Subgroups — #36574127

Order 22784.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
2	#26028920	C₂	1	2	yes	maximal, complemented
3	#26021138	C₂	1	2	yes	maximal
4	#26020899	C₂	1	4	yes	maximal, complemented
5	#26020269	C₂	1	2	yes	maximal, complemented
6	#26020134	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	C₄×#13083574	C₂²	1	2	yes	—
11	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic, complemented
13	C₄×#13083563	C₄	1	2	yes	complemented
14	C₄×#13083563	C₄	1	2	yes	complemented
15	C₄×D₇₁₂	C₄	1	1	yes	characteristic, complemented
16	C₄×D₇₁₂	C₄	1	1	yes	characteristic, complemented
17	#19151658	—	2	4	no	—
18	#19151654	—	2	2	no	—
19	#19151650	—	2	2	no	—
20	#19151636	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
22	C₂×#15667938	C₂²	1	4	yes	—
23	#19151595	C₂²	1	2	yes	—
24	#19151566	C₂²	1	4	yes	—
25	#19151556	C₂²	1	4	yes	—
26	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
28	#19151359	C₂²	1	4	yes	—
29	#19151352	C₂²	1	2	yes	—
30	C₂×#108×C₈₉	C₂²	1	2	yes	—
31	C₂×#108×C₈₉	C₂²	1	2	yes	—
32	#423×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#13083574	C₂³	1	2	yes	—
35	#15668034	—	2	2	no	—
36	#15668034	—	2	2	no	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083563	C₂×C₄	1	2	yes	—
39	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
40	C₂²×#10623494	C₂³	1	2	yes	—
41	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
43	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
44	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₈₉	—	2	2	no	—
46	C₄×C₂²⋊D₈₉	—	2	2	no	—
47	#15667971	C₂³	1	2	yes	—
48	#15667969	—	2	2	no	—
49	C₂×C₄⋊Dic₈₉	—	2	1	no	—
50	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
51	C₄²×D₈₉	—	2	2	no	—
52	#15667946	C₂×C₄	1	1	yes	characteristic
53	#15667946	C₂×C₄	1	1	yes	characteristic
54	#15667944	C₂×C₄	1	2	yes	—
55	C₄⋊#10623494	D₄	1	1	yes	characteristic
56	C₄⋊#10623494	D₄	1	1	yes	characteristic
57	C₄⋊#10623494	D₄	1	2	yes	—
58	#15667938	—	4	4	no	—
59	#15667937	C₂×C₄	1	2	yes	—
60	#15667937	C₂×C₄	1	2	yes	—
61	#15667934	C₂×C₄	1	2	yes	—
62	#15667934	C₂×C₄	1	2	yes	—
63	#15667924	—	2	4	no	—
64	#15667924	—	2	4	no	—
65	#15667922	—	2	4	no	—
66	C₈×Dic₈₉	—	2	4	no	—
67	C₈×Dic₈₉	—	2	4	no	—
68	#15667919	—	2	1	no	—
69	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	#108×C₈₉	—	2	4	no	—
74	#108×C₈₉	—	4	2	no	—
75	C₄⋊C₈×C₈₉	—	2	2	no	—
76	C₄⋊C₈×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
81	#13083574	—	2	2	no	—
82	#13083574	—	2	2	no	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
86	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
87	C₂×C₂²⋊D₈₉	—	2	2	no	—
88	C₂²×Dic₈₉	—	2	2	no	—
89	C₂×C₄×D₈₉	—	2	2	no	—
90	C₂×C₄×D₈₉	—	2	2	no	—
91	C₂×C₄×D₈₉	—	2	2	no	—
92	Dic₃₅₆	—	2	2	no	—
93	#13083563	—	2	4	no	—
94	D₇₁₂	—	2	2	no	—
95	C₂×#10623494	—	2	2	no	—
96	C₂×#10623494	—	2	2	no	—
97	C₂×#10623494	C₂×D₄	1	2	yes	—
98	C₂×#10623494	C₂×D₄	1	2	yes	—
99	C₄⋊Dic₈₉	—	2	4	no	—
100	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
101	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
102	#13083553	—	2	2	no	—
103	#13083553	—	2	2	no	—
104	C₂²⋊Dic₈₉	—	2	2	no	—
105	#13083551	—	2	2	no	—
106	#13083551	—	2	2	no	—
107	C₄×Dic₈₉	—	2	2	no	—
108	C₄×Dic₈₉	—	2	2	no	—
109	C₄×Dic₈₉	—	2	2	no	—
110	C₄×Dic₈₉	—	2	2	no	—
111	C₄×Dic₈₉	—	2	2	no	—
112	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
115	M₁₆×C₈₉	—	2	4	no	—
116	M₁₆×C₈₉	—	2	4	no	—
117	C₂×C₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	—	2	1	no	—
119	C₂×C₈×C₈₉	—	2	4	no	—
120	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₈₉	#41	1	2	yes	—
