Subgroups — #36707880

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033388	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151721	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029054	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022152	C₂	1	4	yes	maximal, complemented
5	#26022145	C₂	1	4	yes	maximal, complemented
6	#26022025	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022002	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021989	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#347×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	C₂²	1	1	yes	characteristic
11	C₂×#15668034	C₂²	1	1	yes	characteristic
12	C₂×#15668018	—	2	1	no	—
13	C₄×#13083574	—	2	1	no	—
14	#19152041	C₄	1	2	yes	complemented
15	#19152041	C₂²	1	2	yes	—
16	#19152036	C₄	1	2	yes	complemented
17	#19152036	C₂²	1	2	yes	—
18	#19152030	C₄	1	4	yes	complemented
19	#19151721	—	2	2	no	—
20	#19151721	C₂²	1	4	yes	—
21	#19151711	C₂²	1	2	yes	complemented
22	#19151706	C₂²	1	2	yes	complemented
23	#19151700	—	2	2	no	—
24	C₂×#15667948	C₂²	1	1	yes	characteristic
25	#19151636	C₂²	1	1	yes	characteristic
26	C₂×#15667940	—	2	1	no	—
27	#19151595	—	2	1	no	—
28	#19151421	C₂²	1	4	yes	complemented
29	#19151415	C₂²	1	2	yes	complemented
30	#19151412	C₂²	1	2	yes	complemented
31	#19151409	C₂²	1	4	yes	—
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
35	#347×C₈₉	C₂²	1	4	yes	complemented
36	C₂×#13083574	—	2	1	no	—
37	C₂×#13083574	—	2	1	no	—
38	#15668034	—	2	1	no	—
39	#15668034	—	2	1	no	—
40	#15668034	—	2	1	no	—
41	#15668034	—	2	1	no	—
42	#15668034	C₂×C₄	1	2	yes	—
43	#15668034	C₂×C₄	1	2	yes	—
44	#15668034	C₂×C₄	1	2	yes	complemented
45	#15668034	C₂³	1	2	yes	—
46	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
48	C₂×#13083563	—	2	1	no	—
49	C₂×#13083563	—	2	1	no	—
50	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
52	#15668018	—	2	4	no	—
53	#15668018	—	4	2	no	—
54	C₂×#13083558	—	2	1	no	—
55	C₂×#13083557	—	4	1	no	—
56	C₂×C₈×D₈₉	—	4	1	no	—
57	C₄×Dic₁₇₈	—	2	2	no	—
58	C₄×D₃₅₆	—	2	2	no	—
59	C₄×C₂²⋊D₈₉	—	4	2	no	—
60	#15667971	—	2	1	no	—
61	#15667969	—	4	1	no	—
62	C₄²×D₈₉	—	4	1	no	—
63	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
64	#15667957	C₂×C₄	1	4	yes	complemented
65	#15667957	C₂×C₄	1	4	yes	complemented
66	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
67	#15667948	—	2	1	no	—
68	#15667948	C₂³	1	2	yes	—
69	#15667946	—	2	1	no	—
70	#15667944	C₂³	1	2	yes	—
71	#15667940	—	2	4	no	—
72	#15667937	—	2	2	no	—
73	#15667934	—	2	2	no	—
74	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
77	M₃₂×C₈₉	—	2	1	no	—
78	M₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
79	M₃₂×C₈₉	C₂³	1	2	yes	—
80	M₃₂×C₈₉	C₂³	1	2	yes	—
81	C₂×C₁₆×C₈₉	—	2	1	no	—
82	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
83	#97×C₈₉	D₄	1	2	yes	—
84	#97×C₈₉	C₂³	1	2	yes	—
85	#13083574	—	2	2	no	—
86	#13083574	—	2	2	no	—
87	#13083574	—	2	2	no	—
88	#13083574	—	2	2	no	—
89	#13083574	—	4	1	no	—
90	#13083574	—	4	1	no	—
91	#13083574	—	4	1	no	—
92	#13083574	—	4	1	no	—
93	C₂×Dic₁₇₈	—	2	1	no	—
94	C₂×Dic₁₇₈	—	2	1	no	—
95	C₂×D₃₅₆	—	2	1	no	—
96	C₂×D₃₅₆	—	2	1	no	—
