Subgroups — #36709144

Order 22784.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029129	C₂	1	1	yes	maximal, characteristic
2	#26022413	C₂	1	2	yes	maximal, complemented
3	#26022406	C₂	1	2	yes	maximal, complemented
4	#26022396	C₂	1	2	yes	maximal, complemented
5	#26022391	C₂	1	2	yes	maximal, complemented
6	#26022041	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151425	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022011	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021962	C₂	1	1	yes	maximal, characteristic
10	#26021957	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₂²	1	1	yes	characteristic
13	#19152078	—	2	1	no	—
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂²×#13083549	C₂²	1	1	yes	characteristic
16	#19151700	—	2	2	no	—
17	C₄×Dic₃₅₆	C₂²	1	2	yes	—
18	C₄×D₇₁₂	C₂²	1	2	yes	—
19	C₂×#15667946	C₂²	1	1	yes	characteristic
20	#19151600	—	2	1	no	—
21	#19151425	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151425	C₂²	1	2	yes	—
24	#19151423	C₂²	1	2	yes	—
25	C₄⋊#13083549	C₂²	1	2	yes	—
26	C₄⋊#13083549	C₂²	1	2	yes	—
27	#19151401	C₂²	1	1	yes	characteristic
28	#19151401	C₂²	1	2	yes	—
29	#19151401	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	1	yes	characteristic
31	#19151398	C₂²	1	2	yes	—
32	#19151398	C₂²	1	1	yes	characteristic
33	#544×C₈₉	—	2	1	no	—
34	#501×C₈₉	—	2	2	no	—
35	#500×C₈₉	C₂²	1	2	yes	—
36	#499×C₈₉	C₂²	1	2	yes	—
37	#428×C₈₉	—	2	1	no	—
38	#427×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
40	#405×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	—	2	1	no	—
42	#15668034	—	2	2	no	—
43	#15668034	Q₈	1	4	yes	complemented
44	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
45	C₂×#13083563	—	2	1	no	—
46	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₇₈	—	2	2	no	—
48	C₄×D₃₅₆	—	2	2	no	—
49	#15667972	—	4	2	no	—
50	#15667970	—	4	1	no	—
51	C₂×C₄⋊Dic₈₉	—	2	1	no	—
52	C₄⋊C₄×D₈₉	—	4	1	no	—
53	#15667952	—	2	2	no	—
54	C₂×#13083549	—	2	2	no	—
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	#15667946	—	2	1	no	—
59	#15667946	C₂³	1	2	yes	—
60	#15667937	—	2	2	no	—
61	#15667934	—	2	2	no	—
62	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	—	4	1	no	—
66	#125×C₈₉	—	2	2	no	—
67	#125×C₈₉	—	4	1	no	—
68	C₂²⋊Q₈×C₈₉	—	4	2	no	—
69	#117×C₈₉	—	2	1	no	—
70	#117×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	—	2	2	no	—
74	#107×C₈₉	C₂³	1	2	yes	—
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#106×C₈₉	—	2	1	no	—
78	#106×C₈₉	—	2	2	no	—
79	#97×C₈₉	D₄	1	2	yes	—
80	C₄×C₈×C₈₉	C₂³	1	2	yes	—
81	#13083574	—	2	4	no	—
82	#13083574	—	4	2	no	—
83	C₂×Dic₁₇₈	—	2	1	no	—
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂×C₂²⋊D₈₉	—	4	1	no	—
86	C₂²×Dic₈₉	—	4	1	no	—
87	C₂×C₄×D₈₉	—	4	1	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	Dic₃₅₆	—	2	1	no	—
90	Dic₃₅₆	C₂×Q₈	1	2	yes	—
91	#13083563	—	2	4	no	—
92	D₇₁₂	—	2	1	no	—
93	D₇₁₂	C₂×Q₈	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	#13083553	—	4	2	no	—
97	#13083551	—	4	2	no	—
98	C₄×Dic₈₉	—	4	2	no	—
99	#13083549	—	2	1	no	—
100	#13083549	—	2	1	no	—
101	#13083549	—	2	2	no	—
102	C₂×Q₈×C₈₉	—	4	2	no	—
103	C₂²×C₄×C₈₉	—	4	1	no	—
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	M₁₆×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₈₉	#41	1	2	yes	—
