Subgroups — #36707673

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033354	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029153	C₂	1	1	yes	maximal, characteristic
3	#26022092	C₂	1	4	yes	maximal, complemented
4	#26022087	C₂	1	4	yes	maximal, complemented
5	#26022017	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151382	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021851	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3407	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152396	C₄	1	1	yes	characteristic
11	C₂×#15668018	C₂²	1	1	yes	characteristic
12	#19152016	—	2	2	no	—
13	#19152014	C₂²	1	4	yes	complemented
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	M₃₂×D₈₉	—	2	2	no	—
16	C₂²×#13083549	C₂²	1	1	yes	characteristic
17	#19151718	C₄	1	4	yes	—
18	#19151713	C₂²	1	4	yes	—
19	#19151646	C₄	1	1	yes	characteristic
20	#19151615	C₄	1	1	yes	characteristic
21	#19151614	C₄	1	1	yes	characteristic
22	#19151553	C₂²	1	1	yes	characteristic
23	#19151418	C₂²	1	4	yes	complemented
24	#19151383	C₂²	1	2	yes	complemented
25	#19151383	C₂²	1	2	yes	complemented
26	#19151382	C₂²	1	2	yes	complemented
27	#19151382	C₂²	1	2	yes	complemented
28	#19151373	C₂²	1	2	yes	complemented
29	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
30	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
31	#504×C₈₉	—	2	2	no	—
32	#504×C₈₉	C₂²	1	4	yes	complemented
33	C₂×M₃₂×C₈₉	—	2	2	no	—
34	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
37	#15668043	C₈	1	2	yes	—
38	#15668042	C₈	1	2	yes	—
39	#15668038	C₈	1	2	yes	—
40	#15668037	C₈	1	2	yes	—
41	#15668032	C₂×C₄	1	4	yes	—
42	#15668018	—	2	1	no	—
43	#15668018	—	2	1	no	—
44	#15668018	C₂×C₄	1	2	yes	—
45	#15668018	C₂³	1	2	yes	—
46	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
48	C₂×#13083557	—	2	1	no	—
49	C₂×C₈×D₈₉	—	2	1	no	—
50	#15667971	C₂×C₄	1	1	yes	characteristic
51	#15667952	—	2	1	no	—
52	#15667952	C₂³	1	2	yes	—
53	#15667951	—	2	4	no	—
54	C₂×#13083549	—	2	1	no	—
55	C₂×#13083549	—	2	1	no	—
56	C₂×#13083549	—	2	1	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₁₆×D₈₉	—	2	4	no	—
60	#15667947	C₂×C₄	1	4	yes	—
61	#15667939	C₂×C₄	1	4	yes	—
62	#15667938	C₂×C₄	1	4	yes	—
63	#15667922	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₉	C₂³	1	2	yes	—
65	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
66	#131×C₈₉	—	2	1	no	—
67	#131×C₈₉	—	2	1	no	—
68	#131×C₈₉	C₂×C₄	1	2	yes	—
69	#131×C₈₉	C₂³	1	2	yes	—
70	C₂×M₁₆×C₈₉	—	2	1	no	—
71	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	—	2	1	no	—
73	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	M₃₂×C₈₉	—	2	1	no	—
75	M₃₂×C₈₉	—	2	4	no	—
76	M₃₂×C₈₉	D₄	1	2	yes	—
77	M₃₂×C₈₉	D₄	1	2	yes	—
78	M₃₂×C₈₉	C₂³	1	2	yes	—
79	C₂×C₁₆×C₈₉	—	2	1	no	—
80	C₂×C₁₆×C₈₉	—	2	4	no	—
81	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
82	#13083574	—	2	1	no	—
83	#13083574	—	2	1	no	—
84	#13083574	C₂×C₈	1	2	yes	—
85	#13083574	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
