Subgroups — #36536699

Order 22784.

694 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030261	C₂	1	2	yes	maximal
3	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#26019652	C₂	1	2	yes	maximal, complemented
5	#26019642	C₂	1	2	yes	maximal, complemented
6	#26019628	C₂	1	4	yes	maximal, complemented
7	#26018907	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₈₉	—	2	1	no	—
10	#19152155	C₂²	1	2	yes	—
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	#19152003	C₂²	1	1	yes	characteristic, complemented
13	#19152003	C₂²	1	1	yes	characteristic, complemented
14	#19152000	—	2	1	no	—
15	#19152000	—	2	1	no	—
16	#19151996	C₂²	1	2	yes	complemented
17	#19151929	—	2	2	no	—
18	#19151889	C₂²	1	4	yes	—
19	#19151804	C₂²	1	4	yes	—
20	#19151751	—	2	2	no	—
21	C₄×#13083563	—	2	4	no	—
22	C₈×D₃₅₆	—	2	1	no	—
23	C₈×D₃₅₆	—	2	1	no	—
24	C₈×D₃₅₆	C₂²	1	2	yes	—
25	C₈×D₃₅₆	C₂²	1	2	yes	—
26	#19151636	—	2	1	no	—
27	#19151636	—	2	1	no	—
28	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
29	C₂×#15667924	—	2	1	no	—
30	#19151346	C₂²	1	4	yes	complemented
31	#19151345	C₂²	1	4	yes	complemented
32	#431×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#335×C₈₉	C₂²	1	2	yes	complemented
35	#334×C₈₉	C₂²	1	2	yes	complemented
36	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₈₉	—	2	1	no	—
38	C₂×#13083563	—	2	4	no	—
39	C₂²×#10623494	—	2	1	no	—
40	C₂×C₈×D₈₉	—	2	2	no	—
41	C₂×C₈×D₈₉	—	2	2	no	—
42	C₂×C₈×D₈₉	—	4	4	no	—
43	#15668012	C₂³	1	2	yes	—
44	#15668012	C₂³	1	2	yes	—
45	#15668004	—	2	2	no	—
46	#15667991	—	2	2	no	—
47	#15667991	—	2	2	no	—
48	C₄×Dic₁₇₈	—	2	2	no	—
49	C₄×Dic₁₇₈	—	2	2	no	—
50	C₄×D₃₅₆	—	2	1	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	C₄×D₃₅₆	D₄	1	2	yes	—
53	C₄×D₃₅₆	C₂³	1	2	yes	—
54	#15667971	—	2	2	no	—
55	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
56	C₂×#13083551	—	2	1	no	—
57	#15667946	—	2	1	no	—
58	#15667946	—	2	1	no	—
59	#15667946	—	2	1	no	—
60	#15667946	—	2	1	no	—
61	#15667944	—	2	4	no	—
62	#15667944	—	2	2	no	—
63	#15667944	—	2	2	no	—
64	C₄⋊#10623494	—	2	1	no	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	#15667937	—	2	4	no	—
67	#15667934	—	2	4	no	—
68	#15667924	—	2	2	no	—
69	#15667924	—	2	2	no	—
70	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
74	C₂²⋊C₈×C₈₉	—	2	2	no	—
75	C₄×C₈×C₈₉	—	2	2	no	—
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂³×D₈₉	—	2	1	no	—
81	C₂×Dic₁₇₈	—	2	2	no	—
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×D₃₅₆	—	2	4	no	—
85	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
86	C₂²×Dic₈₉	—	2	1	no	—
87	C₂×C₄×D₈₉	—	2	2	no	—
88	C₂×C₄×D₈₉	—	2	2	no	—
89	C₂×C₄×D₈₉	—	4	2	no	—
90	C₂×C₄×D₈₉	—	4	2	no	—
91	#13083563	—	4	8	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	4	2	no	—
94	C₈×D₈₉	—	4	4	no	—
95	C₈×D₈₉	—	4	4	no	—
96	C₄⋊Dic₈₉	—	2	1	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	2	2	no	—
101	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
102	#13083553	—	4	2	no	—
103	#13083553	—	4	2	no	—
104	C₂²⋊Dic₈₉	—	4	2	no	—
105	#13083551	—	2	2	no	—
106	#13083551	—	2	2	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	4	no	—
113	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	#41	1	1	yes	characteristic
121	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂³×C₈₉	#121	1	1	yes	characteristic
123	C₂²×D₈₉	—	2	4	no	—
124	C₂²×D₈₉	—	2	2	no	—
125	C₂²×D₈₉	—	4	2	no	—