97	C₂×C₂²⋊D₈₉	—	4	1	no	—
98	C₂×C₂²⋊D₈₉	—	4	1	no	—
99	C₂×C₄×D₈₉	—	4	1	no	—
100	C₂×C₄×D₈₉	—	4	1	no	—
101	C₂×C₄×D₈₉	—	4	2	no	—
102	Dic₃₅₆	—	2	1	no	—
103	Dic₃₅₆	—	2	1	no	—
104	Dic₃₅₆	C₂²×C₄	1	2	yes	—
105	Dic₃₅₆	C₂²×C₄	1	2	yes	—
106	#13083563	—	2	2	no	—
107	#13083563	—	2	2	no	—
108	#13083563	—	2	2	no	—
109	#13083563	—	2	2	no	—
110	D₇₁₂	—	2	1	no	—
111	D₇₁₂	—	2	1	no	—
112	D₇₁₂	C₂²×C₄	1	2	yes	—
113	D₇₁₂	C₂²×C₄	1	2	yes	—
114	#13083558	—	2	2	no	—
115	#13083558	—	4	1	no	—
116	#13083557	—	4	4	no	—
117	C₂×#10623494	—	4	1	no	—
118	C₈×D₈₉	—	4	4	no	—
119	C₄⋊Dic₈₉	—	2	2	no	—
120	#13083553	—	4	2	no	—
121	C₂²⋊Dic₈₉	—	4	1	no	—
122	#13083551	—	4	2	no	—
123	C₄×Dic₈₉	—	4	2	no	—
124	C₄×Dic₈₉	—	4	1	no	—
125	C₂²×C₄×C₈₉	—	2	1	no	—
126	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
134	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₄²×C₈₉	—	2	1	no	—
137	C₄²×C₈₉	C₂×D₄	1	2	yes	—
138	C₁₆×C₈₉	—	2	2	no	—
139	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
140	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂²×D₈₉	—	4	1	no	—
142	C₂²×D₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	8	1	no	—
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₈₉	—	4	2	no	—
152	C₂²⋊D₈₉	—	4	2	no	—
153	C₂²⋊D₈₉	—	4	2	no	—
154	C₂²⋊D₈₉	—	4	2	no	—
155	C₂×Dic₈₉	—	4	1	no	—
156	C₂×Dic₈₉	—	4	1	no	—
157	C₂×Dic₈₉	—	4	2	no	—
158	D₃₅₆	—	2	2	no	—
159	D₃₅₆	—	2	2	no	—
160	D₃₅₆	—	4	1	no	—
161	D₃₅₆	—	4	1	no	—
162	C₄×D₈₉	—	4	2	no	—
163	C₄×D₈₉	—	4	2	no	—
164	C₄×D₈₉	—	4	2	no	—
165	C₄×D₈₉	—	4	2	no	—
166	C₄×D₈₉	—	8	4	no	—
167	Dic₁₇₈	—	2	2	no	—
168	Dic₁₇₈	—	2	2	no	—
169	Dic₁₇₈	—	4	1	no	—
170	Dic₁₇₈	—	4	1	no	—
171	C₈×C₈₉	—	2	2	no	—
172	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
173	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
174	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
175	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
176	#10623494	—	4	2	no	—
177	C₂²×C₈₉	—	4	1	no	—
178	C₂²×C₈₉	#442	1	1	yes	characteristic
179	C₂²×C₈₉	#442	1	1	yes	characteristic
180	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×D₈₉	—	4	2	no	—
182	C₂×D₈₉	—	4	2	no	—
183	C₂×D₈₉	—	8	2	no	—
184	C₂×D₈₉	—	8	2	no	—
185	C₄×C₈₉	—	4	2	no	—
186	C₄×C₈₉	#442	1	1	yes	characteristic
187	C₄×C₈₉	#442	1	1	yes	characteristic
188	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	Dic₈₉	—	4	2	no	—
191	Dic₈₉	—	4	2	no	—
192	Dic₈₉	—	8	2	no	—
193	#33468	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	1	no	—
196	C₂×C₈₉	—	2	1	no	—
197	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
198	D₈₉	—	8	2	no	—
199	D₈₉	—	8	2	no	—
200	#3414	—	89	1	no	—
201	C₂×#444	—	89	1	no	—
202	#2943	—	89	1	no	—
203	AGL(1,ℤ/16)	—	89	4	no	—
204	#2180	—	89	4	no	—
205	#2155	—	89	1	no	—
206	#2148	—	89	1	no	—
207	#2145	—	89	1	no	—
208	C₂×#347	D₈₉	1	1	yes	characteristic, complemented
209	C₈₉	#33468	1	1	yes	characteristic, complemented, Sylow
210	C₂×#135	—	89	1	no	—
211	C₂×#135	—	89	1	no	—
212	C₂×#131	—	178	1	no	—
213	C₄×#41	—	178	1	no	—
214	#510	—	89	2	no	—
215	#510	—	89	2	no	—
216	#509	—	89	2	no	—
217	#509	—	89	2	no	—
218	#508	—	89	4	no	—
219	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