87	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
88	C₂×C₂²⋊D₈₉	—	2	1	no	—
89	C₂×C₄×D₈₉	—	2	1	no	—
90	#13083564	C₂×C₈	1	4	yes	—
91	#13083562	C₂×C₈	1	4	yes	—
92	#13083561	C₂×C₈	1	4	yes	—
93	#13083559	C₂×C₈	1	4	yes	—
94	#13083558	—	2	1	no	—
95	#13083558	C₂×C₈	1	2	yes	—
96	#13083558	C₂×C₈	1	2	yes	—
97	#13083558	C₂²×C₄	1	2	yes	—
98	#13083557	—	2	2	no	—
99	#13083557	—	2	2	no	—
100	C₂×#10623494	—	2	1	no	—
101	C₂×#10623494	C₂²×C₄	1	2	yes	—
102	C₈×D₈₉	—	2	2	no	—
103	C₈×D₈₉	—	2	2	no	—
104	C₄⋊Dic₈₉	—	2	1	no	—
105	C₂²⋊Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
107	#13083549	—	2	2	no	—
108	#13083549	—	2	1	no	—
109	#13083549	—	2	1	no	—
110	#41×C₈₉	—	2	1	no	—
111	#41×C₈₉	—	2	1	no	—
112	#41×C₈₉	C₂×C₈	1	2	yes	—
113	#41×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
115	C₂×D₄×C₈₉	—	2	1	no	—
116	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	—	2	1	no	—
118	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	M₁₆×C₈₉	—	2	1	no	—
120	M₁₆×C₈₉	—	2	2	no	—
121	M₁₆×C₈₉	—	2	2	no	—
122	M₁₆×C₈₉	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	2	no	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
130	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₁₆×C₈₉	—	2	2	no	—
133	C₁₆×C₈₉	C₂×D₄	1	4	yes	—
134	C₂³×C₈₉	—	2	1	no	—
135	C₂³×C₈₉	#131	1	1	yes	characteristic
136	C₂²×D₈₉	—	2	1	no	—
137	Q₈×C₈₉	—	2	1	no	—
138	Q₈×C₈₉	C₂²×C₈	1	2	yes	—
139	D₄×C₈₉	—	2	2	no	—
140	D₄×C₈₉	—	2	2	no	—
141	D₄×C₈₉	—	2	1	no	—
142	D₄×C₈₉	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
152	C₂²⋊D₈₉	—	2	2	no	—
153	C₂²⋊D₈₉	—	2	2	no	—
154	C₂×Dic₈₉	—	2	1	no	—
155	C₂×Dic₈₉	—	2	2	no	—
156	D₃₅₆	—	2	1	no	—
157	D₃₅₆	C₂²×C₈	1	2	yes	—
158	C₄×D₈₉	—	2	2	no	—
159	C₄×D₈₉	—	2	2	no	—
160	Dic₁₇₈	—	2	1	no	—
161	Dic₁₇₈	C₂²×C₈	1	2	yes	—
162	C₈×C₈₉	—	2	2	no	—
163	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
164	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
165	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
166	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
167	#10623494	—	2	2	no	—
168	#10623494	C₂²×C₈	1	4	yes	—
169	C₂²×C₈₉	—	2	2	no	—
170	C₂²×C₈₉	—	4	1	no	—
171	C₂²×C₈₉	—	8	1	no	—
172	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
173	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
174	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
175	C₂×D₈₉	—	2	2	no	—
176	C₂×D₈₉	—	8	1	no	—
177	C₄×C₈₉	—	2	2	no	—
178	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
179	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
180	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
182	Dic₈₉	—	2	2	no	—
183	Dic₈₉	—	4	2	no	—
184	#33442	—	89	1	no	maximal, Sylow
185	C₂×C₈₉	—	2	1	no	—
186	C₂×C₈₉	—	2	1	no	—
187	C₂×C₈₉	—	2	1	no	—
188	C₂×C₈₉	—	4	2	no	—
189	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
190	D₈₉	—	4	2	no	—
191	#3407	—	89	1	no	—
192	#3407	D₈₉	1	1	yes	characteristic, complemented
193	#2955	—	89	1	no	—
194	#2171	—	89	4	no	—
195	#2170	—	89	4	no	—
