Subgroups — #36471498

Order 22784.

554 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667957	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151403	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151400	C₂	1	1	yes	maximal, characteristic
5	#26021113	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₅₆	—	2	1	no	—
11	C₂×#15667957	—	2	2	no	—
12	C₂×#15667957	—	2	2	no	—
13	C₂×#15667957	C₄	1	4	yes	complemented
14	C₄×D₇₁₂	—	2	2	no	—
15	C₄×D₇₁₂	—	2	2	no	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667937	—	2	1	no	—
19	C₂×#15667934	—	2	1	no	—
20	#19151504	—	2	1	no	—
21	#19151403	—	2	2	no	—
22	#19151400	—	2	2	no	—
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	1	no	—
28	C₂²×C₄×D₈₉	—	4	1	no	—
29	C₂×Dic₃₅₆	—	2	1	no	—
30	C₂×Dic₃₅₆	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	1	no	—
33	C₂×Dic₃₅₆	—	2	1	no	—
34	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	complemented
35	C₂×D₇₁₂	—	2	1	no	—
36	C₂×D₇₁₂	—	2	1	no	—
37	C₂×D₇₁₂	—	2	1	no	—
38	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
39	C₂×D₇₁₂	D₄	1	2	yes	—
40	C₂×D₇₁₂	D₄	1	2	yes	—
41	C₄×D₃₅₆	—	2	4	no	—
42	C₄×D₃₅₆	—	4	2	no	—
43	C₂×C₄⋊Dic₈₉	—	2	1	no	—
44	C₂×C₄⋊Dic₈₉	—	2	1	no	—
45	C₂×#13083551	—	4	1	no	—
46	#15667957	—	2	8	no	—
47	#15667946	—	2	1	no	—
48	#15667946	—	2	1	no	—
49	#15667946	—	2	1	no	—
50	#15667946	D₄	1	2	yes	—
51	#15667937	—	4	2	no	—
52	#15667934	—	2	4	no	—
53	#15667919	—	4	1	no	—
54	#15667919	—	4	1	no	—
55	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₈₉	—	2	1	no	—
59	C₂×C₁₆×C₈₉	—	2	1	no	—
60	C₂×C₁₆×C₈₉	—	2	1	no	—
61	C₂×C₁₆×C₈₉	—	2	1	no	—
62	C₂×C₁₆×C₈₉	—	2	1	no	—
63	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
64	C₄×C₈×C₈₉	—	2	1	no	—
65	C₄×C₈×C₈₉	—	2	1	no	—
66	C₂³×D₈₉	—	4	1	no	—
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×Dic₁₇₈	—	4	1	no	—
70	C₂×Dic₁₇₈	—	4	1	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×D₃₅₆	—	2	2	no	—
75	C₂×D₃₅₆	—	2	2	no	—
76	C₂×D₃₅₆	—	4	1	no	—
77	C₂×D₃₅₆	—	4	1	no	—
78	C₂×D₃₅₆	—	4	1	no	—
79	C₂²×Dic₈₉	—	4	1	no	—
80	C₂²×Dic₈₉	—	4	1	no	—
81	C₂²×Dic₈₉	—	4	1	no	—
82	C₂×C₄×D₈₉	—	4	4	no	—
83	C₂×C₄×D₈₉	—	8	2	no	—
84	C₂×C₄×D₈₉	—	8	2	no	—
85	Dic₃₅₆	—	2	2	no	—
86	Dic₃₅₆	—	2	2	no	—
87	Dic₃₅₆	—	4	1	no	—
88	Dic₃₅₆	—	4	1	no	—
89	D₇₁₂	—	2	2	no	—
90	D₇₁₂	—	2	2	no	—
91	D₇₁₂	—	2	2	no	—
92	D₇₁₂	C₂²⋊C₄	1	4	yes	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	4	1	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	#13083551	—	4	4	no	—
98	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	2	1	no	—
103	C₂×C₈×C₈₉	—	2	1	no	—
104	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₁₆×C₈₉	—	2	2	no	—
113	C₁₆×C₈₉	—	4	1	no	—
114	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	2	no	—
116	C₂²×D₈₉	—	4	2	no	—
117	C₂²×D₈₉	—	4	2	no	—
118	C₂²×D₈₉	—	8	1	no	—
119	C₂²×D₈₉	—	8	1	no	—
120	C₂²×D₈₉	—	8	1	no	—
121	C₂²×D₈₉	—	8	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	SD₃₂	1	2	yes	complemented
128	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
132	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	4	2	no	—
136	C₂×Dic₈₉	—	8	1	no	—
137	C₂×Dic₈₉	—	8	1	no	—
138	C₂×Dic₈₉	—	8	1	no	—
139	C₂×Dic₈₉	—	8	1	no	—
140	C₂×Dic₈₉	—	8	1	no	—
141	C₂×Dic₈₉	—	8	1	no	—
142	C₂×Dic₈₉	—	8	1	no	—
143	D₃₅₆	—	2	4	no	—
144	D₃₅₆	—	4	2	no	—
145	D₃₅₆	—	4	2	no	—
146	D₃₅₆	—	4	2	no	—
147	C₄×D₈₉	—	8	8	no	—
148	Dic₁₇₈	—	4	2	no	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	—	8	1	no	—
151	Dic₁₇₈	—	8	1	no	—
152	C₈×C₈₉	—	4	1	no	—
153	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
