Subgroups — #36711239

Order 22784.

636 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033403	C₂	1	2	yes	maximal, complemented
2	#26032057	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022504	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022375	C₂	1	2	yes	maximal, complemented
5	#26022333	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022330	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022230	C₂	1	2	yes	maximal, complemented
8	#26022228	C₂	1	2	yes	maximal, complemented
9	#26022131	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022089	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3293	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152415	C₂²	1	2	yes	—
13	#19152412	—	2	1	no	—
14	#19152039	—	2	2	no	—
15	#19152039	C₂²	1	2	yes	—
16	#19152038	—	2	2	no	—
17	#19152038	C₂²	1	2	yes	—
18	#19152028	—	2	2	no	—
19	#19152028	C₂²	1	1	yes	characteristic
20	C₂×#15667958	—	2	2	no	—
21	C₂×#15667958	C₂²	1	2	yes	—
22	C₂×#15667958	C₂²	1	2	yes	—
23	#19152015	C₂²	1	2	yes	—
24	#19151978	C₂²	1	1	yes	characteristic
25	#19151874	—	2	1	no	—
26	#19151873	C₂²	1	2	yes	—
27	#19151782	—	2	2	no	—
28	#19151716	C₂²	1	1	yes	characteristic
29	#19151427	C₂²	1	1	yes	characteristic
30	#19151419	C₂²	1	1	yes	characteristic
31	#19151414	C₂²	1	2	yes	—
32	#19151413	C₂²	1	2	yes	—
33	#19151405	C₂²	1	1	yes	characteristic
34	#19151404	C₂²	1	1	yes	characteristic
35	#19151382	C₂²	1	2	yes	—
36	C₄×#13083549	C₂²	1	1	yes	characteristic
37	#578×C₈₉	—	2	1	no	—
38	#578×C₈₉	C₂²	1	2	yes	—
39	#494×C₈₉	C₂²	1	1	yes	characteristic
40	#473×C₈₉	—	2	1	no	—
41	#473×C₈₉	C₂²	1	2	yes	—
42	#456×C₈₉	—	2	2	no	—
43	#443×C₈₉	C₂²	1	1	yes	characteristic
44	#443×C₈₉	C₂²	1	1	yes	characteristic
45	#15668059	—	2	2	no	—
46	#15668045	—	2	2	no	—
47	#15668045	—	4	2	no	—
48	#15668042	—	2	2	no	—
49	#15668041	—	4	2	no	—
50	#15668034	—	2	2	no	—
51	#15668034	C₂³	1	1	yes	characteristic
52	#15668031	—	4	1	no	—
53	C₂×Dic₃₅₆	—	2	2	no	—
54	C₂×Dic₃₅₆	—	2	1	no	—
55	C₂×Dic₃₅₆	C₂³	1	2	yes	—
56	Q₁₆×D₈₉	—	4	1	no	—
57	#15668018	—	2	1	no	—
58	#15668012	—	2	1	no	—
59	#15667958	—	2	2	no	—
60	#15667958	—	2	2	no	—
61	#15667958	—	2	2	no	—
62	#15667958	—	2	2	no	—
63	#15667958	—	4	2	no	—
64	#15667958	D₄	1	1	yes	characteristic
65	#15667956	—	2	2	no	—
66	#15667956	D₄	1	1	yes	characteristic
67	#15667955	—	2	2	no	—
68	#15667955	D₄	1	1	yes	characteristic
69	#15667953	D₄	1	1	yes	characteristic
70	#15667952	—	2	2	no	—
71	#15667952	C₂³	1	2	yes	—
