Subgroups — #36711304

Order 22784.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032600	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022480	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022450	C₂	1	2	yes	maximal, complemented
5	#26022395	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022348	C₂	1	2	yes	maximal, complemented
7	#26022298	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022171	C₂	1	2	yes	maximal, complemented
9	C₄×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151403	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
13	D₄×Dic₁₇₈	—	2	1	no	—
14	#19152252	—	2	1	no	—
15	C₂×#15667957	C₂²	1	1	yes	characteristic
16	C₂×#15667957	C₂²	1	2	yes	—
17	#19152003	C₂²	1	1	yes	characteristic
18	#19151978	C₂²	1	1	yes	characteristic
19	#19151912	—	2	2	no	—
20	#19151879	—	2	1	no	—
21	#19151839	C₂²	1	2	yes	—
22	#19151834	C₂²	1	2	yes	—
23	#19151809	—	2	1	no	—
24	#19151756	—	2	2	no	—
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	C₂×#15667946	C₂²	1	2	yes	—
28	#19151426	C₂²	1	1	yes	characteristic
29	#19151426	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151403	C₂²	1	2	yes	—
32	#19151403	C₂²	1	2	yes	—
33	#19151403	C₂²	1	1	yes	characteristic
34	#19151403	C₂²	1	2	yes	—
35	#19151400	C₂²	1	1	yes	characteristic
36	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	—	4	1	no	—
42	C₂²×Dic₁₇₈	—	2	2	no	—
43	C₂×Dic₃₅₆	—	2	4	no	—
44	C₂×Dic₃₅₆	—	2	2	no	—
45	C₂×Dic₃₅₆	—	2	1	no	—
46	C₂×Dic₃₅₆	—	4	2	no	—
47	C₂×Dic₃₅₆	C₂³	1	2	yes	—
48	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
49	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
50	#15668012	—	2	1	no	—
51	#15668012	—	2	1	no	—
52	#15668011	—	4	1	no	—
53	#15667996	—	4	2	no	—
54	#15667992	—	4	1	no	—
55	#15667991	—	2	2	no	—
56	#15667989	—	4	2	no	—
57	#15667986	—	2	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	1	no	—
70	#15667937	—	2	2	no	—
71	#15667934	—	2	2	no	—
72	#15667934	—	2	1	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	#13083573	—	8	4	no	—
83	C₂×Dic₁₇₈	—	2	1	no	—
84	C₂×Dic₁₇₈	—	2	2	no	—
85	C₂×Dic₁₇₈	—	4	4	no	—
86	C₂×Dic₁₇₈	—	4	2	no	—
87	C₂×Dic₁₇₈	—	4	1	no	—
88	C₂×Dic₁₇₈	—	8	2	no	—
89	C₂×D₃₅₆	—	2	1	no	—
90	C₂×C₂²⋊D₈₉	—	4	2	no	—
91	C₂²×Dic₈₉	—	4	2	no	—
92	C₂²×Dic₈₉	—	4	2	no	—
93	C₂×C₄×D₈₉	—	4	1	no	—
94	Dic₃₅₆	—	2	2	no	—
95	Dic₃₅₆	—	2	1	no	—
96	Dic₃₅₆	—	2	2	no	—
97	Dic₃₅₆	—	4	4	no	—
98	Dic₃₅₆	—	8	1	no	—
99	Dic₃₅₆	#41	1	2	yes	complemented
100	D₇₁₂	—	2	1	no	—
101	D₇₁₂	—	2	1	no	—
102	C₄⋊Dic₈₉	—	2	1	no	—
103	C₄⋊Dic₈₉	—	2	1	no	—
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	4	1	no	—
106	C₄⋊Dic₈₉	—	4	1	no	—
107	C₄⋊Dic₈₉	—	4	2	no	—
108	C₄⋊Dic₈₉	—	4	1	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	D₄×C₈₉	SD₃₂	1	4	yes	complemented
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	8	4	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	1	no	—
140	C₂×Dic₈₉	—	4	1	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	8	4	no	—
144	C₂×Dic₈₉	—	8	4	no	—
145	D₃₅₆	—	4	1	no	—
146	D₃₅₆	—	4	1	no	—
147	C₄×D₈₉	—	8	2	no	—
148	Dic₁₇₈	—	2	2	no	—
149	Dic₁₇₈	—	4	1	no	—
150	Dic₁₇₈	—	4	2	no	—
151	Dic₁₇₈	—	4	2	no	—
152	Dic₁₇₈	—	8	4	no	—