155	C₈×C₈₉	#121	1	1	yes	characteristic
156	C₈×C₈₉	#123	1	1	yes	characteristic
157	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
158	C₂²×C₈₉	#447	1	1	yes	characteristic
159	C₂²×C₈₉	#459	1	1	yes	characteristic
160	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
161	C₂²×C₈₉	#508	1	1	yes	characteristic
162	C₂²×C₈₉	#509	1	1	yes	characteristic
163	C₂²×C₈₉	#510	1	1	yes	characteristic
164	C₂×D₈₉	—	4	4	no	—
165	C₂×D₈₉	—	8	2	no	—
166	C₂×D₈₉	—	8	2	no	—
167	C₂×D₈₉	—	8	2	no	—
168	C₂×D₈₉	—	8	2	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	#386	1	1	yes	characteristic
174	C₄×C₈₉	#442	1	1	yes	characteristic
175	C₄×C₈₉	#449	1	1	yes	characteristic
176	C₄×C₈₉	#463	1	1	yes	characteristic
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	—	8	2	no	—
179	Dic₈₉	—	16	1	no	—
180	#13014	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	#1879	1	1	yes	characteristic
182	C₂×C₈₉	C₄×SD₃₂	1	1	yes	characteristic
183	C₂×C₈₉	#2175	1	1	yes	characteristic
184	C₂×C₈₉	#2187	1	1	yes	characteristic
185	C₂×C₈₉	#2189	1	1	yes	characteristic
186	C₂×C₈₉	#2208	1	1	yes	characteristic
187	C₂×C₈₉	#2210	1	1	yes	characteristic
188	D₈₉	—	8	4	no	—
189	C₂²×SD₃₂	—	89	1	no	—
190	C₂×C₄×D₈	—	89	1	no	—
191	C₂×#358	—	89	1	no	—
192	C₂×#357	—	89	1	no	—
193	#1969	—	89	1	no	—
194	#1380	—	89	1	no	—
195	#1374	D₈₉	1	1	yes	characteristic, complemented
196	C₈₉	#13014	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₁₆	—	89	1	no	—
198	C₂²×D₈	—	89	1	no	—
199	C₂×C₄×D₄	—	178	1	no	—
200	C₂×SD₃₂	—	89	4	no	—
201	C₂×SD₃₂	—	178	2	no	—
202	C₂×SD₃₂	—	178	2	no	—
203	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
204	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₄×D₈	—	178	2	no	—
206	C₄×D₈	—	178	2	no	—
207	C₂×#107	—	89	1	no	—
208	C₂×#107	—	89	1	no	—
209	C₂×#103	—	178	1	no	—
210	C₂×#102	—	178	1	no	—
211	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
212	#391	—	178	1	no	—
213	#358	—	178	2	no	—
214	#357	—	178	2	no	—
215	C₂²×Q₈	—	178	1	no	—
216	C₂²×D₄	—	178	1	no	—
217	C₂³×C₄	—	356	1	no	—
218	C₂×Q₁₆	—	89	2	no	—
219	C₂×Q₁₆	—	178	1	no	—
220	C₂×Q₁₆	—	178	1	no	—
221	C₂×Q₁₆	—	178	1	no	—
222	C₂×Q₁₆	—	178	1	no	—
223	C₂×Q₁₆	—	178	1	no	—
224	C₂×D₈	—	89	2	no	—
225	C₂×D₈	—	89	2	no	—
226	C₂×D₈	—	89	2	no	—
227	C₂×D₈	—	178	1	no	—
228	C₂×D₈	—	178	1	no	—
229	C₂×D₈	—	178	1	no	—
230	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄×D₄	—	356	2	no	—
233	C₄×D₄	—	178	4	no	—
234	C₂×C₄⋊C₄	—	178	1	no	—
235	C₂×C₄⋊C₄	—	178	1	no	—
236	C₂×C₂²⋊C₄	—	356	1	no	—
237	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
238	SD₃₂	—	178	8	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	2	1	no	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	C₄×D₈₉	1	2	yes	complemented
245	#107	—	89	2	no	—
246	#107	—	178	1	no	—
247	#107	—	178	1	no	—
248	#107	—	178	1	no	—
249	#103	—	356	2	no	—
250	#102	—	178	4	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	#95	—	356	1	no	—
254	#95	—	356	1	no	—
255	C₂⁴	—	356	1	no	—
256	C₂×Q₈	—	356	1	no	—
257	C₂×Q₈	—	356	1	no	—
258	C₂×Q₈	—	356	1	no	—
259	C₂×Q₈	—	356	1	no	—
260	C₂×Q₈	—	356	1	no	—
261	C₂×Q₈	—	178	2	no	—
262	C₂×D₄	—	356	1	no	—
263	C₂×D₄	—	356	1	no	—
264	C₂×D₄	—	356	1	no	—
265	C₂×D₄	—	178	2	no	—
266	C₂×D₄	—	178	2	no	—
267	C₂×D₄	—	178	2	no	—
268	C₂²×C₄	—	356	4	no	—
269	C₂²×C₄	—	356	1	no	—
270	C₂²×C₄	—	356	1	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	—	712	2	no	—
273	C₂²×C₄	—	712	2	no	—
274	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