196	#2153	—	89	1	no	—
197	C₂×#350	—	89	1	no	—
198	C₂×#350	—	89	1	no	—
199	#2108	—	89	1	no	—
200	C₈₉	#33442	1	1	yes	characteristic, complemented, Sylow
201	#575	—	89	1	no	—
202	C₂×#131	—	89	1	no	—
203	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
204	#504	—	2	2	no	—
205	#504	—	89	4	no	—
206	#504	—	178	2	no	—
207	#504	C₂×D₈₉	1	4	yes	complemented
208	C₂×M₃₂	—	2	2	no	—
209	C₂×M₃₂	—	89	1	no	—
210	C₂×M₃₂	—	178	2	no	—
211	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
212	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
213	C₂²×C₁₆	—	89	1	no	—
214	#444	—	89	4	no	—
215	#443	—	89	4	no	—
216	#430	—	89	1	no	—
217	#421	—	89	1	no	—
218	#421	—	89	1	no	—
219	#405	—	89	1	no	—
220	#364	—	89	4	no	—
221	#350	—	89	2	no	—
222	#350	—	89	2	no	—
223	#350	—	89	2	no	—
224	#350	—	89	2	no	—
225	#346	—	89	2	no	—
226	C₄×C₁₆	—	89	2	no	—
227	C₂×#41	—	89	1	no	—
228	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
229	#137	—	89	2	no	—
230	#137	—	89	2	no	—
231	AGL(1,ℤ/8)	—	89	2	no	—
232	AGL(1,ℤ/8)	—	89	2	no	—
233	#135	—	89	4	no	—
234	#131	—	2	1	no	—
235	#131	—	2	1	no	—
236	#131	—	89	2	no	—
237	#131	—	89	2	no	—
238	#131	—	178	1	no	—
239	#131	—	178	1	no	—
240	#131	C₄×D₈₉	1	2	yes	—
241	#131	C₂²×D₈₉	1	2	yes	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	89	1	no	—
244	C₂×M₁₆	—	89	1	no	—
245	C₂×M₁₆	—	178	1	no	—
246	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	178	1	no	—
249	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	#117	—	89	1	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	2	4	no	—
253	M₃₂	—	89	2	no	—
254	M₃₂	—	178	1	no	—
255	M₃₂	—	178	4	no	—
256	M₃₂	C₂²⋊D₈₉	1	2	yes	—
257	M₃₂	C₂²⋊D₈₉	1	2	yes	—
258	M₃₂	C₂²×D₈₉	1	2	yes	—
259	C₂×C₁₆	—	2	1	no	—
260	C₂×C₁₆	—	2	4	no	—
261	C₂×C₁₆	—	89	1	no	—
262	C₂×C₁₆	—	89	1	no	—
263	C₂×C₁₆	—	178	1	no	—
264	C₂×C₁₆	—	178	1	no	—
265	C₂×C₁₆	—	178	1	no	—
266	C₂×C₁₆	—	178	4	no	—
267	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
268	#108	—	89	4	no	—
269	#104	—	89	4	no	—
270	#104	—	89	4	no	—
271	#97	—	89	2	no	—
272	C₄×C₈	—	89	2	no	—
273	#41	—	2	1	no	—
274	#41	—	2	1	no	—
275	#41	—	89	2	no	—
276	#41	—	89	2	no	—
277	#41	—	178	1	no	—
278	#41	—	178	1	no	—
279	#41	C₈×D₈₉	1	2	yes	—
280	#41	C₂×C₄×D₈₉	1	2	yes	—
281	C₂×Q₈	—	89	1	no	—
282	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	89	1	no	—
285	C₂×D₄	—	178	1	no	—
286	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	178	1	no	—
289	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
290	Q₁₆	—	89	4	no	—
291	SD₁₆	—	89	4	no	—
292	SD₁₆	—	89	4	no	—
293	D₈	—	89	4	no	—
294	M₁₆	—	2	1	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	89	2	no	—
298	M₁₆	—	89	2	no	—
299	M₁₆	—	89	2	no	—
300	M₁₆	—	178	1	no	—
301	M₁₆	—	178	2	no	—
302	M₁₆	—	178	2	no	—
303	M₁₆	C₂×C₄×D₈₉	1	2	yes	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	89	2	no	—
308	C₂×C₈	—	178	1	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	2	no	—
311	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