126	C₂²×D₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₈₉	#137	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	D₃₅₆	—	2	4	no	—
142	D₃₅₆	—	2	2	no	—
143	D₃₅₆	#135	1	4	yes	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	—	4	2	no	—
146	C₄×D₈₉	—	4	4	no	—
147	Dic₁₇₈	—	4	4	no	—
148	Dic₁₇₈	—	8	2	no	—
149	C₈×C₈₉	—	2	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	4	4	no	—
152	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
153	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
154	#10623494	—	4	2	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	#461	1	2	yes	—
158	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
159	C₂×D₈₉	—	2	4	no	—
160	C₂×D₈₉	—	4	8	no	—
161	C₂×D₈₉	—	4	2	no	—
162	C₂×D₈₉	—	4	2	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	#461	1	2	yes	—
167	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#135	1	2	yes	—
169	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	Dic₈₉	—	8	2	no	—
172	Dic₈₉	—	8	2	no	—
173	#19105	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	—	2	2	no	—
175	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
176	C₂×C₈₉	#3050	1	1	yes	characteristic
177	C₂×C₈₉	#3067	1	1	yes	characteristic
178	D₈₉	—	4	4	no	—
179	#3160	—	89	1	no	—
180	#3101	—	89	2	no	—
181	C₂×C₈×D₄	—	89	1	no	—
182	#1675	—	89	2	no	—
183	#1673	—	89	2	no	—
184	#1671	—	89	4	no	—
185	#1564	D₈₉	1	1	yes	characteristic, complemented
186	#1542	—	89	2	no	—
187	C₈₉	#19105	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₈	—	178	1	no	—
189	#533	—	89	2	no	—
190	C₂×C₄×D₄	—	89	1	no	—
191	#500	—	89	1	no	—
192	#500	—	89	1	no	—
193	#499	—	178	1	no	—
194	#499	—	178	1	no	—
195	#498	—	89	2	no	—
196	#483	—	178	2	no	—
197	#476	—	89	4	no	—
198	#461	—	89	4	no	—
199	#450	—	178	2	no	—
200	C₄×SD₁₆	—	178	4	no	—
201	C₈×D₄	—	89	2	no	—
202	C₈×D₄	—	89	2	no	—
203	C₈×D₄	—	178	1	no	—
204	C₈×D₄	—	178	1	no	—
205	#431	C₂×D₈₉	1	2	yes	—
206	#427	—	178	1	no	—
207	#427	—	178	1	no	—
208	C₂×C₄⋊C₈	—	89	1	no	—
209	C₂×C₂²⋊C₈	—	178	1	no	—
210	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
211	#335	C₂×D₈₉	1	2	yes	complemented
212	#334	C₂×D₈₉	1	2	yes	complemented
213	#333	—	89	4	no	—
214	#332	—	89	4	no	—
215	C₂²×D₄	—	89	1	no	—
216	C₂³×C₄	—	178	1	no	—
217	C₂×SD₁₆	—	178	4	no	—
218	C₂²×C₈	—	356	4	no	—
219	C₂²×C₈	—	178	1	no	—
220	C₂²×C₈	—	178	2	no	—
221	C₂²×C₈	—	178	2	no	—
222	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
223	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
224	C₄⋊Q₈	—	89	2	no	—
225	C₄⋊Q₈	—	89	2	no	—
226	#125	—	178	2	no	—
227	C₂²⋊Q₈	—	178	2	no	—
228	C₂²⋊Q₈	—	178	2	no	—
229	C₄×Q₈	—	178	2	no	—
230	C₄×Q₈	—	178	2	no	—
231	C₄×D₄	—	89	2	no	—
232	C₄×D₄	—	89	2	no	—
233	C₄×D₄	—	178	1	no	—
234	C₄×D₄	—	178	1	no	—
235	#117	—	178	2	no	—
236	C₂×C₄⋊C₄	—	89	1	no	—
237	C₂×C₂²⋊C₄	—	178	1	no	—
238	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
239	#107	—	178	1	no	—
240	#107	—	178	1	no	—
241	#107	—	178	1	no	—
242	#107	—	178	1	no	—
243	#106	—	178	4	no	—
244	#106	—	178	2	no	—
245	#106	—	178	2	no	—
246	C₄⋊C₈	—	89	2	no	—
247	C₄⋊C₈	—	178	1	no	—
248	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
249	#103	—	178	4	no	—
250	#102	—	178	4	no	—
251	C₂²⋊C₈	—	2	2	no	—
252	C₂²⋊C₈	—	178	2	no	—
253	C₂²⋊C₈	—	178	2	no	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	D₃₅₆	1	1	yes	characteristic
256	C₄×C₈	D₃₅₆	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₂⁴	—	178	1	no	—
260	C₂×Q₈	—	178	2	no	—