153	Dic₁₇₈	—	8	2	no	—
154	C₈×C₈₉	—	2	2	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₈₉	—	2	2	no	—
159	C₂²×C₈₉	—	2	2	no	—
160	C₂²×C₈₉	C₂×SD₃₂	1	4	yes	—
161	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₈₉	—	8	1	no	—
163	C₂×D₈₉	—	8	1	no	—
164	C₂×D₈₉	—	8	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
168	C₄×C₈₉	#546	1	1	yes	characteristic
169	C₄×C₈₉	#577	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	Dic₈₉	—	8	2	no	—
172	Dic₈₉	—	8	1	no	—
173	Dic₈₉	—	8	1	no	—
174	Dic₈₉	—	8	2	no	—
175	#33865	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	4	no	—
177	C₂×C₈₉	#3294	1	1	yes	characteristic
178	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
179	C₂×C₈₉	#3417	1	1	yes	characteristic
180	D₈₉	—	16	1	no	—
181	#3334	—	89	1	no	—
182	D₄×Q₁₆	—	89	1	no	—
183	#2241	—	89	1	no	—
184	#2235	—	89	2	no	—
185	#2225	—	89	1	no	—
186	#2217	—	89	2	no	—
187	#2208	—	89	1	no	—
188	#2185	—	89	2	no	—
189	C₄×SD₃₂	—	89	1	no	—
190	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
191	C₂×#358	—	89	2	no	—
192	C₈₉	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	—	89	2	no	—
194	D₄×Q₈	—	178	1	no	—
195	#547	—	178	1	no	—
196	C₂×SD₃₂	—	89	2	no	—
197	C₂×SD₃₂	—	89	1	no	—
198	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
199	#500	—	89	1	no	—
200	#494	—	89	1	no	—
201	#480	—	178	2	no	—
202	#474	—	178	1	no	—
203	#467	—	89	2	no	—
204	#466	—	89	2	no	—
205	#462	—	178	1	no	—
206	#451	—	178	2	no	—
207	C₄×Q₁₆	—	89	1	no	—
208	C₄×D₈	—	89	1	no	—
209	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
210	C₂×#107	—	89	2	no	—
211	#367	—	89	1	no	—
212	#367	—	89	2	no	—
213	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
214	#358	—	89	1	no	—
215	#358	—	89	2	no	—
216	#358	—	89	2	no	—
217	#358	—	89	2	no	—
218	#358	—	89	2	no	—
219	#357	—	89	1	no	—
220	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
221	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
222	C₂×#41	—	356	1	no	—
223	C₂²×Q₈	—	178	2	no	—
224	C₂×Q₁₆	—	89	1	no	—
225	C₂×Q₁₆	—	89	2	no	—
226	C₂×Q₁₆	—	356	2	no	—
227	C₂×Q₁₆	—	178	4	no	—
228	C₂×Q₁₆	—	178	2	no	—
229	C₂×Q₁₆	—	178	1	no	—
230	C₂×D₈	—	89	1	no	—
231	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
232	C₄⋊Q₈	—	356	1	no	—
233	C₄⋊Q₈	—	178	1	no	—
234	C₄⋊Q₈	—	178	1	no	—
235	#123	—	356	2	no	—
236	C₂²⋊Q₈	—	356	1	no	—
237	C₂²⋊Q₈	—	356	2	no	—
238	C₂²⋊Q₈	—	178	2	no	—
239	#121	—	178	2	no	—
240	C₄×Q₈	—	178	1	no	—
241	C₄×D₄	—	356	1	no	—
242	C₄×D₄	—	178	1	no	—
243	C₄×D₄	D₃₅₆	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	178	2	no	—
245	SD₃₂	—	356	2	no	—
246	SD₃₂	—	178	2	no	—
247	C₂×C₁₆	—	4	2	no	—
248	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
251	#107	—	89	1	no	—
252	#107	—	89	1	no	—
253	#107	—	89	2	no	—
254	#107	—	178	2	no	—
255	#107	—	178	1	no	—
256	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
257	#103	—	178	1	no	—
258	#103	—	178	2	no	—
259	#102	—	178	1	no	—
260	#102	—	178	2	no	—
261	C₂²⋊C₈	D₃₅₆	1	2	yes	—
262	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	#41	—	712	4	no	—
264	C₂×Q₈	—	356	4	no	—
265	C₂×Q₈	—	356	2	no	—
266	C₂×Q₈	—	356	1	no	—
267	C₂×Q₈	—	178	2	no	—
268	C₂×Q₈	—	178	1	no	—
269	C₂×Q₈	—	712	2	no	—