72	C₂×#13083549	—	2	2	no	—
73	C₂×#13083549	C₂³	1	1	yes	characteristic
74	C₂×#13083549	C₂³	1	1	yes	characteristic
75	#15667948	C₂³	1	1	yes	characteristic
76	#15667940	—	2	1	no	—
77	#15667930	—	2	2	no	—
78	2₋¹⁺⁴×C₈₉	—	2	2	no	—
79	#137×C₈₉	—	2	2	no	—
80	#137×C₈₉	—	2	2	no	—
81	#137×C₈₉	—	4	2	no	—
82	#137×C₈₉	—	4	2	no	—
83	#135×C₈₉	—	2	2	no	—
84	#135×C₈₉	—	4	1	no	—
85	#135×C₈₉	D₄	1	1	yes	characteristic
86	#135×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×Q₁₆×C₈₉	—	2	2	no	—
88	C₂×Q₁₆×C₈₉	—	2	1	no	—
89	C₂×Q₁₆×C₈₉	—	4	1	no	—
90	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
91	#131×C₈₉	—	2	1	no	—
92	#131×C₈₉	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₈₉	—	2	1	no	—
94	#108×C₈₉	D₄	1	1	yes	characteristic
95	#108×C₈₉	C₂³	1	1	yes	characteristic
96	#104×C₈₉	—	2	1	no	—
97	#104×C₈₉	D₄	1	2	yes	—
98	#101×C₈₉	—	2	2	no	—
99	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	#13083574	—	2	1	no	—
101	#13083574	—	4	2	no	—
102	#13083573	—	4	2	no	—
103	#13083573	—	8	2	no	—
104	C₂×Dic₁₇₈	—	2	2	no	—
105	C₂×Dic₁₇₈	—	4	2	no	—
106	Q₈×D₈₉	—	4	2	no	—
107	Dic₃₅₆	—	2	2	no	—
108	Dic₃₅₆	—	2	2	no	—
109	Dic₃₅₆	—	4	2	no	—
110	Dic₃₅₆	—	4	1	no	—
111	Dic₃₅₆	—	4	1	no	—
112	Dic₃₅₆	C₂×D₄	1	1	yes	characteristic
113	#13083564	—	4	2	no	—
114	#13083563	—	2	1	no	—
115	#13083563	—	4	2	no	—
116	#13083562	—	4	2	no	—
117	D₇₁₂	C₂×D₄	1	1	yes	characteristic
118	#13083558	—	2	1	no	—
119	#13083557	—	4	1	no	—
120	C₈×D₈₉	—	4	1	no	—
121	C₄⋊Dic₈₉	—	8	1	no	—
122	#13083549	—	2	2	no	—
123	#13083549	—	2	1	no	—
124	#13083549	C₂×D₄	1	1	yes	characteristic
125	#13083549	C₂×D₄	1	1	yes	characteristic
126	#41×C₈₉	—	2	1	no	—
127	#41×C₈₉	—	4	2	no	—
128	#41×C₈₉	—	4	2	no	—
129	#41×C₈₉	—	8	2	no	—
130	#41×C₈₉	C₂×D₄	1	2	yes	—
131	C₂×Q₈×C₈₉	—	2	2	no	—
132	C₂×Q₈×C₈₉	—	4	2	no	—
133	C₂×Q₈×C₈₉	—	4	2	no	—
134	Q₁₆×C₈₉	—	2	2	no	—
135	Q₁₆×C₈₉	—	2	2	no	—
136	Q₁₆×C₈₉	—	4	2	no	—
137	Q₁₆×C₈₉	—	4	2	no	—
138	Q₁₆×C₈₉	—	4	1	no	—
139	Q₁₆×C₈₉	—	4	1	no	—
140	Q₁₆×C₈₉	D₈	1	1	yes	characteristic
141	Q₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
142	SD₁₆×C₈₉	—	2	1	no	—
143	SD₁₆×C₈₉	—	4	2	no	—
144	SD₁₆×C₈₉	—	4	2	no	—
145	SD₁₆×C₈₉	D₈	1	2	yes	—
146	D₈×C₈₉	D₈	1	1	yes	characteristic
147	D₈×C₈₉	C₂×D₄	1	1	yes	characteristic
148	M₁₆×C₈₉	—	2	2	no	—
149	M₁₆×C₈₉	—	2	1	no	—
150	M₁₆×C₈₉	—	4	1	no	—
151	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₈₉	—	2	2	no	—
153	C₂×C₈×C₈₉	—	4	1	no	—
154	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