261	C₂×Q₈	—	178	2	no	—
262	C₂×D₄	—	89	2	no	—
263	C₂×D₄	—	178	4	no	—
264	C₂×D₄	—	178	1	no	—
265	C₂²×C₄	—	356	2	no	—
266	C₂²×C₄	—	356	2	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	2	no	—
269	C₂²×C₄	—	178	2	no	—
270	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
272	C₂²×C₄	#13083575	1	1	yes	characteristic
273	SD₁₆	—	356	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	356	2	no	—
277	C₂×C₈	—	356	4	no	—
278	C₂×C₈	—	356	4	no	—
279	C₂×C₈	—	178	2	no	—
280	C₂×C₈	#13083563	1	4	yes	complemented
281	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
282	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
283	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
284	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
285	C₄⋊C₄	—	89	2	no	—
286	C₄⋊C₄	—	356	2	no	—
287	C₄⋊C₄	—	356	2	no	—
288	C₄⋊C₄	—	178	2	no	—
289	C₄⋊C₄	—	178	2	no	—
290	C₄⋊C₄	—	178	2	no	—
291	C₄⋊C₄	—	178	2	no	—
292	C₄⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	356	2	no	—
294	C₂²⋊C₄	—	178	2	no	—
295	C₂²⋊C₄	—	178	2	no	—
296	C₄²	—	356	2	no	—
297	C₄²	D₄×D₈₉	1	1	yes	characteristic
298	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
299	C₄²	#13083575	1	1	yes	characteristic
300	C₄²	C₂³×D₈₉	1	1	yes	characteristic
301	C₂³	—	356	2	no	—
302	C₂³	—	356	2	no	—
303	C₂³	—	178	4	no	—
304	C₂³	—	178	2	no	—
305	C₂³	#15667987	1	1	yes	characteristic
306	Q₈	—	356	4	no	—
307	Q₈	—	712	2	no	—
308	D₄	—	89	4	no	—
309	D₄	—	178	4	no	—
310	D₄	—	178	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	356	4	no	—
316	C₂×C₄	—	356	2	no	—
317	C₂×C₄	—	356	2	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	178	2	no	—
320	C₂×C₄	#15667987	1	1	yes	characteristic
321	C₂×C₄	#15667987	1	1	yes	characteristic
322	C₂×C₄	#15667987	1	1	yes	characteristic
323	C₂×C₄	C₂×#13083563	1	2	yes	—
324	C₂×C₄	#15668045	1	2	yes	—
325	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
327	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	4	4	no	—
331	C₈	—	356	2	no	—
332	C₈	C₂×#13083563	1	2	yes	—
333	C₈	C₂×#13083563	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	356	8	no	—
337	C₂²	—	356	2	no	—
338	C₂²	—	356	2	no	—
339	C₂²	—	178	4	no	—
340	C₂²	#19151804	1	2	yes	—
341	C₂²	C₂×#15667987	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	356	2	no	—
346	C₄	—	712	2	no	—
347	C₄	—	712	2	no	—
348	C₄	#19151804	1	2	yes	—
349	C₄	C₂²×#13083563	1	1	yes	characteristic
350	C₄	#135×D₈₉	1	2	yes	—
351	C₄	C₂×#15668045	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	356	4	no	—
354	C₂	C₂×#19151804	1	1	yes	characteristic
355	C₂	#26029834	1	1	yes	characteristic
356	C₂	#26029973	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#19105
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #26030766	C₂	semidirect
4. #26019652	C₂	semidirect
5. #26019642	C₂	semidirect
6. #26019628	C₂	semidirect
7. #26018907	C₂	semidirect
8. C₈₉×#1564	C₂	semidirect
12. #19152003	C₂²	semidirect
13. #19152003	C₂²	semidirect
16. #19151996	C₂²	semidirect
30. #19151346	C₂²	semidirect
31. #19151345	C₂²	semidirect
34. #335×C₈₉	C₂²	semidirect
35. #334×C₈₉	C₂²	semidirect
113. C₂×C₈×C₈₉	SD₁₆	semidirect
185. #1564	D₈₉	semidirect
187. C₈₉	#19105	semidirect
211. #335	C₂×D₈₉	semidirect
212. #334	C₂×D₈₉	semidirect
280. C₂×C₈	#13083563	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26030261	2
C₂×C₈×D₃₅₆	2
#26019652	2
#26019642	2
#26019628	2
#26018907	2
C₈₉×#1564	2
#19105	89