275	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
277	Q₁₆	—	356	1	no	—
278	Q₁₆	—	356	1	no	—
279	Q₁₆	—	178	2	no	—
280	Q₁₆	—	178	2	no	—
281	D₈	—	89	4	no	—
282	D₈	—	178	2	no	—
283	D₈	—	178	2	no	—
284	D₈	—	178	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
290	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	#13083574	1	1	yes	characteristic
293	C₂×C₈	#13083575	1	1	yes	characteristic
294	C₄⋊C₄	—	356	1	no	—
295	C₄⋊C₄	—	356	1	no	—
296	C₄⋊C₄	—	356	1	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₂²⋊C₄	—	356	4	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₁₆	—	2	2	no	—
302	C₁₆	—	4	1	no	—
303	C₂³	—	356	2	no	—
304	C₂³	—	356	2	no	—
305	C₂³	—	356	2	no	—
306	C₂³	—	712	1	no	—
307	C₂³	—	712	1	no	—
308	C₂³	—	712	1	no	—
309	C₂³	—	712	1	no	—
310	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
311	Q₈	—	356	2	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	712	1	no	—
314	Q₈	—	712	1	no	—
315	D₄	—	356	2	no	—
316	D₄	—	356	2	no	—
317	D₄	—	356	2	no	—
318	D₄	—	178	4	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	712	8	no	—
327	C₂×C₄	—	712	1	no	—
328	C₂×C₄	—	712	1	no	—
329	C₂×C₄	—	712	1	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	—	712	1	no	—
332	C₂×C₄	—	712	1	no	—
333	C₂×C₄	—	712	1	no	—
334	C₂×C₄	#15667957	1	2	yes	complemented
335	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
336	C₂×C₄	#15667982	1	1	yes	characteristic
337	C₂×C₄	#15667987	1	1	yes	characteristic
338	C₂×C₄	#15668034	1	1	yes	characteristic
339	C₂×C₄	#15668040	1	1	yes	characteristic
340	C₂×C₄	#15668040	1	1	yes	characteristic
341	C₈	—	4	1	no	—
342	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
343	C₈	#15667974	1	1	yes	characteristic
344	C₈	#15667985	1	1	yes	characteristic
345	C₈	#15667997	1	1	yes	characteristic
346	C₂²	—	356	4	no	—
347	C₂²	—	712	2	no	—
348	C₂²	—	712	2	no	—
349	C₂²	—	712	2	no	—
350	C₂²	—	712	2	no	—
351	C₂²	—	712	2	no	—
352	C₂²	—	712	2	no	—
353	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
354	C₂²	#19151734	1	1	yes	characteristic
355	C₂²	#19151794	1	1	yes	characteristic
356	C₂²	C₂×#15667957	1	1	yes	characteristic
357	C₂²	#19152030	1	1	yes	characteristic
358	C₂²	#19152036	1	1	yes	characteristic
359	C₂²	#19152041	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	4	1	no	—
362	C₄	—	1424	1	no	—
363	C₄	—	712	2	no	—
364	C₄	—	712	2	no	—
365	C₄	#19151476	1	1	yes	characteristic
366	C₄	#19151711	1	1	yes	characteristic
367	C₄	#19151745	1	1	yes	characteristic
368	C₄	#19151814	1	1	yes	characteristic
369	C₂	—	712	4	no	—
370	C₂	#26020599	1	1	yes	characteristic
371	C₂	C₄×#15667957	1	1	yes	characteristic
372	C₂	#26022118	1	1	yes	characteristic
373	C₂	#26022183	1	1	yes	characteristic
374	C₂	#26022194	1	1	yes	characteristic
375	C₂	#26022298	1	1	yes	characteristic
376	C₂	#26022308	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1374	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1374	D₈₉
hypocenter	C₈₉	#13014
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. C₂×#19151403	C₂	semidirect
5. #26021113	C₂	semidirect
6. #26018344	C₂	semidirect
7. C₈₉×#1374	C₂	semidirect
13. C₂×#15667957	C₄	semidirect
34. C₂×Dic₃₅₆	C₂×C₄	semidirect
63. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
127. C₂×C₄×C₈₉	SD₃₂	semidirect
195. #1374	D₈₉	semidirect
196. C₈₉	#13014	semidirect
244. C₂×C₁₆	C₄×D₈₉	semidirect
334. C₂×C₄	#15667957	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667957	2
C₂×C₄×D₇₁₂	2
C₂×#19151403	2
C₂×#19151400	2
#26021113	2
#26018344	2
C₈₉×#1374	2
#13014	89