²	1	2	yes	complemented
29	#19151399	C₂²	1	2	yes	complemented
30	#19151399	C₂²	1	2	yes	complemented
31	C₂²⋊#13083549	C₂²	1	2	yes	complemented
32	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
33	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
34	D₄²×C₈₉	—	2	1	no	—
35	#493×C₈₉	C₂²	1	1	yes	characteristic
36	#466×C₈₉	C₂²	1	2	yes	—
37	#459×C₈₉	—	2	1	no	—
38	#447×C₈₉	—	2	2	no	—
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668003	—	4	1	no	—
44	#15667991	—	2	2	no	—
45	#15667989	—	4	2	no	—
46	C₄×Dic₁₇₈	—	2	1	no	—
47	D₄×Dic₈₉	—	4	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	#15667956	—	2	2	no	—
50	#15667956	—	4	2	no	—
51	C₂×#13083549	—	4	2	no	—
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	C₂×#13083549	C₂³	1	2	yes	—
55	#15667946	—	2	2	no	—
56	#15667946	—	2	1	no	—
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667946	C₂³	1	2	yes	—
59	#15667937	—	2	2	no	—
60	#15667937	—	2	1	no	—
61	C₂²×D₄×C₈₉	—	2	2	no	—
62	C₂²×D₄×C₈₉	—	4	1	no	—
63	C₂×D₈×C₈₉	—	2	4	no	—
64	C₂×D₈×C₈₉	—	2	2	no	—
65	C₂×D₈×C₈₉	—	2	1	no	—
66	C₂×D₈×C₈₉	—	4	2	no	—
67	C₂×D₈×C₈₉	C₂³	1	2	yes	—
68	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
70	#127×C₈₉	—	2	1	no	—
71	#121×C₈₉	—	2	2	no	—
72	#121×C₈₉	—	4	1	no	—
73	#120×C₈₉	—	4	2	no	—
74	C₄×D₄×C₈₉	—	2	1	no	—
75	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#102×C₈₉	—	2	1	no	—
79	#102×C₈₉	—	2	2	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	1	no	—
83	C₂×Dic₁₇₈	—	4	1	no	—
84	C₂²×Dic₈₉	—	4	2	no	—
85	Dic₃₅₆	—	2	1	no	—
86	Dic₃₅₆	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	2	no	—
91	#13083553	—	4	1	no	—
92	#13083553	—	4	2	no	—
93	C₂²⋊Dic₈₉	—	4	2	no	—
94	C₄×Dic₈₉	—	4	1	no	—
95	#13083549	—	2	1	no	—
96	#13083549	—	2	1	no	—
97	#13083549	—	2	2	no	—
98	C₂⁴×C₈₉	—	4	2	no	—
99	C₂×D₄×C₈₉	—	2	1	no	—
100	C₂×D₄×C₈₉	—	2	2	no	—
101	C₂×D₄×C₈₉	—	4	4	no	—
102	C₂×D₄×C₈₉	—	4	2	no	—
103	C₂×D₄×C₈₉	—	4	2	no	—
104	C₂×D₄×C₈₉	—	4	1	no	—
105	C₂×D₄×C₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	—	8	2	no	—
107	C₂×D₄×C₈₉	—	8	4	no	—
108	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	—	4	1	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
111	D₈×C₈₉	—	2	2	no	—
112	D₈×C₈₉	—	2	1	no	—
113	D₈×C₈₉	—	2	2	no	—
114	D₈×C₈₉	—	4	4	no	—
115	D₈×C₈₉	—	8	1	no	—
116	D₈×C₈₉	#41	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	—	2	1	no	—
123	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₂²⋊C₄×C₈₉	—	4	2	no	—
126	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₈₉	—	4	1	no	—
129	C₂³×C₈₉	—	4	2	no	—
130	C₂³×C₈₉	—	8	4	no	—
131	C₂³×C₈₉	—	8	4	no	—
132	C₂³×C₈₉	—	16	2	no	—
133	C₂³×C₈₉	C₂×D₈	1	2	yes	—
134	D₄×C₈₉	—	2	2	no	—
135	D₄×C₈₉	—	4	2	no	—
136	D₄×C₈₉	—	4	1	no	—
137	D₄×C₈₉	—	4	2	no	—
138	D₄×C₈₉	—	8	4	no	—
139	D₄×C₈₉	—	8	4	no	—
140	D₄×C₈₉	—	8	2	no	—
141	D₄×C₈₉	—	16	1	no	—
142	D₄×C₈₉	SD₃₂	1	4	yes	complemented
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	8	2	no	—
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₈₉	—	4	1	no	—
152	C₂×Dic₈₉	—	4	1	no	—
153	C₂×Dic₈₉	—	4	2	no	—
154	C₂×Dic₈₉	—	16	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	—	4	2	no	—
165	C₂²×C₈₉	—	8	1	no	—
166	C₂²×C₈₉	—	8	1	no	—
167	C₂²×C₈₉	—	8	2	no	—
168	C₂²×C₈₉	—	8	2	no	—
169	C₂²×C₈₉	—	8	2	no	—
170	C₂²×C₈₉	—	16	8	no	—
171	C₂²×C₈₉	C₂×SD₃₂	1	4	yes	—
172	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	2	no	—
175	C₄×C₈₉	—	8	1	no	—
176	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
177	C₄×C₈₉	#546	1	1	yes	characteristic
178	C₄×C₈₉	#577	1	1	yes	characteristic
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	—	8	2	no	—
185	C₂×C₈₉	—	16	2	no	—
186	C₂×C₈₉	#3294	1	1	yes	characteristic
187	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
188	C₂×C₈₉	#3415	1	1	yes	characteristic
189	#3338	—	89	1	no	—
190	D₄×D₈	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
199	C₂×#357	—	89	2	no	—
200	C₈₉	#33862	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₈	C₂×D₈₉	1	2	yes	—
202	#554	—	178	1	no	—
203	D₄²	—	2	1	no	—
