Subgroups — #36709168

Order 22784.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030764	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151715	C₂	1	1	yes	maximal, characteristic
3	#26022426	C₂	1	4	yes	maximal, complemented
4	#26022421	C₂	1	4	yes	maximal, complemented
5	#26022025	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022024	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151418	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021988	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2145	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668019	—	2	1	no	—
11	#19152151	—	2	1	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	#19152002	—	2	1	no	—
14	#19151999	—	2	1	no	—
15	#19151998	C₂²	1	2	yes	complemented
16	#19151995	C₂²	1	2	yes	complemented
17	#19151715	—	2	1	no	—
18	#19151715	—	2	1	no	—
19	#19151715	C₂²	1	2	yes	—
20	#19151715	C₂²	1	2	yes	—
21	C₂×#15667948	C₂²	1	1	yes	characteristic
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151634	—	2	1	no	—
24	C₂×#15667939	—	2	1	no	—
25	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	#19151421	C₂²	1	4	yes	complemented
27	#19151420	C₂²	1	4	yes	complemented
28	#19151418	C₂²	1	2	yes	complemented
29	#19151418	C₂²	1	2	yes	complemented
30	#19151408	C₂²	1	4	yes	—
31	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
33	#427×C₈₉	C₂²	1	1	yes	characteristic
34	#360×C₈₉	C₂²	1	4	yes	—
35	#15668019	—	2	2	no	—
36	#15668019	—	2	2	no	—
37	#15668019	—	4	1	no	—
38	#15668019	—	4	1	no	—
39	C₂×#13083558	—	2	1	no	—
40	C₂×#13083558	—	4	1	no	—
41	C₂²×#10623494	—	4	1	no	—
42	C₂²×#10623494	D₄	1	1	yes	characteristic
43	#15668011	—	2	2	no	—
44	#15668003	—	2	2	no	—
45	#15667989	—	4	2	no	—
46	C₄×Dic₁₇₈	—	4	2	no	—
47	#15667971	—	2	1	no	—
48	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
49	#15667952	—	2	1	no	—
50	#15667952	C₂³	1	2	yes	—
51	C₂×#13083549	—	2	1	no	—
52	#15667948	—	2	1	no	—
53	#15667948	C₂³	1	2	yes	—
54	#15667946	—	2	1	no	—
55	#15667945	—	2	2	no	—
56	#15667944	C₂³	1	2	yes	—
57	#15667943	—	2	2	no	—
58	#15667939	—	2	2	no	—
59	#15667939	—	2	2	no	—
60	C₈×Dic₈₉	D₄	1	1	yes	characteristic
61	C₈×Dic₈₉	D₄	1	1	yes	characteristic
62	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
63	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₂×#41×C₈₉	—	2	1	no	—
65	#135×C₈₉	—	2	1	no	—
66	#135×C₈₉	—	2	1	no	—
67	#135×C₈₉	Q₈	1	2	yes	complemented
68	#135×C₈₉	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic, complemented
70	C₂×SD₁₆×C₈₉	—	2	1	no	—
71	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic, complemented
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	#117×C₈₉	—	2	1	no	—
74	M₃₂×C₈₉	—	2	1	no	—
75	M₃₂×C₈₉	C₂³	1	2	yes	—
76	C₂×C₁₆×C₈₉	—	2	1	no	—
77	#107×C₈₉	—	2	1	no	—
78	#106×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	C₂²×Dic₈₉	—	2	1	no	—
81	#13083558	—	2	2	no	—
82	#13083558	—	4	1	no	—
83	#13083558	—	4	2	no	—
84	#13083558	—	4	2	no	—
85	C₂×#10623494	—	4	1	no	—
86	C₂×#10623494	—	4	2	no	—
87	C₂×#10623494	—	4	2	no	—
88	C₂×#10623494	SD₁₆	1	1	yes	characteristic
89	C₂×#10623494	SD₁₆	1	1	yes	characteristic
90	C₂×#10623494	SD₁₆	1	1	yes	characteristic
91	C₂×#10623494	SD₁₆	1	1	yes	characteristic
92	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
93	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	C₂²⋊Dic₈₉	—	4	1	no	—
99	C₄×Dic₈₉	—	2	1	no	—
100	C₄×Dic₈₉	—	4	1	no	—
101	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
103	#13083549	—	2	2	no	—
104	#41×C₈₉	—	2	2	no	—
105	#41×C₈₉	—	2	2	no	—
106	#41×C₈₉	—	4	1	no	—