270	C₂×D₄	—	356	2	no	—
271	C₂×D₄	—	178	1	no	—
272	C₂×D₄	D₇₁₂	1	1	yes	characteristic
273	C₂²×C₄	—	356	2	no	—
274	C₂²×C₄	—	356	2	no	—
275	C₂²×C₄	—	356	1	no	—
276	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
277	Q₁₆	—	89	2	no	—
278	Q₁₆	—	356	4	no	—
279	Q₁₆	—	178	2	no	—
280	Q₁₆	—	178	2	no	—
281	Q₁₆	—	178	1	no	—
282	Q₁₆	—	712	1	no	—
283	D₈	—	178	1	no	—
284	D₈	—	178	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
288	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
289	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	356	2	no	—
291	C₄⋊C₄	—	356	1	no	—
292	C₄⋊C₄	—	356	1	no	—
293	C₄⋊C₄	—	356	1	no	—
294	C₄⋊C₄	—	356	2	no	—
295	C₄⋊C₄	—	356	1	no	—
296	C₄⋊C₄	—	178	1	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	1	no	—
299	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
300	C₂²⋊C₄	—	356	2	no	—
301	C₂²⋊C₄	—	356	1	no	—
302	C₂²⋊C₄	—	356	2	no	—
303	C₂²⋊C₄	D₇₁₂	1	2	yes	—
304	C₄²	—	356	1	no	—
305	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₂³	—	356	1	no	—
310	C₂³	C₂×D₇₁₂	1	2	yes	—
311	Q₈	—	356	1	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	356	2	no	—
314	Q₈	—	178	2	no	—
315	Q₈	—	712	4	no	—
316	Q₈	—	712	2	no	—
317	D₄	—	356	1	no	—
318	D₄	—	356	1	no	—
319	D₄	—	712	4	no	—
320	D₄	#15667957	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	356	1	no	—
324	C₂×C₄	—	356	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	1	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	712	4	no	—
329	C₂×C₄	—	712	4	no	—
330	C₂×C₄	—	712	2	no	—
331	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
332	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
333	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	2	no	—
337	C₈	#41×D₈₉	1	1	yes	characteristic
338	C₈	#15668057	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	712	1	no	—
342	C₂²	—	712	1	no	—
343	C₂²	—	712	1	no	—
344	C₂²	C₂×#15667957	1	4	yes	—
345	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	356	2	no	—
349	C₄	—	712	2	no	—
350	C₄	—	712	1	no	—
351	C₄	—	712	2	no	—
352	C₄	—	712	1	no	—
353	C₄	C₂×#15667957	1	2	yes	—
354	C₄	#19152243	1	1	yes	characteristic
355	C₄	#19152410	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	1424	1	no	—
358	C₂	#26032066	1	1	yes	characteristic
359	C₂	C₂²×#15667957	1	1	yes	characteristic
360	C₂	#26033405	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	#26032600	C₂
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33865
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
2. D₄×Dic₃₅₆	C₂	semidirect
3. #26022480	C₂	semidirect
4. #26022450	C₂	semidirect
5. #26022395	C₂	semidirect
6. #26022348	C₂	semidirect
7. #26022298	C₂	semidirect
8. #26022171	C₂	semidirect
9. C₄×#15667957	C₂	semidirect
10. C₂×#19151403	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
99. Dic₃₅₆	#41	semidirect
130. D₄×C₈₉	SD₃₂	semidirect
190. D₄×C₁₆	D₈₉	semidirect
192. C₈₉	#33865	semidirect
320. D₄	#15667957	semidirect

Maximal subgroups

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

subgroup	index
#26032600	2
D₄×Dic₃₅₆	2
#26022480	2
#26022450	2
#26022395	2
#26022348	2
#26022298	2
#26022171	2
C₄×#15667957	2
C₂×#19151403	2
D₄×C₁₆×C₈₉	2
#33865	89