Subgroups — #36539611

Order 22784.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030045	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151875	C₂	1	1	yes	maximal, characteristic
3	#26029275	C₂	1	1	yes	maximal, characteristic
4	#26019700	C₂	1	4	yes	maximal, complemented
5	#26019614	C₂	1	4	yes	maximal, complemented
6	#26019027	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018937	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018909	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3209	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
11	#19151895	—	2	2	no	—
12	#19151885	C₂²	1	2	yes	—
13	#19151875	—	2	2	no	—
14	#19151875	C₂²	1	2	yes	—
15	#19151875	C₂²	1	4	yes	—
16	C₈×D₃₅₆	C₂²	1	4	yes	—
17	#19151663	—	2	2	no	—
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151641	—	2	1	no	—
20	C₂×#15667945	—	2	1	no	—
21	#19151622	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
23	#19151602	—	2	1	no	—
24	C₂×#15667932	—	2	1	no	—
25	C₂²⋊C₈×D₈₉	—	2	2	no	—
26	#19151351	C₂²	1	4	yes	complemented
27	#19151347	C₂²	1	2	yes	complemented
28	#19151344	C₂²	1	2	yes	complemented
29	#19151344	C₂²	1	4	yes	complemented
30	#533×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	#494×C₈₉	C₂²	1	2	yes	complemented
33	#487×C₈₉	C₂²	1	2	yes	complemented
34	#462×C₈₉	—	2	2	no	—
35	#451×C₈₉	—	2	2	no	—
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₈₉	—	2	1	no	—
39	C₂×#13083558	—	2	1	no	—
40	C₂²×#10623494	—	2	1	no	—
41	C₂×C₈×D₈₉	—	2	4	no	—
42	C₄×D₃₅₆	—	2	1	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	C₄×D₃₅₆	D₄	1	2	yes	—
45	C₄×D₃₅₆	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
47	C₂×#13083551	—	2	1	no	—
48	#15667945	—	2	4	no	—
49	#15667945	—	2	2	no	—
50	#15667943	—	2	2	no	—
51	C₄⋊#10623494	—	2	1	no	—
52	C₄⋊#10623494	—	2	1	no	—
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	C₄⋊#10623494	C₂³	1	2	yes	—
55	C₄⋊#10623494	C₂³	1	2	yes	—
56	#15667932	—	2	4	no	—
57	#15667932	—	2	2	no	—
58	#15667932	—	2	2	no	—
59	C₂²⋊#10623494	—	2	1	no	—
60	#15667924	—	2	4	no	—
61	C₄×#10623494	—	2	1	no	—
62	C₂²×Q₈×C₈₉	—	2	1	no	—
63	C₂×Q₁₆×C₈₉	—	2	4	no	—
64	C₄⋊Q₈×C₈₉	—	2	1	no	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	C₄⋊Q₈×C₈₉	—	2	1	no	—
67	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
70	#125×C₈₉	—	2	1	no	—
71	C₂²⋊Q₈×C₈₉	—	2	1	no	—
72	C₂²⋊Q₈×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	—	2	1	no	—
74	C₄×Q₈×C₈₉	—	2	1	no	—
75	#117×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
80	#103×C₈₉	—	2	4	no	—
81	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
82	C₄×C₈×C₈₉	C₂³	1	2	yes	—
83	C₂³×D₈₉	—	2	1	no	—
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	Q₁₆	1	4	yes	complemented
87	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
88	C₂²×Dic₈₉	—	2	1	no	—
89	C₂×C₄×D₈₉	—	2	2	no	—
90	C₂×C₄×D₈₉	—	2	2	no	—
91	C₂×C₄×D₈₉	—	8	2	no	—
92	#13083558	—	4	2	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	—	2	2	no	—
96	C₂×#10623494	—	4	2	no	—
97	C₈×D₈₉	—	8	4	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
100	#13083551	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×Q₈×C₈₉	—	4	1	no	—
106	C₂×Q₈×C₈₉	—	8	2	no	—
107	C₂²×C₄×C₈₉	—	4	1	no	—
108	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
111	Q₁₆×C₈₉	—	8	4	no	—
112	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
113	C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	4	1	no	—
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	#41	1	1	yes	characteristic