107	#41×C₈₉	—	4	1	no	—
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	—	2	1	no	—
110	C₂×D₄×C₈₉	—	4	1	no	—
111	C₂²×C₄×C₈₉	—	4	1	no	—
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₈₉	—	2	1	no	—
114	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
115	SD₁₆×C₈₉	—	2	2	no	—
116	SD₁₆×C₈₉	—	2	2	no	—
117	D₈×C₈₉	—	2	1	no	—
118	D₈×C₈₉	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
122	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
123	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
124	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₂²⋊C₄×C₈₉	—	4	1	no	—
127	C₄²×C₈₉	—	4	1	no	—
128	C₁₆×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	—	4	1	no	—
130	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₈₉	—	2	2	no	—
132	Q₈×C₈₉	—	4	1	no	—
133	D₄×C₈₉	—	2	2	no	—
134	D₄×C₈₉	—	4	2	no	—
135	D₄×C₈₉	—	4	2	no	—
136	D₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	Dic₁₇₈	—	8	4	no	—
152	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
155	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
156	#10623494	—	4	2	no	—
157	#10623494	C₂×SD₁₆	1	1	yes	characteristic
158	#10623494	C₂×SD₁₆	1	1	yes	characteristic
159	#10623494	C₂×SD₁₆	1	1	yes	characteristic
160	#10623494	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	2	no	—
163	C₂²×C₈₉	—	8	2	no	—
164	C₂²×C₈₉	#476	1	1	yes	characteristic
165	C₂²×C₈₉	#476	1	1	yes	characteristic
166	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₈₉	—	4	2	no	—
168	C₄×C₈₉	—	8	2	no	—
169	C₄×C₈₉	#476	1	1	yes	characteristic
170	C₄×C₈₉	#476	1	1	yes	characteristic
171	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
173	Dic₈₉	—	2	2	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	8	2	no	—
176	#33637	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	2	1	no	—
179	C₂×C₈₉	—	2	1	no	—
180	C₂×C₈₉	—	8	2	no	—
181	C₂×C₈₉	C₂×#476	1	1	yes	characteristic
182	#3160	—	89	1	no	—
183	C₂×#443	—	89	1	no	—
184	#2231	—	89	4	no	—
185	#2230	—	89	4	no	—
186	#2155	—	89	1	no	—
187	#2155	—	89	1	no	—
188	C₂×#364	—	89	1	no	—
189	#2145	—	89	1	no	—
190	#2145	D₈₉	1	1	yes	characteristic
191	C₈₉	#33637	1	1	yes	characteristic, complemented, Sylow
192	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
193	C₂×#131	—	178	1	no	—
194	#533	—	178	1	no	—
195	C₂×M₃₂	—	89	1	no	—
196	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
197	#500	—	178	1	no	—
198	#499	—	89	2	no	—
199	#499	—	178	1	no	—
200	#498	—	89	2	no	—
201	#443	—	89	2	no	—
202	#443	—	89	2	no	—
203	#443	—	178	1	no	—
204	#443	—	178	1	no	—
205	C₂×#108	—	89	1	no	—
206	#427	—	89	1	no	—
207	#427	—	178	1	no	—
208	#427	C₂×D₈₉	1	1	yes	characteristic
209	C₂×#104	—	178	1	no	—
210	C₂×C₄×C₈	—	89	1	no	—
211	#365	—	89	4	no	—
212	#365	—	89	4	no	—
213	#364	—	89	2	no	—
214	#364	—	89	2	no	—
215	#360	—	89	4	no	—
216	#360	C₂×D₈₉	1	4	yes	—
217	C₂×#41	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	#135	Dic₁₇₈	1	2	yes	complemented
221	#135	C₂²×D₈₉	1	2	yes	—
222	C₂×Q₁₆	Dic₁₇₈	1	1	yes	characteristic, complemented
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×D₈	Dic₁₇₈	1	1	yes	characteristic, complemented
225	#131	—	356	1	no	—
226	#131	—	356	1	no	—
227	#131	—	178	2	no	—
228	#131	—	178	2	no	—
229	C₂×M₁₆	—	356	1	no	—
230	C₂×M₁₆	—	178	1	no	—
231	C₂²×C₈	—	89	1	no	—
232	C₂²×C₈	—	356	1	no	—
233	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₄⋊Q₈	—	178	2	no	—
235	#125	—	178	2	no	—
236	C₂²⋊Q₈	—	356	2	no	—
237	C₄×Q₈	—	356	2	no	—
238	#117	—	2	1	no	—
239	#117	—	178	1	no	—
240	C₂×C₄²	—	89	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	—	89	2	no	—