156	C₄⋊C₄×C₈₉	—	8	1	no	—
157	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
158	Q₈×C₈₉	—	2	1	no	—
159	Q₈×C₈₉	—	4	2	no	—
160	Q₈×C₈₉	—	4	2	no	—
161	Q₈×C₈₉	—	8	2	no	—
162	Q₈×C₈₉	C₂×D₈	1	2	yes	—
163	D₄×C₈₉	—	2	2	no	—
164	D₄×C₈₉	—	2	1	no	—
165	D₄×C₈₉	—	4	1	no	—
166	D₄×C₈₉	—	8	2	no	—
167	D₄×C₈₉	C₂×D₈	1	2	yes	—
168	C₂×C₄×C₈₉	—	2	1	no	—
169	C₂×C₄×C₈₉	—	2	2	no	—
170	C₂×C₄×C₈₉	—	4	1	no	—
171	C₂×C₄×C₈₉	—	4	2	no	—
172	C₂×C₄×C₈₉	—	8	2	no	—
173	C₂×C₄×C₈₉	—	8	2	no	—
174	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₈₉	—	4	1	no	—
176	C₂²⋊D₈₉	—	8	2	no	—
177	C₂×Dic₈₉	—	4	2	no	—
178	C₂×Dic₈₉	—	8	2	no	—
179	D₃₅₆	—	2	1	no	—
180	C₄×D₈₉	—	4	1	no	—
181	C₄×D₈₉	—	8	2	no	—
182	Dic₁₇₈	—	2	1	no	—
183	Dic₁₇₈	—	4	2	no	—
184	Dic₁₇₈	—	4	2	no	—
185	Dic₁₇₈	—	8	2	no	—
186	C₈×C₈₉	—	2	2	no	—
187	C₈×C₈₉	—	4	1	no	—
188	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
189	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
190	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
192	#10623494	—	4	1	no	—
193	C₂²×C₈₉	—	2	2	no	—
194	C₂²×C₈₉	—	4	1	no	—
195	C₂²×C₈₉	#576	1	1	yes	characteristic
196	C₂×D₈₉	—	4	1	no	—
197	C₄×C₈₉	—	2	2	no	—
198	C₄×C₈₉	—	2	2	no	—
199	C₄×C₈₉	—	4	1	no	—
200	C₄×C₈₉	—	8	2	no	—
201	C₄×C₈₉	D₄²	1	1	yes	characteristic
202	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
203	Dic₈₉	—	4	1	no	—
204	Dic₈₉	—	8	2	no	—
205	#33860	—	89	1	no	maximal, Sylow
206	C₂×C₈₉	—	2	1	no	—
207	C₂×C₈₉	—	4	2	no	—
208	C₂×C₈₉	—	8	1	no	—
209	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
210	D₈₉	—	8	1	no	—
211	#3417	—	89	2	no	—
212	#3293	—	89	1	no	—
213	#3293	D₈₉	1	1	yes	characteristic, complemented
214	#2247	—	89	1	no	—
215	#2222	—	89	2	no	—
216	#2214	—	89	1	no	—
217	#2214	—	89	1	no	—
218	#2195	—	89	2	no	—
219	#2195	—	89	2	no	—
220	#2178	—	89	1	no	—
221	#2170	—	89	1	no	—
222	C₈₉	#33860	1	1	yes	characteristic, complemented, Sylow
223	#578	—	2	1	no	—
224	#578	—	89	2	no	—
225	#578	—	178	1	no	—
226	#578	C₂×D₈₉	1	2	yes	—
227	#510	—	89	2	no	—
228	#510	—	89	2	no	—
229	#510	—	178	2	no	—
230	#510	—	178	2	no	—
231	#508	—	89	1	no	—
232	#508	—	178	2	no	—
233	C₂×Q₃₂	—	89	2	no	—
234	C₂×Q₃₂	—	89	2	no	—
235	C₂×Q₃₂	—	178	2	no	—
236	#504	—	89	2	no	—
237	#494	—	89	1	no	—
238	#494	C₂×D₈₉	1	1	yes	characteristic
239	#473	—	2	1	no	—
240	#473	—	89	2	no	—
241	#473	—	178	1	no	—
242	#473	C₂×D₈₉	1	2	yes	—
243	#456	—	2	2	no	—
244	#456	—	178	2	no	—
245	#443	—	89	1	no	—
246	#443	C₂×D₈₉	1	1	yes	characteristic
247	#443	C₂×D₈₉	1	1	yes	characteristic