204	C₂×SD₃₂	—	89	2	no	—
205	C₂×SD₃₂	—	89	1	no	—
206	C₂²×C₁₆	—	89	2	no	—
207	#500	—	89	1	no	—
208	#493	C₂×D₈₉	1	1	yes	characteristic
209	#484	—	178	2	no	—
210	#477	—	178	1	no	—
211	#467	—	89	2	no	—
212	#466	C₂×D₈₉	1	2	yes	—
213	#459	—	2	1	no	—
214	#447	—	2	2	no	—
215	C₄×Q₁₆	—	89	1	no	—
216	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
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	—	89	2	no	—
229	C₄×C₁₆	—	89	1	no	—
230	C₂²×D₄	—	2	2	no	—
231	C₂²×D₄	—	4	1	no	—
232	C₂×Q₁₆	—	89	1	no	—
233	C₂×D₈	—	2	4	no	—
234	C₂×D₈	—	2	2	no	—
235	C₂×D₈	—	2	1	no	—
236	C₂×D₈	—	4	2	no	—
237	C₂×D₈	C₂²×D₈₉	1	2	yes	—
238	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
240	C₄⋊Q₈	—	178	1	no	—
241	#127	—	2	1	no	—
242	#125	—	356	1	no	—
243	C₂²⋊Q₈	—	356	2	no	—
244	C₂²⋊Q₈	—	178	2	no	—
245	#121	—	2	2	no	—
246	#121	—	4	1	no	—
247	#120	—	4	2	no	—
248	C₄×Q₈	—	178	1	no	—
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	356	1	no	—
251	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	178	2	no	—
253	SD₃₂	—	356	2	no	—
254	SD₃₂	—	178	2	no	—
255	C₂×C₁₆	—	89	1	no	—
256	C₂×C₁₆	—	89	1	no	—
257	C₂×C₁₆	—	89	2	no	—
258	C₂×C₁₆	—	356	2	no	—
259	#107	—	89	1	no	—
260	#107	—	89	2	no	—
261	#107	—	178	2	no	—
262	#107	—	178	1	no	—
263	#107	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
265	#103	—	178	1	no	—
266	#103	—	178	2	no	—
267	#102	—	2	1	no	—
268	#102	—	2	2	no	—
269	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
270	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₂⁴	—	4	2	no	—
272	C₂×Q₈	—	356	1	no	—
273	C₂×Q₈	—	178	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	4	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	8	2	no	—
282	C₂×D₄	—	8	4	no	—
283	C₂×D₄	#13083559	1	1	yes	characteristic
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	356	2	no	—
286	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
287	Q₁₆	—	178	1	no	—
288	Q₁₆	—	178	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	1	no	—
291	D₈	—	2	2	no	—
292	D₈	—	4	4	no	—
293	D₈	—	8	1	no	—
294	D₈	#13083573	1	2	yes	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
299	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	356	2	no	—
302	C₄⋊C₄	—	356	1	no	—
303	C₄⋊C₄	—	356	1	no	—
304	C₄⋊C₄	—	356	2	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	#13083559	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	4	2	no	—
310	C₂²⋊C₄	—	356	2	no	—
311	C₂²⋊C₄	#13083559	1	2	yes	—
312	C₄²	—	356	1	no	—
313	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₁₆	—	178	1	no	—
315	C₁₆	—	178	1	no	—
316	C₁₆	—	178	2	no	—
317	C₂³	—	4	1	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	8	4	no	—
320	C₂³	—	8	4	no	—
321	C₂³	—	16	2	no	—
322	C₂³	C₂×#13083559	1	2	yes	—
323	Q₈	—	356	1	no	—
324	Q₈	—	356	1	no	—
325	Q₈	—	1424	1	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	4	no	—
331	D₄	—	8	4	no	—
332	D₄	—	8	2	no	—
333	D₄	—	16	1	no	—
334	D₄	#15667956	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	356	1	no	—
340	C₂×C₄	—	356	1	no	—
341	C₂×C₄	—	356	2	no	—
342	C₂×C₄	—	1424	2	no	—
343	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
344	C₂×C₄	C₂×#13083559	1	2	yes	—
345	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	2	no	—
349	C₈	C₂×#13083573	1	1	yes	characteristic
350	C₈	#15668057	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	16	8	no	—
360	C₂²	C₂×#15667956	1	4	yes	—
361	C₂²	C₂²×#13083559	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	712	1	no	—
366	C₄	—	712	1	no	—
367	C₄	—	712	2	no	—
368	C₄	C₂×#15667956	1	2	yes	—
369	C₄	#19152242	1	1	yes	characteristic
370	C₄	#19152409	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	8	2	no	—
373	C₂	—	16	2	no	—
374	C₂	#26032065	1	1	yes	characteristic
375	C₂	C₂²×#15667956	1	1	yes	characteristic
376	C₂	#26033391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	D₄×D₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	D₄×D₈	D₈₉
hypocenter	C₈₉	#33862
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×D₈