220	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
221	#444	—	89	4	no	—
222	#444	—	178	2	no	—
223	#442	—	89	2	no	—
224	#441	—	89	2	no	—
225	#440	—	178	2	no	—
226	C₂×#108	—	89	1	no	—
227	#427	—	89	1	no	—
228	C₂×#104	—	178	1	no	—
229	#416	—	178	1	no	—
230	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
231	#365	—	89	4	no	—
232	#362	—	89	2	no	—
233	#361	—	89	2	no	—
234	#360	—	89	4	no	—
235	#347	C₂×D₈₉	1	4	yes	complemented
236	C₂×#41	—	178	1	no	—
237	C₂×#41	—	178	1	no	—
238	#135	—	89	2	no	—
239	#135	—	89	2	no	—
240	#135	—	89	2	no	—
241	#135	—	89	2	no	—
242	#135	—	178	1	no	—
243	#135	—	178	1	no	—
244	#135	—	178	1	no	—
245	#135	—	178	1	no	—
246	C₂×Q₁₆	—	89	1	no	—
247	C₂×Q₁₆	—	89	1	no	—
248	C₂×SD₁₆	—	178	1	no	—
249	C₂×SD₁₆	—	178	1	no	—
250	C₂×D₈	—	89	1	no	—
251	C₂×D₈	—	89	1	no	—
252	#131	—	356	2	no	—
253	#131	—	178	4	no	—
254	C₂×M₁₆	—	356	1	no	—
255	C₂×M₁₆	—	178	1	no	—
256	C₂²×C₈	—	356	1	no	—
257	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
258	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄×Q₈	—	178	2	no	—
260	C₄×D₄	—	356	2	no	—
261	C₄×D₄	—	178	2	no	—
262	#117	—	356	1	no	—
263	#117	—	178	1	no	—
264	C₂×C₄²	—	356	1	no	—
265	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
266	Q₃₂	—	89	4	no	—
267	SD₃₂	—	89	4	no	—
268	SD₃₂	—	89	4	no	—
269	D₁₆	—	89	4	no	—
270	M₃₂	—	2	1	no	—
271	M₃₂	C₄×D₈₉	1	2	yes	complemented
272	M₃₂	C₂²×D₈₉	1	2	yes	—
273	M₃₂	C₂²×D₈₉	1	2	yes	—
274	C₂×C₁₆	—	2	1	no	—
275	C₂×C₁₆	C₄×D₈₉	1	2	yes	complemented
276	#108	—	89	2	no	—
277	#108	—	178	1	no	—
278	#107	—	178	1	no	—
279	#106	—	89	2	no	—
280	#104	—	178	4	no	—
281	#103	—	178	2	no	—
282	#102	—	178	2	no	—
283	#97	D₃₅₆	1	2	yes	—
284	#97	C₂²×D₈₉	1	2	yes	—
285	#41	—	356	1	no	—
286	#41	—	356	1	no	—
287	#41	—	356	1	no	—
288	#41	—	356	1	no	—
289	#41	—	178	2	no	—
290	#41	—	178	2	no	—
291	#41	—	178	2	no	—
292	#41	—	178	2	no	—
293	C₂×Q₈	—	178	1	no	—
294	C₂×Q₈	—	178	1	no	—
295	C₂×D₄	—	356	1	no	—
296	C₂×D₄	—	356	1	no	—
297	C₂×D₄	—	178	1	no	—
298	C₂×D₄	—	178	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	356	2	no	—
301	C₂²×C₄	—	356	1	no	—
302	C₂²×C₄	—	356	1	no	—
303	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
304	Q₁₆	—	89	2	no	—
305	Q₁₆	—	89	2	no	—
306	Q₁₆	—	178	1	no	—
307	Q₁₆	—	178	1	no	—
308	SD₁₆	—	178	2	no	—
309	SD₁₆	—	178	2	no	—
310	SD₁₆	—	178	2	no	—
311	SD₁₆	—	178	2	no	—
312	D₈	—	89	2	no	—
313	D₈	—	89	2	no	—
314	D₈	—	178	1	no	—
315	D₈	—	178	1	no	—
316	M₁₆	—	356	4	no	—
317	M₁₆	—	356	1	no	—
318	M₁₆	—	178	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	356	4	no	—
322	C₂×C₈	—	356	1	no	—
323	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
325	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
326	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
327	C₂×C₈	#13083574	1	1	yes	characteristic
328	C₂×C₈	#13083574	1	1	yes	characteristic
329	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
330	C₄⋊C₄	—	356	2	no	—
331	C₄⋊C₄	—	178	2	no	—
332	C₂²⋊C₄	—	356	2	no	—
333	C₂²⋊C₄	—	356	1	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	356	2	no	—
336	C₄²	—	356	1	no	—
337	C₄²	C₂×D₃₅₆	1	2	yes	—