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	2	no	—
116	C₄⋊C₄×C₈₉	—	4	2	no	—
117	C₄⋊C₄×C₈₉	—	4	2	no	—
118	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₈₉	—	4	1	no	—
121	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
122	C₄²×C₈₉	—	4	1	no	—
123	C₄²×C₈₉	C₂×D₄	1	2	yes	—
124	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₈₉	—	4	1	no	—
126	Q₈×C₈₉	—	16	2	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	16	1	no	—
131	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#137	1	2	yes	—
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	4	4	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	16	1	no	—
141	D₃₅₆	—	2	2	no	—
142	D₃₅₆	—	4	1	no	—
143	C₄×D₈₉	—	4	4	no	—
144	C₄×D₈₉	—	16	2	no	—
145	Dic₁₇₈	—	2	2	no	—
146	Dic₁₇₈	—	4	1	no	—
147	C₈×C₈₉	—	2	2	no	—
148	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
149	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	#474	1	2	yes	—
154	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₈₉	—	4	2	no	—
156	C₂×D₈₉	—	8	1	no	—
157	C₂×D₈₉	—	8	1	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	—	8	1	no	—
163	C₄×C₈₉	#474	1	2	yes	—
164	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
166	Dic₈₉	—	4	2	no	—
167	Dic₈₉	—	8	2	no	—
168	#33634	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	2	2	no	—
170	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
171	C₂×C₈₉	#3416	1	1	yes	characteristic
172	C₂×C₈₉	#3416	1	1	yes	characteristic
173	D₈₉	—	8	2	no	—
174	#3164	D₈₉	1	1	yes	characteristic, complemented
175	#2951	—	89	1	no	—
176	#2229	—	89	2	no	—
177	#2228	—	89	2	no	—
178	#2226	—	89	2	no	—
179	#2225	—	89	2	no	—
180	#2159	—	89	1	no	—
181	C₂×#367	—	89	1	no	—
182	#2151	—	89	1	no	—
183	#2139	—	89	1	no	—
184	#2138	—	89	1	no	—
185	C₈₉	#33634	1	1	yes	characteristic, complemented, Sylow
186	C₂×#135	—	89	1	no	—
187	#544	—	2	1	no	—
188	#520	—	178	1	no	—
189	C₂×M₃₂	—	89	1	no	—
190	C₂²×C₁₆	—	89	1	no	—
191	#501	—	2	2	no	—
192	#500	C₂×D₈₉	1	2	yes	—
193	#499	C₂×D₈₉	1	2	yes	—
194	#440	—	178	2	no	—
195	C₄×Q₁₆	—	89	2	no	—
196	C₄×D₈	—	89	2	no	—
197	#428	—	2	1	no	—
198	#427	C₂×D₈₉	1	1	yes	characteristic
199	C₂×#107	—	89	1	no	—
200	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
201	#417	—	178	1	no	—
202	#405	C₂×D₈₉	1	1	yes	characteristic
203	#367	—	89	2	no	—
204	#367	—	89	2	no	—
205	#367	—	89	2	no	—
206	#366	—	89	2	no	—
207	C₄⋊C₁₆	—	89	2	no	—
208	C₄⋊C₁₆	—	89	2	no	—
209	#358	—	89	1	no	—
210	#358	—	89	1	no	—
211	#358	—	89	2	no	—
212	#357	—	89	1	no	—
213	#357	—	89	1	no	—
214	#357	—	89	2	no	—
215	C₂×#41	—	178	1	no	—
216	#135	—	89	4	no	—
217	#135	—	178	2	no	—
218	C₂×Q₁₆	—	89	1	no	—
219	C₂×SD₁₆	—	178	1	no	—
220	C₂×D₈	—	89	1	no	—
221	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	4	1	no	—
225	#125	—	2	2	no	—
226	#125	—	4	1	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	C₄×Q₈	—	178	2	no	—
229	C₄×D₄	—	356	2	no	—
230	C₄×D₄	—	178	2	no	—
231	#117	—	2	1	no	—
232	#117	—	356	1	no	—
233	#117	C₂²⋊D₈₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	356	1	no	—
236	C₂×C₄⋊C₄	—	178	1	no	—
237	M₃₂	—	178	2	no	—
238	C₂×C₁₆	—	89	2	no	—
239	C₂×C₁₆	—	89	1	no	—
240	C₂×C₁₆	—	89	1	no	—
241	C₂×C₁₆	—	178	2	no	—
242	#107	—	2	1	no	—