Complements

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

N	H ≅ G/N	type
1. #26032645	C₂	semidirect
2. #26022475	C₂	semidirect
3. #26022435	C₂	semidirect
4. #26022399	C₂	semidirect
5. #26022346	C₂	semidirect
6. #26022289	C₂	semidirect
7. #26022181	C₂	semidirect
8. C₄×#15667956	C₂	semidirect
9. D₄×#13083549	C₂	semidirect
10. C₂×#19151399	C₂	semidirect
14. C₂×#15667956	C₂²	semidirect
15. C₂²×#13083549	C₂²	semidirect
16. #19152003	C₂²	semidirect
19. #19151839	C₂²	semidirect
20. C₄×Dic₃₅₆	C₂²	semidirect
21. C₂×#15667946	C₂²	semidirect
22. #19151425	C₂²	semidirect
23. #19151425	C₂²	semidirect
24. C₄⋊#13083549	C₂²	semidirect
25. #19151401	C₂²	semidirect
28. #19151399	C₂²	semidirect
29. #19151399	C₂²	semidirect
30. #19151399	C₂²	semidirect
31. C₂²⋊#13083549	C₂²	semidirect
32. C₄×#13083549	C₂²	semidirect
142. D₄×C₈₉	SD₃₂	semidirect
200. C₈₉	#33862	semidirect
334. D₄	#15667956	semidirect

Maximal subgroups

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

subgroup	index
#26032645	2
#26022475	2
#26022435	2
#26022399	2
#26022346	2
#26022289	2
#26022181	2
C₄×#15667956	2
D₄×#13083549	2
C₂×#19151399	2
D₄×D₈×C₈₉	2
#33862	89