312	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₂×C₈	#13083574	1	1	yes	characteristic
315	C₂×C₈	#13083574	1	1	yes	characteristic
316	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
317	C₄⋊C₄	—	178	1	no	—
318	C₂²⋊C₄	—	178	1	no	—
319	C₄²	—	89	2	no	—
320	C₁₆	—	2	2	no	—
321	C₁₆	—	178	2	no	—
322	C₁₆	—	178	1	no	—
323	C₁₆	—	178	1	no	—
324	C₁₆	C₂×C₂²⋊D₈₉	1	4	yes	—
325	C₂³	—	2	1	no	—
326	C₂³	—	178	1	no	—
327	C₂³	#15668018	1	1	yes	characteristic
328	Q₈	—	2	1	no	—
329	Q₈	—	89	2	no	—
330	Q₈	—	178	1	no	—
331	Q₈	C₂×C₈×D₈₉	1	2	yes	—
332	D₄	—	2	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	1	no	—
335	D₄	—	89	2	no	—
336	D₄	—	178	2	no	—
337	D₄	—	178	2	no	—
338	D₄	—	178	1	no	—
339	D₄	C₂×C₈×D₈₉	1	2	yes	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	—	178	1	no	—
345	C₂×C₄	—	178	2	no	—
346	C₂×C₄	—	178	2	no	—
347	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
350	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
351	C₂×C₄	#15668018	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	89	4	no	—
355	C₈	—	178	2	no	—
356	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
357	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
358	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
359	C₈	C₂×#13083574	1	1	yes	characteristic
360	C₂²	—	2	2	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	178	2	no	—
364	C₂²	—	712	1	no	—
365	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
366	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
367	C₂²	C₂×#15668018	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	356	2	no	—
370	C₄	—	178	2	no	—
371	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
372	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
373	C₄	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
374	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	356	2	no	—
380	C₂	C₂×C₈×C₂²⋊D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3407	C₂
socle	C₂×C₈₉	C₂×C₈×D₄
prime radical	#19152396	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#3407	D₈₉
hypocenter	C₈₉	#33442
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #3407

Complements

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

N	H ≅ G/N	type
1. #26033354	C₂	semidirect
3. #26022092	C₂	semidirect
4. #26022087	C₂	semidirect
5. #26022017	C₂	semidirect
6. C₂×#19151383	C₂	semidirect
7. C₂×#19151382	C₂	semidirect
8. #26021851	C₂	semidirect
9. C₈₉×#3407	C₂	semidirect
13. #19152014	C₂²	semidirect
23. #19151418	C₂²	semidirect
24. #19151383	C₂²	semidirect
25. #19151383	C₂²	semidirect
26. #19151382	C₂²	semidirect
27. #19151382	C₂²	semidirect
28. #19151373	C₂²	semidirect
29. C₁₆×Dic₈₉	C₂²	semidirect
32. #504×C₈₉	C₂²	semidirect
192. #3407	D₈₉	semidirect
200. C₈₉	#33442	semidirect
207. #504	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033354	2
#26029153	2
#26022092	2
#26022087	2
#26022017	2
C₂×#19151383	2
C₂×#19151382	2
#26021851	2
C₈₉×#3407	2
#33442	89