125	C₂×C₈×C₈₉	#41	1	2	yes	—
126	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
131	C₂²×D₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²⋊D₈₉	—	4	2	no	—
145	C₂²⋊D₈₉	—	4	2	no	—
146	C₂×Dic₈₉	—	2	4	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	D₃₅₆	—	2	2	no	—
153	D₃₅₆	—	2	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₄×D₈₉	—	4	4	no	—
157	C₄×D₈₉	—	8	2	no	—
158	Dic₁₇₈	—	2	2	no	—
159	Dic₁₇₈	—	2	2	no	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	4	4	no	—
163	C₈×C₈₉	#117	1	2	yes	—
164	C₈×C₈₉	#117	1	2	yes	—
165	#10623494	—	2	2	no	—
166	#10623494	—	2	2	no	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₈₉	#555	1	1	yes	characteristic
171	C₂²×C₈₉	#556	1	1	yes	characteristic
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	#443	1	2	yes	—
179	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
180	C₄×C₈₉	#520	1	1	yes	characteristic
181	C₄×C₈₉	#547	1	1	yes	characteristic
182	C₄×C₈₉	#550	1	1	yes	characteristic
183	Dic₈₉	—	4	4	no	—
184	Dic₈₉	—	4	2	no	—
185	Dic₈₉	—	4	2	no	—
186	Dic₈₉	—	4	2	no	—
187	#22802	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	2	1	no	—
189	C₂×C₈₉	—	2	1	no	—
190	C₂×C₈₉	#2354	1	1	yes	characteristic
191	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
192	C₂×C₈₉	#2957	1	1	yes	characteristic
193	D₈₉	—	8	2	no	—
194	C₄×#135	—	89	1	no	—
195	#2925	—	89	2	no	—
196	#1974	—	89	2	no	—
197	#1929	—	89	4	no	—
198	#1848	D₈₉	1	1	yes	characteristic, complemented
199	#1806	—	89	2	no	—
200	#1772	—	89	2	no	—
201	C₈×C₄⋊C₄	—	89	1	no	—
202	C₈₉	#22802	1	1	yes	characteristic, complemented, Sylow
203	C₂×#135	—	89	1	no	—
204	C₄×#41	—	89	2	no	—
205	C₄×Q₁₆	—	89	1	no	—
206	C₄×Q₁₆	—	89	1	no	—
207	C₄×SD₁₆	—	89	2	no	—
208	C₄×SD₁₆	—	89	2	no	—
209	C₄×D₈	—	89	1	no	—
210	C₄×D₈	—	89	1	no	—
211	#433	—	178	4	no	—
212	#432	—	178	2	no	—
213	#431	—	178	2	no	—
214	C₂×#108	C₂×D₈₉	1	2	yes	—
215	C₂×#108	C₂×D₈₉	1	2	yes	—
216	#427	—	89	1	no	—
217	#423	C₂×D₈₉	1	2	yes	—
218	C₂×C₄⋊C₈	—	89	1	no	—
219	C₂×#104	—	89	4	no	—
220	#416	—	89	2	no	—
221	#408	—	89	4	no	—
222	#405	—	89	4	no	—
223	C₂×C₄×C₈	—	89	2	no	—
224	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
225	C₄×C₄⋊C₄	—	89	1	no	—
226	#339	—	89	4	no	—
227	#336	—	89	2	no	—
228	C₂×#41	—	89	2	no	—
229	#135	—	178	2	no	—
230	#135	—	178	2	no	—
231	C₂×Q₁₆	—	89	1	no	—
232	C₂×SD₁₆	—	89	2	no	—
233	C₂×D₈	—	89	1	no	—
234	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
235	C₂²×C₈	—	89	2	no	—
236	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₄×Q₈	—	89	2	no	—
239	C₄×Q₈	—	89	2	no	—
240	C₄×D₄	—	89	2	no	—
241	C₄×D₄	—	89	2	no	—
242	C₄×D₄	—	178	2	no	—
243	C₄×D₄	—	178	2	no	—
244	#117	—	89	2	no	—
245	#117	—	178	2	no	—
246	C₂×C₄⋊C₄	—	178	1	no	—
247	C₂×C₄²	—	89	2	no	—
248	C₂×C₄²	—	178	2	no	—
249	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
250	#108	—	2	4	no	—
251	#108	—	4	2	no	—
252	#107	—	89	1	no	—
253	#107	—	89	1	no	—
254	#106	—	89	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	89	1	no	—
258	C₄⋊C₈	—	89	1	no	—
259	C₄⋊C₈	—	89	2	no	—
260	#104	—	356	4	no	—
261	#103	—	89	2	no	—
262	#103	—	89	2	no	—
263	#102	—	89	2	no	—
264	#102	—	89	2	no	—
265	C₂²⋊C₈	—	178	4	no	—
266	C₂²⋊C₈	—	178	4	no	—
267	#97	—	178	4	no	—
268	C₄×C₈	—	178	4	no	—
269	C₄×C₈	—	178	4	no	—
270	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
271	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
272	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
273	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
274	#95	—	178	1	no	—
275	#41	—	178	2	no	—