243	#107	—	2	2	no	—
244	#107	—	89	2	no	—
245	#107	—	178	1	no	—
246	#107	C₂²×D₈₉	1	1	yes	characteristic
247	#107	C₂²×D₈₉	1	2	yes	—
248	#107	C₂²×D₈₉	1	1	yes	characteristic
249	#106	—	2	1	no	—
250	#106	—	2	2	no	—
251	#103	—	178	2	no	—
252	#102	—	178	2	no	—
253	#97	C₂²⋊D₈₉	1	2	yes	—
254	C₄×C₈	C₂²×D₈₉	1	2	yes	—
255	#41	—	356	2	no	—
256	#41	—	178	4	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	178	1	no	—
259	C₂×D₄	—	356	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	356	1	no	—
263	C₂²×C₄	—	356	1	no	—
264	C₂²×C₄	—	356	2	no	—
265	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	Q₁₆	—	89	2	no	—
267	Q₁₆	—	178	1	no	—
268	SD₁₆	—	178	4	no	—
269	D₈	—	89	2	no	—
270	D₈	—	178	1	no	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₈₉	1	2	yes	—
275	C₂×C₈	#13083575	1	2	yes	—
276	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	356	1	no	—
285	C₄⋊C₄	—	356	2	no	—
286	C₄⋊C₄	—	178	2	no	—
287	C₄⋊C₄	#13083559	1	1	yes	characteristic
288	C₄⋊C₄	#13083559	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	356	2	no	—
291	C₂²⋊C₄	#13083559	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	356	2	no	—
294	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
295	C₁₆	—	178	1	no	—
296	C₁₆	—	178	1	no	—
297	C₁₆	—	178	2	no	—
298	C₂³	—	356	1	no	—
299	C₂³	C₂×#13083559	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	Q₈	—	356	1	no	—
302	Q₈	—	178	2	no	—
303	D₄	—	356	4	no	—
304	D₄	—	356	1	no	—
305	D₄	—	178	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	356	1	no	—
311	C₂×C₄	—	356	4	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	1424	2	no	—
314	C₂×C₄	—	1424	1	no	—
315	C₂×C₄	#15667988	1	2	yes	—
316	C₂×C₄	C₂×#13083559	1	2	yes	—
317	C₂×C₄	C₂×#13083559	1	2	yes	—
318	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
319	C₂×C₄	#15668043	1	2	yes	—
320	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#15667988	1	2	yes	—
323	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
324	C₈	C₂×#13083575	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	356	2	no	—
328	C₂²	—	712	1	no	—
329	C₂²	—	712	1	no	—
330	C₂²	#19151875	1	2	yes	—
331	C₂²	C₂²×#13083559	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	356	2	no	—
338	C₄	—	712	2	no	—
339	C₄	#19151875	1	2	yes	—
340	C₄	C₂×#15667988	1	1	yes	characteristic
341	C₄	C₂×#15668043	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	712	2	no	—
344	C₂	C₂×#19151875	1	1	yes	characteristic
345	C₂	#26033396	1	1	yes	characteristic
346	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3164	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15668034	C₂²
hypercenter	#3164	D₈₉
hypocenter	C₈₉	#33634
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #26022413	C₂	semidirect
3. #26022406	C₂	semidirect
4. #26022396	C₂	semidirect
5. #26022391	C₂	semidirect
6. #26022041	C₂	semidirect
7. C₂×#19151425	C₂	semidirect
8. #26022011	C₂	semidirect
11. C₈₉×#3164	C₂	semidirect
43. #15668034	Q₈	semidirect
174. #3164	D₈₉	semidirect
185. C₈₉	#33634	semidirect

Maximal subgroups

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

subgroup	index
#26029129	2
#26022413	2
#26022406	2
#26022396	2
#26022391	2
#26022041	2
C₂×#19151425	2
#26022011	2
#26021962	2
#26021957	2
C₈₉×#3164	2
#33634	89