128	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₈₉	#121	1	1	yes	characteristic
130	C₂²×D₈₉	—	2	2	no	—
131	C₂²×D₈₉	—	2	4	no	—
132	C₂²×D₈₉	—	8	1	no	—
133	C₂²×D₈₉	—	8	1	no	—
134	Q₈×C₈₉	—	4	2	no	—
135	Q₈×C₈₉	—	4	4	no	—
136	Q₈×C₈₉	—	8	1	no	—
137	Q₈×C₈₉	—	8	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	4	1	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	8	2	no	—
144	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
148	C₂×C₄×C₈₉	#135	1	2	yes	—
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	8	1	no	—
154	D₃₅₆	—	2	2	no	—
155	D₃₅₆	—	2	4	no	—
156	D₃₅₆	C₂×Q₁₆	1	4	yes	—
157	C₄×D₈₉	—	4	4	no	—
158	C₄×D₈₉	—	8	2	no	—
159	C₈×C₈₉	—	2	4	no	—
160	#10623494	—	4	2	no	—
161	#10623494	—	4	2	no	—
162	#10623494	—	4	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	#467	1	2	yes	—
166	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
167	C₂×D₈₉	—	2	4	no	—
168	C₂×D₈₉	—	4	2	no	—
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	—	8	4	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	8	2	no	—
175	C₄×C₈₉	—	8	1	no	—
176	C₄×C₈₉	—	8	1	no	—
177	C₄×C₈₉	#467	1	2	yes	—
178	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
179	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
180	C₄×C₈₉	#576	1	2	yes	—
181	Dic₈₉	—	4	2	no	—
182	#19387	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	2	no	—
184	C₂×C₈₉	#3041	1	1	yes	characteristic
185	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
186	C₂×C₈₉	#3069	1	1	yes	characteristic
187	D₈₉	—	4	4	no	—
188	#3209	D₈₉	1	1	yes	characteristic, complemented
189	#3077	—	89	1	no	—
190	C₂×#474	—	89	1	no	—
191	#2972	—	89	1	no	—
192	#1683	—	89	4	no	—
193	#1669	—	89	4	no	—
194	#1571	—	89	1	no	—
195	#1549	—	89	1	no	—
196	#1542	—	89	1	no	—
197	C₈₉	#19387	1	1	yes	characteristic, complemented, Sylow
198	#533	C₂×D₈₉	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
200	C₂×C₄×D₄	—	89	1	no	—
201	#494	C₂×D₈₉	1	2	yes	complemented
202	#487	C₂×D₈₉	1	2	yes	complemented
203	#478	—	178	2	no	—
204	#476	—	89	2	no	—
205	#474	—	89	2	no	—
206	#474	—	89	4	no	—
207	#474	—	178	2	no	—
208	#462	—	2	2	no	—
209	#451	—	2	2	no	—
210	C₈×D₄	—	89	4	no	—
211	C₈×D₄	—	178	2	no	—
212	#431	—	89	1	no	—
213	#431	C₂×D₈₉	1	1	yes	characteristic
214	#428	—	178	1	no	—
215	C₂×#107	—	178	1	no	—
216	#423	—	89	1	no	—
217	C₂×C₄⋊C₈	—	89	1	no	—
218	#418	—	178	1	no	—
219	C₂×#102	—	178	1	no	—
220	C₂×C₂²⋊C₈	—	178	2	no	—
221	#335	—	89	4	no	—
222	#333	—	89	2	no	—
223	#332	—	89	2	no	—
224	#332	—	89	4	no	—
225	C₂²×Q₈	—	2	1	no	—
226	C₂²×D₄	—	89	1	no	—
227	C₂³×C₄	—	178	1	no	—
228	C₂×Q₁₆	—	2	4	no	—
229	C₂×M₁₆	—	178	1	no	—
230	C₂²×C₈	—	178	1	no	—
231	C₂²×C₈	—	178	4	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
236	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
238	#125	—	2	1	no	—
239	C₂²⋊Q₈	—	2	1	no	—
240	C₂²⋊Q₈	—	2	1	no	—
241	C₄×Q₈	—	2	1	no	—
242	C₄×Q₈	—	2	1	no	—
243	C₄×D₄	—	89	2	no	—
244	C₄×D₄	—	89	2	no	—
245	C₄×D₄	—	178	1	no	—
246	C₄×D₄	—	178	1	no	—
247	#117	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	89	1	no	—
251	C₂×C₂²⋊C₄	—	178	1	no	—
252	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
253	#107	—	178	4	no	—
254	#107	—	178	2	no	—
255	#106	—	178	2	no	—
256	C₄⋊C₈	—	89	2	no	—
257	C₄⋊C₈	—	89	2	no	—
258	C₄⋊C₈	—	89	2	no	—
259	C₄⋊C₈	—	178	1	no	—
260	C₄⋊C₈	—	178	1	no	—
261	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
262	#103	—	2	4	no	—
263	#102	—	178	4	no	—
264	#102	—	178	2	no	—