243	M₃₂	—	178	1	no	—
244	M₃₂	C₂²×D₈₉	1	2	yes	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	178	1	no	—
247	#108	—	89	2	no	—
248	#108	—	178	1	no	—
249	#107	—	2	1	no	—
250	#107	—	178	2	no	—
251	#107	—	178	1	no	—
252	#106	—	89	2	no	—
253	#106	—	178	2	no	—
254	#106	C₂²×D₈₉	1	2	yes	—
255	#104	—	178	2	no	—
256	#104	—	178	2	no	—
257	C₄×C₈	—	89	1	no	—
258	C₄×C₈	—	89	1	no	—
259	C₄×C₈	—	89	1	no	—
260	C₄×C₈	—	89	1	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	4	1	no	—
264	#41	—	4	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	356	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	178	1	no	—
271	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	C₂×Dic₁₇₈	1	2	yes	—
278	M₁₆	—	356	2	no	—
279	M₁₆	—	356	2	no	—
280	M₁₆	—	356	1	no	—
281	M₁₆	—	178	2	no	—
282	C₂×C₈	—	89	1	no	—
283	C₂×C₈	—	89	1	no	—
284	C₂×C₈	—	89	1	no	—
285	C₂×C₈	—	89	1	no	—
286	C₂×C₈	—	89	1	no	—
287	C₂×C₈	—	89	1	no	—
288	C₂×C₈	—	356	2	no	—
289	C₂×C₈	—	356	2	no	—
290	C₂×C₈	—	356	1	no	—
291	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
293	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
294	C₂×C₈	#13083573	1	1	yes	characteristic
295	C₂×C₈	#13083573	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	356	2	no	—
299	C₄⋊C₄	—	356	2	no	—
300	C₄⋊C₄	—	356	2	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	356	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	89	1	no	—
306	C₄²	—	89	1	no	—
307	C₄²	—	356	1	no	—
308	C₄²	—	178	1	no	—
309	C₁₆	—	2	2	no	—
310	C₁₆	—	178	2	no	—
311	C₂³	—	4	1	no	—
312	C₂³	#15668036	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	712	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	#15667989	1	1	yes	characteristic
329	C₂×C₄	#15667989	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
332	C₂×C₄	#15668036	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
334	C₈	—	89	1	no	—
335	C₈	—	89	1	no	—
336	C₈	—	89	1	no	—
337	C₈	—	89	1	no	—
338	C₈	—	356	2	no	—
339	C₈	#15667989	1	1	yes	characteristic
340	C₈	#15667989	1	1	yes	characteristic
341	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
342	C₈	C₂×#13083573	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#19151886	1	1	yes	characteristic
347	C₂²	#19151886	1	1	yes	characteristic
348	C₂²	C₂×#15668036	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	356	1	no	—
352	C₄	—	178	2	no	—
353	C₄	—	712	2	no	—
354	C₄	#19151886	1	1	yes	characteristic
355	C₄	#19151886	1	1	yes	characteristic
356	C₄	C₂×#15667989	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#19151886	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151886
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2145	C₂
socle	C₂×C₈₉	C₂×#476
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	#2145	D₈₉
hypocenter	C₈₉	#33637
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
1. #26030764	C₂	semidirect
3. #26022426	C₂	semidirect
4. #26022421	C₂	semidirect
5. #26022025	C₂	semidirect
6. #26022024	C₂	semidirect
7. C₂×#19151418	C₂	semidirect
15. #19151998	C₂²	semidirect
16. #19151995	C₂²	semidirect
26. #19151421	C₂²	semidirect
27. #19151420	C₂²	semidirect
28. #19151418	C₂²	semidirect
29. #19151418	C₂²	semidirect
67. #135×C₈₉	Q₈	semidirect
69. C₂×Q₁₆×C₈₉	Q₈	semidirect
71. C₂×D₈×C₈₉	Q₈	semidirect
191. C₈₉	#33637	semidirect
220. #135	Dic₁₇₈	semidirect
222. C₂×Q₁₆	Dic₁₇₈	semidirect
224. C₂×D₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030764	2
C₂×#19151715	2
#26022426	2
#26022421	2
#26022025	2
#26022024	2
C₂×#19151418	2
#26021988	2
C₈₉×#2145	2
#33637	89