Subgroups — #26031606

Order 11392.

360 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152135	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152133	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151904	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151835	C₂	1	1	yes	maximal, characteristic
8	#19151830	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151828	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151823	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151820	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#479×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668007	—	2	1	no	—
23	#15668007	—	2	1	no	—
24	#15668003	C₂²	1	1	yes	characteristic
25	#15668001	C₂²	1	1	yes	characteristic
26	#15668000	C₂²	1	1	yes	characteristic
27	#15667997	—	2	1	no	—
28	#15667997	—	2	1	no	—
29	#15667992	C₂²	1	1	yes	characteristic
30	#15667988	C₂²	1	1	yes	characteristic
31	#15667987	C₂²	1	1	yes	characteristic
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667977	C₂²	1	1	yes	characteristic
34	C₄×D₃₅₆	—	2	1	no	—
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	#15667974	—	2	1	no	—
37	#15667974	C₂²	1	1	yes	characteristic
38	#15667970	—	2	1	no	—
39	#15667969	—	2	1	no	—
40	C₄²×D₈₉	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667932	C₂²	1	1	yes	characteristic
51	#15667932	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
67	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
69	#13083564	—	4	1	no	—
70	#13083563	—	4	1	no	—
71	#13083561	—	4	1	no	—
72	D₇₁₂	—	4	1	no	—
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₈×D₈₉	—	4	1	no	—
76	C₈×D₈₉	—	4	1	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂²×D₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	D₃₅₆	—	2	1	no	—
126	D₃₅₆	—	2	1	no	—
127	D₃₅₆	—	2	1	no	—
128	D₃₅₆	—	2	1	no	—
129	D₃₅₆	—	4	2	no	—
130	D₃₅₆	—	4	2	no	—
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	4	1	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	—	4	1	no	—
147	C₂×D₈₉	—	4	1	no	—
148	C₂×D₈₉	—	4	1	no	—
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	#135	1	2	yes	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
160	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	C₂×C₈₉	#537	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
168	C₂×C₈₉	#575	1	1	yes	characteristic
169	D₈₉	—	2	2	no	—
170	D₈₉	—	8	1	no	—
171	D₈₉	—	8	1	no	—
172	#3250	—	89	1	no	maximal, Sylow
173	C₈₉	#3250	1	1	yes	characteristic, complemented, Sylow
174	#529	—	89	1	no	—
175	#529	—	89	1	no	—
176	#479	—	89	1	no	—
177	#479	D₈₉	1	1	yes	characteristic, complemented
178	#478	—	89	1	no	—
179	#478	—	89	1	no	—
180	#467	—	89	1	no	—
181	#466	—	89	1	no	—
182	#465	—	89	1	no	—
183	#465	—	89	1	no	—
184	#464	—	89	1	no	—
185	#464	—	89	1	no	—
186	#463	—	89	1	no	—
187	#463	—	89	1	no	—
188	C₂×C₄⋊C₈	—	89	1	no	—
189	C₂×Q₁₆	—	89	1	no	—
190	C₂×SD₁₆	—	89	1	no	—
191	C₂×SD₁₆	—	89	1	no	—
192	C₂×D₈	—	89	1	no	—
193	C₂²×C₈	—	89	1	no	—
194	C₂²×C₈	—	89	1	no	—
195	#126	—	178	1	no	—
196	#126	—	178	1	no	—
197	#125	—	89	1	no	—
198	#125	C₂×D₈₉	1	1	yes	characteristic
199	#124	—	89	1	no	—
200	#124	—	89	1	no	—
201	#123	—	178	1	no	—
202	#123	—	178	1	no	—
203	C₂²⋊Q₈	—	89	1	no	—
204	C₂²⋊Q₈	—	89	1	no	—
205	#121	—	89	1	no	—
206	#121	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
209	C₄×D₄	—	89	1	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	—	178	1	no	—
212	C₄×D₄	—	178	1	no	—
213	#117	—	178	1	no	—
214	#117	—	178	1	no	—
215	C₂×C₄²	—	89	1	no	—
216	#107	—	89	1	no	—
217	#107	C₂×D₈₉	1	1	yes	characteristic
218	#106	—	89	1	no	—
219	#106	C₂×D₈₉	1	1	yes	characteristic
220	C₄⋊C₈	—	89	1	no	—
221	C₄⋊C₈	—	89	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
224	#103	—	89	1	no	—
225	#103	—	89	1	no	—
226	#103	C₂×D₈₉	1	1	yes	characteristic
227	#103	C₂×D₈₉	1	1	yes	characteristic
228	#102	—	89	1	no	—
229	#102	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	—	89	1	no	—
232	C₂×Q₈	—	89	1	no	—
233	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×D₄	—	89	1	no	—
235	C₂×D₄	—	89	1	no	—
236	C₂×D₄	—	178	1	no	—
237	C₂×D₄	—	178	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	178	1	no	—
242	C₂²×C₄	—	178	1	no	—
243	Q₁₆	—	356	1	no	—
244	SD₁₆	—	356	1	no	—
245	SD₁₆	—	356	1	no	—
246	D₈	—	356	1	no	—
247	C₂×C₈	—	89	1	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	356	1	no	—
250	C₂×C₈	—	356	1	no	—
251	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₂²⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	89	1	no	—
279	C₄²	—	178	1	no	—
280	C₄²	C₂²×D₈₉	1	1	yes	characteristic
281	C₂³	—	89	1	no	—
282	C₂³	—	178	1	no	—
283	C₂³	—	178	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	178	1	no	—
287	Q₈	—	178	1	no	—
288	D₄	—	356	2	no	—
289	D₄	—	356	2	no	—
290	D₄	—	178	1	no	—
291	D₄	—	178	1	no	—
292	D₄	—	178	1	no	—
293	D₄	—	178	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	1	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	178	1	no	—
315	C₈	—	178	1	no	—
316	C₂²	—	89	2	no	—
317	C₂²	—	356	1	no	—
318	C₂²	—	356	1	no	—
319	C₂²	—	356	1	no	—
320	C₂²	—	356	1	no	—
321	C₂²	—	356	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	178	1	no	—
324	C₂²	—	178	1	no	—
325	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	356	1	no	—
333	C₄	—	356	1	no	—
334	C₄	—	178	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	1	no	—
337	C₄	#15668058	1	1	yes	characteristic
338	C₄	#15668060	1	1	yes	characteristic
339	C₂	—	178	2	no	—
340	C₂	—	712	1	no	—
341	C₂	—	712	1	no	—
342	C₂	#19152178	1	1	yes	characteristic
343	C₂	#135×D₈₉	1	1	yes	characteristic
344	C₂	#19152397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#479×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151835	C₂
hypercenter	#479	D₈₉
hypocenter	C₈₉	#3250
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #19152135	C₂	semidirect
2. #19152133	C₂	semidirect
3. #19151904	C₂	semidirect
4. #19151897	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151838	C₂	semidirect
8. #19151830	C₂	semidirect
9. #19151828	C₂	semidirect
10. #19151823	C₂	semidirect
11. #19151820	C₂	semidirect
12. #19151816	C₂	semidirect
13. #19151814	C₂	semidirect
14. C₄⋊C₈×D₈₉	C₂	semidirect
15. #479×C₈₉	C₂	semidirect
173. C₈₉	#3250	semidirect
177. #479	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152135	2
#19152133	2
#19151904	2
#19151897	2
#19151895	2
#19151838	2
#19151835	2
#19151830	2
#19151828	2
#19151823	2
#19151820	2
#19151816	2
#19151814	2
C₄⋊C₈×D₈₉	2
#479×C₈₉	2
#3250	89