248	#368	—	89	1	no	—
249	#364	—	89	1	no	—
250	#362	—	89	2	no	—
251	#362	—	89	2	no	—
252	#359	—	89	1	no	—
253	#359	—	89	1	no	—
254	#350	—	89	2	no	—
255	C₄×C₁₆	—	89	1	no	—
256	2₋¹⁺⁴	—	2	2	no	—
257	2₋¹⁺⁴	—	178	2	no	—
258	#137	—	2	2	no	—
259	#137	—	2	2	no	—
260	#137	—	4	2	no	—
261	#137	—	4	2	no	—
262	#137	—	356	2	no	—
263	#137	—	356	2	no	—
264	#137	—	178	2	no	—
265	#137	—	178	2	no	—
266	#135	—	2	2	no	—
267	#135	—	4	1	no	—
268	#135	—	89	1	no	—
269	#135	—	356	1	no	—
270	#135	—	178	2	no	—
271	#135	C₂²⋊D₈₉	1	1	yes	characteristic
272	#135	C₂²×D₈₉	1	1	yes	characteristic
273	C₂×Q₁₆	—	2	2	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	4	1	no	—
276	C₂×Q₁₆	—	89	2	no	—
277	C₂×Q₁₆	—	356	1	no	—
278	C₂×Q₁₆	—	178	2	no	—
279	C₂×Q₁₆	—	178	1	no	—
280	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
281	#131	—	2	1	no	—
282	#131	—	178	1	no	—
283	#131	C₂²×D₈₉	1	2	yes	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	178	1	no	—
286	Q₃₂	—	89	1	no	—
287	Q₃₂	—	356	2	no	—
288	Q₃₂	—	178	2	no	—
289	Q₃₂	—	178	2	no	—
290	Q₃₂	—	178	2	no	—
291	Q₃₂	—	178	2	no	—
292	SD₃₂	—	89	1	no	—
293	SD₃₂	—	89	1	no	—
294	SD₃₂	—	178	2	no	—
295	SD₃₂	—	178	2	no	—
296	D₁₆	—	89	1	no	—
297	M₃₂	—	89	2	no	—
298	M₃₂	—	178	2	no	—
299	C₂×C₁₆	—	89	1	no	—
300	C₂×C₁₆	—	89	1	no	—
301	C₂×C₁₆	—	178	2	no	—
302	#108	—	89	1	no	—
303	#108	C₂²⋊D₈₉	1	1	yes	characteristic
304	#108	C₂²×D₈₉	1	1	yes	characteristic
305	#104	—	2	1	no	—
306	#104	—	178	1	no	—
307	#104	C₂²⋊D₈₉	1	2	yes	—
308	#101	—	2	2	no	—
309	#101	—	178	2	no	—
310	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
311	#41	—	2	1	no	—
312	#41	—	4	2	no	—
313	#41	—	4	2	no	—
314	#41	—	8	2	no	—
315	#41	—	356	2	no	—
316	#41	—	356	2	no	—
317	#41	—	178	1	no	—
318	#41	—	712	2	no	—
319	#41	C₂×C₂²⋊D₈₉	1	2	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×Q₈	—	356	2	no	—
324	C₂×Q₈	—	356	2	no	—
325	C₂×Q₈	—	178	2	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	4	1	no	—
331	Q₁₆	—	4	1	no	—
332	Q₁₆	—	89	1	no	—
333	Q₁₆	—	356	2	no	—
334	Q₁₆	—	356	2	no	—
335	Q₁₆	—	356	1	no	—
336	Q₁₆	—	356	1	no	—
337	Q₁₆	—	178	2	no	—
338	Q₁₆	—	178	2	no	—
339	Q₁₆	#13083559	1	1	yes	characteristic
340	Q₁₆	D₄×D₈₉	1	1	yes	characteristic
341	SD₁₆	—	2	1	no	—
342	SD₁₆	—	4	2	no	—
343	SD₁₆	—	4	2	no	—
344	SD₁₆	—	356	2	no	—
345	SD₁₆	—	356	2	no	—
346	SD₁₆	—	178	1	no	—
347	SD₁₆	#13083559	1	2	yes	—
348	D₈	—	89	1	no	—
349	D₈	#13083559	1	1	yes	characteristic
350	D₈	D₄×D₈₉	1	1	yes	characteristic