338	C₁₆	—	2	2	no	—
339	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
340	C₂³	—	356	1	no	—
341	C₂³	—	356	1	no	—
342	C₂³	#15668040	1	1	yes	characteristic
343	Q₈	—	356	1	no	—
344	Q₈	—	356	1	no	—
345	Q₈	—	178	2	no	—
346	Q₈	—	178	2	no	—
347	D₄	—	356	2	no	—
348	D₄	—	356	2	no	—
349	D₄	—	356	2	no	—
350	D₄	—	356	2	no	—
351	D₄	—	356	1	no	—
352	D₄	—	356	1	no	—
353	D₄	—	178	2	no	—
354	D₄	—	178	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	8	1	no	—
357	C₂×C₄	—	356	1	no	—
358	C₂×C₄	—	356	1	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	356	2	no	—
363	C₂×C₄	—	356	2	no	—
364	C₂×C₄	—	712	4	no	—
365	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
366	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
367	C₂×C₄	#15668040	1	1	yes	characteristic
368	C₂×C₄	#15668040	1	1	yes	characteristic
369	C₂×C₄	#15668040	1	1	yes	characteristic
370	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
371	C₈	—	2	2	no	—
372	C₈	—	356	2	no	—
373	C₈	C₄×D₃₅₆	1	1	yes	characteristic
374	C₈	C₄×D₃₅₆	1	1	yes	characteristic
375	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
376	C₈	C₂×#13083574	1	1	yes	characteristic
377	C₂²	—	4	1	no	—
378	C₂²	—	356	2	no	—
379	C₂²	—	356	2	no	—
380	C₂²	—	712	2	no	—
381	C₂²	—	712	2	no	—
382	C₂²	#19151711	1	1	yes	characteristic
383	C₂²	#19151711	1	1	yes	characteristic
384	C₂²	C₂×#15668040	1	1	yes	characteristic
385	C₄	—	4	2	no	—
386	C₄	—	356	2	no	—
387	C₄	—	356	2	no	—
388	C₄	—	712	2	no	—
389	C₄	#19151711	1	1	yes	characteristic
390	C₄	#19151711	1	1	yes	characteristic
391	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
392	C₄	C₂×#15668040	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	712	2	no	—
397	C₂	—	712	2	no	—
398	C₂	C₂×#19151711	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151711
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#347×C₈₉	C₂
socle	C₂×C₈₉	C₂×#442
prime radical	#26033388	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#347	D₈₉
hypocenter	C₈₉	#33468
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
2. C₂×#19151721	C₂	semidirect
3. #26029054	C₂	semidirect
4. #26022152	C₂	semidirect
5. #26022145	C₂	semidirect
6. #26022025	C₂	semidirect
7. #26022002	C₂	semidirect
8. #26021989	C₂	semidirect
9. C₂×#347×C₈₉	C₂	semidirect
14. #19152041	C₄	semidirect
16. #19152036	C₄	semidirect
18. #19152030	C₄	semidirect
21. #19151711	C₂²	semidirect
22. #19151706	C₂²	semidirect
28. #19151421	C₂²	semidirect
29. #19151415	C₂²	semidirect
30. #19151412	C₂²	semidirect
35. #347×C₈₉	C₂²	semidirect
44. #15668034	C₂×C₄	semidirect
46. C₂×Dic₃₅₆	C₂×C₄	semidirect
50. C₂×D₇₁₂	C₂×C₄	semidirect
63. Dic₇₁₂	C₂×C₄	semidirect
64. #15667957	C₂×C₄	semidirect
65. #15667957	C₂×C₄	semidirect
66. D₁₄₂₄	C₂×C₄	semidirect
78. M₃₂×C₈₉	C₂×C₄	semidirect
82. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
208. C₂×#347	D₈₉	semidirect
209. C₈₉	#33468	semidirect
235. #347	C₂×D₈₉	semidirect
271. M₃₂	C₄×D₈₉	semidirect
275. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033388	2
C₂×#19151721	2
#26029054	2
#26022152	2
#26022145	2
#26022025	2
#26022002	2
#26021989	2
C₂×#347×C₈₉	2
#33468	89