265	#102	—	178	2	no	—
266	C₂²⋊C₈	—	178	1	no	—
267	C₂²⋊C₈	—	178	4	no	—
268	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
269	C₄×C₈	—	178	1	no	—
270	C₄×C₈	C₂²×D₈₉	1	2	yes	—
271	C₂⁴	—	178	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	—	8	2	no	—
277	C₂×D₄	—	89	4	no	—
278	C₂×D₄	—	89	2	no	—
279	C₂×D₄	—	178	1	no	—
280	C₂×D₄	—	178	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	178	2	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	—	178	2	no	—
285	C₂²×C₄	—	712	2	no	—
286	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂²×C₄	#13083573	1	1	yes	characteristic
289	Q₁₆	—	8	4	no	—
290	M₁₆	—	356	2	no	—
291	C₂×C₈	—	356	2	no	—
292	C₂×C₈	—	178	2	no	—
293	C₂×C₈	—	178	2	no	—
294	C₂×C₈	—	178	2	no	—
295	C₂×C₈	—	712	4	no	—
296	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	89	2	no	—
308	C₄⋊C₄	—	178	1	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	178	2	no	—
311	C₂²⋊C₄	—	178	2	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₄×D₈₉	1	1	yes	characteristic
314	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₄²	#13083573	1	1	yes	characteristic
316	C₄²	C₂³×D₈₉	1	1	yes	characteristic
317	C₂³	—	178	2	no	—
318	C₂³	—	178	4	no	—
319	C₂³	—	712	1	no	—
320	C₂³	—	712	1	no	—
321	C₂³	#15667983	1	1	yes	characteristic
322	Q₈	—	4	2	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	8	1	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	89	4	no	—
327	D₄	—	178	4	no	—
328	D₄	—	178	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	356	4	no	—
336	C₂×C₄	—	178	2	no	—
337	C₂×C₄	—	712	2	no	—
338	C₂×C₄	—	712	1	no	—
339	C₂×C₄	#15667983	1	1	yes	characteristic
340	C₂×C₄	#15667983	1	1	yes	characteristic
341	C₂×C₄	#15667983	1	1	yes	characteristic
342	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
343	C₂×C₄	#15668035	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
346	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
347	C₈	—	2	4	no	—
348	C₈	—	356	2	no	—
349	C₈	—	356	2	no	—
350	C₈	—	356	2	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	356	2	no	—
354	C₂²	—	356	2	no	—
355	C₂²	—	178	4	no	—
356	C₂²	—	712	4	no	—
357	C₂²	#19151836	1	2	yes	—
358	C₂²	C₂×#15667983	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	356	2	no	—
366	C₄	#19151836	1	2	yes	—
367	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
368	C₄	C₂×#15668035	1	1	yes	characteristic
369	C₄	#19152401	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	356	4	no	—
372	C₂	#26029751	1	1	yes	characteristic
373	C₂	C₂×#19151836	1	1	yes	characteristic
374	C₂	#26029985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3209	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	#3209	D₈₉
hypocenter	C₈₉	#19387
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3209

Complements

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

N	H ≅ G/N	type
4. #26019700	C₂	semidirect
5. #26019614	C₂	semidirect
6. #26019027	C₂	semidirect
7. #26018937	C₂	semidirect
8. #26018909	C₂	semidirect
9. C₈₉×#3209	C₂	semidirect
26. #19151351	C₂²	semidirect
27. #19151347	C₂²	semidirect
28. #19151344	C₂²	semidirect
29. #19151344	C₂²	semidirect
32. #494×C₈₉	C₂²	semidirect
33. #487×C₈₉	C₂²	semidirect
86. C₂×D₃₅₆	Q₁₆	semidirect
188. #3209	D₈₉	semidirect
197. C₈₉	#19387	semidirect
201. #494	C₂×D₈₉	semidirect
202. #487	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030045	2
C₂×#19151875	2
#26029275	2
#26019700	2
#26019614	2
#26019027	2
#26018937	2
#26018909	2
C₈₉×#3209	2
#19387	89