276	#41	—	178	2	no	—
277	#41	—	178	2	no	—
278	#41	—	178	2	no	—
279	C₂×Q₈	—	89	2	no	—
280	C₂×D₄	—	89	2	no	—
281	C₂×D₄	—	178	2	no	—
282	C₂²×C₄	—	178	2	no	—
283	C₂²×C₄	—	178	2	no	—
284	C₂²×C₄	—	178	2	no	—
285	C₂²×C₄	—	178	2	no	—
286	C₂²×C₄	#13083575	1	1	yes	characteristic
287	C₂²×C₄	#13083575	1	1	yes	characteristic
288	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
289	Q₁₆	—	178	2	no	—
290	SD₁₆	—	178	4	no	—
291	D₈	—	178	2	no	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	2	4	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	89	2	no	—
299	C₂×C₈	—	178	2	no	—
300	C₂×C₈	—	178	2	no	—
301	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₈	#13083573	1	2	yes	—
306	C₂×C₈	#13083575	1	2	yes	—
307	C₄⋊C₄	—	89	2	no	—
308	C₄⋊C₄	—	89	2	no	—
309	C₄⋊C₄	—	178	4	no	—
310	C₄⋊C₄	—	178	2	no	—
311	C₄⋊C₄	—	178	2	no	—
312	C₂²⋊C₄	—	178	2	no	—
313	C₂²⋊C₄	—	178	2	no	—
314	C₂²⋊C₄	—	178	2	no	—
315	C₄²	—	178	2	no	—
316	C₄²	—	178	2	no	—
317	C₄²	—	178	2	no	—
318	C₄²	—	178	2	no	—
319	C₄²	—	178	2	no	—
320	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
321	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
322	C₄²	D₄×D₈₉	1	1	yes	characteristic
323	C₄²	D₄×D₈₉	1	1	yes	characteristic
324	C₂³	—	178	2	no	—
325	C₂³	C₂×#13083575	1	1	yes	characteristic
326	Q₈	—	178	2	no	—
327	Q₈	—	178	2	no	—
328	D₄	—	356	2	no	—
329	D₄	—	356	2	no	—
330	D₄	—	178	2	no	—
331	D₄	—	178	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	356	4	no	—
337	C₂×C₄	—	356	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	178	4	no	—
340	C₂×C₄	—	178	2	no	—
341	C₂×C₄	—	178	2	no	—
342	C₂×C₄	—	178	2	no	—
343	C₂×C₄	—	712	2	no	—
344	C₂×C₄	#15667974	1	1	yes	characteristic
345	C₂×C₄	#15667974	1	1	yes	characteristic
346	C₂×C₄	#15667974	1	1	yes	characteristic
347	C₂×C₄	#15667974	1	1	yes	characteristic
348	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
350	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
351	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
352	C₂×C₄	#15668057	1	1	yes	characteristic
353	C₂×C₄	#15668060	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	4	no	—
357	C₈	—	178	2	no	—
358	C₈	—	178	2	no	—
359	C₈	#15667970	1	2	yes	—
360	C₈	#15667970	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	356	2	no	—
364	C₂²	—	356	2	no	—
365	C₂²	—	356	2	no	—
366	C₂²	C₂×#15667974	1	1	yes	characteristic
367	C₂²	#19152314	1	1	yes	characteristic
368	C₂²	#19152322	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	356	4	no	—
373	C₄	—	356	2	no	—
374	C₄	—	356	2	no	—
375	C₄	—	356	2	no	—
376	C₄	#19151714	1	2	yes	—
377	C₄	C₂×#15667970	1	1	yes	characteristic
378	C₄	#19152079	1	1	yes	characteristic
379	C₄	#19152251	1	1	yes	characteristic
380	C₄	#19152268	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	712	2	no	—
384	C₂	#26023203	1	1	yes	characteristic
385	C₂	C₂×#19151714	1	1	yes	characteristic
386	C₂	#26029168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#22802
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #26028920	C₂	semidirect
4. #26020899	C₂	semidirect
5. #26020269	C₂	semidirect
6. #26020134	C₂	semidirect
7. C₈×C₄⋊Dic₈₉	C₂	semidirect
8. C₈₉×#1848	C₂	semidirect
11. C₄×Dic₃₅₆	C₄	semidirect
12. C₄×Dic₃₅₆	C₄	semidirect
13. C₄×#13083563	C₄	semidirect
14. C₄×#13083563	C₄	semidirect
15. C₄×D₇₁₂	C₄	semidirect
16. C₄×D₇₁₂	C₄	semidirect
198. #1848	D₈₉	semidirect
202. C₈₉	#22802	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668034	2
#26028920	2
#26021138	2
#26020899	2
#26020269	2
#26020134	2
C₈×C₄⋊Dic₈₉	2
C₈₉×#1848	2
#22802	89