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	4	1	no	—
354	M₁₆	—	356	1	no	—
355	M₁₆	—	178	1	no	—
356	M₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	4	1	no	—
359	C₂×C₈	—	356	1	no	—
360	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
361	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
362	C₄⋊C₄	—	8	1	no	—
363	C₄⋊C₄	—	712	1	no	—
364	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
365	C₁₆	—	89	1	no	—
366	C₁₆	—	89	1	no	—
367	C₁₆	—	178	2	no	—
368	C₁₆	—	178	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	8	2	no	—
373	Q₈	—	356	2	no	—
374	Q₈	—	356	2	no	—
375	Q₈	—	178	1	no	—
376	Q₈	—	712	2	no	—
377	Q₈	C₂×#13083559	1	2	yes	—
378	D₄	—	2	2	no	—
379	D₄	—	2	1	no	—
380	D₄	—	4	1	no	—
381	D₄	—	8	2	no	—
382	D₄	—	356	1	no	—
383	D₄	—	178	1	no	—
384	D₄	—	712	2	no	—
385	D₄	C₂×#13083559	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	356	1	no	—
393	C₂×C₄	—	356	2	no	—
394	C₂×C₄	—	712	2	no	—
395	C₂×C₄	—	712	2	no	—
396	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	4	1	no	—
399	C₈	—	356	1	no	—
400	C₈	C₂×#13083559	1	1	yes	characteristic
401	C₈	C₂×#13083559	1	1	yes	characteristic
402	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
403	C₈	#15668057	1	1	yes	characteristic
404	C₂²	—	2	2	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	356	1	no	—
407	C₂²	#19152401	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	356	1	no	—
413	C₄	—	712	2	no	—
414	C₄	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
415	C₄	C₂²×#13083559	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	8	1	no	—
419	C₂	—	712	1	no	—
420	C₂	D₄×#13083559	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3293	C₂
socle	C₂×C₈₉	D₄×D₈
hypercenter	#3293	D₈₉
hypocenter	C₈₉	#33860
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #3293

Complements

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

N	H ≅ G/N	type
1. #26033403	C₂	semidirect
2. #26032057	C₂	semidirect
3. #26022504	C₂	semidirect
4. #26022375	C₂	semidirect
5. #26022333	C₂	semidirect
6. #26022330	C₂	semidirect
7. #26022230	C₂	semidirect
8. #26022228	C₂	semidirect
9. #26022131	C₂	semidirect
10. #26022089	C₂	semidirect
11. C₈₉×#3293	C₂	semidirect
213. #3293	D₈₉	semidirect
222. C₈₉	#33860	semidirect

Maximal subgroups

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

subgroup	index
#26033403	2
#26032057	2
#26022504	2
#26022375	2
#26022333	2
#26022330	2
#26022230	2
#26022228	2
#26022131	2
#26022089	2
C₈₉×#3293	2
#33860	89