Subgroups — #26033306

Order 11392.

362 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	#19152331	C₂	1	1	yes	maximal, characteristic
2	#19152326	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151992	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151989	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151969	C₂	1	1	yes	maximal, characteristic
6	#19151820	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151819	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151803	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151797	C₂	1	1	yes	maximal, characteristic
12	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151692	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083563	C₂²	1	1	yes	characteristic
21	C₂×#13083562	C₂²	1	1	yes	characteristic
22	C₂×#13083562	C₂²	1	1	yes	characteristic
23	C₂×#13083561	C₂²	1	1	yes	characteristic
24	#15668012	C₂²	1	1	yes	characteristic
25	#15668010	C₂²	1	1	yes	characteristic
26	#15668010	C₂²	1	1	yes	characteristic
27	#15668008	C₂²	1	1	yes	characteristic
28	#15668004	—	2	1	no	—
29	#15668002	—	2	1	no	—
30	#15668002	—	2	1	no	—
31	#15668000	C₂²	1	1	yes	characteristic
32	#15667999	C₂²	1	1	yes	characteristic
33	#15667998	—	2	1	no	—
34	#15667995	—	2	1	no	—
35	#15667994	—	2	1	no	—
36	C₄×Dic₁₇₈	—	2	1	no	—
37	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
38	#15667977	C₂²	1	1	yes	characteristic
39	#15667977	C₂²	1	1	yes	characteristic
40	#15667977	C₂²	1	1	yes	characteristic
41	#15667974	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊D₈₉	—	2	1	no	—
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	Dic₃₅₆	—	2	2	no	—
68	Dic₃₅₆	—	4	1	no	—
69	#13083564	—	4	1	no	—
70	#13083563	—	2	2	no	—
71	#13083563	—	4	1	no	—
72	#13083562	—	4	1	no	—
73	#13083562	—	4	1	no	—
74	#13083561	—	4	1	no	—
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	D₄	1	1	yes	characteristic
88	#13083553	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	SD₁₆×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	Q₈×C₈₉	—	2	1	no	—
114	Q₈×C₈₉	—	2	1	no	—
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	8	1	no	—
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₈₉	—	4	2	no	—
125	C₂²⋊D₈₉	—	8	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	C₄×D₈₉	—	8	1	no	—
136	Dic₁₇₈	—	2	2	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	—	2	1	no	—
139	Dic₁₇₈	—	2	1	no	—
140	Dic₁₇₈	—	2	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	#41	1	2	yes	—
145	#10623494	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₉	H(5,𝔽₂)	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	Dic₈₉	—	4	1	no	—
167	Dic₈₉	#137	1	2	yes	—
168	C₂×C₈₉	—	8	1	no	—
169	C₂×C₈₉	#550	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
171	C₂×C₈₉	#577	1	1	yes	characteristic
172	D₈₉	—	8	1	no	—
173	#3400	—	89	1	no	maximal, Sylow
174	C₈₉	#3400	1	1	yes	characteristic, complemented, Sylow
175	#559	—	89	1	no	—
176	#557	—	89	1	no	—
177	#497	—	89	1	no	—
178	#497	—	89	1	no	—
179	#492	—	89	1	no	—
180	#464	—	89	1	no	—
181	#464	—	89	1	no	—
182	#462	—	89	1	no	—
183	#461	—	89	1	no	—
184	#461	—	89	1	no	—
185	#460	—	89	1	no	—
186	#446	—	89	1	no	—
187	#440	—	89	1	no	—
188	#440	D₈₉	1	1	yes	characteristic, complemented
189	C₄×Q₁₆	—	89	1	no	—
190	C₂×Q₁₆	—	89	1	no	—
191	C₂×Q₁₆	—	89	1	no	—
192	C₂×Q₁₆	—	89	1	no	—
193	C₂×SD₁₆	—	89	1	no	—
194	C₂×SD₁₆	—	89	1	no	—
195	C₂×SD₁₆	—	89	1	no	—
196	C₂×SD₁₆	—	89	1	no	—
197	C₂×SD₁₆	—	89	1	no	—
198	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	89	1	no	—
200	C₄⋊Q₈	—	89	1	no	—
201	C₄⋊Q₈	—	89	1	no	—
202	#127	—	89	1	no	—
203	#125	—	178	1	no	—
204	#125	—	178	1	no	—
205	#125	—	178	1	no	—
206	#124	—	89	1	no	—
207	#124	—	89	1	no	—
208	#124	—	178	1	no	—
209	#123	—	178	1	no	—
210	#123	—	178	1	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×Q₈	—	89	1	no	—
213	C₄×Q₈	—	89	1	no	—
214	C₄×Q₈	—	89	1	no	—
215	C₄×Q₈	—	178	1	no	—
216	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	178	1	no	—
219	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
220	#107	C₂×D₈₉	1	1	yes	characteristic
221	C₄⋊C₈	—	89	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₄⋊C₈	—	89	1	no	—
224	#103	—	89	1	no	—
225	#103	—	89	1	no	—
226	#103	—	89	1	no	—
227	#103	C₂×D₈₉	1	1	yes	characteristic
228	#102	—	89	1	no	—
229	#102	C₂×D₈₉	1	1	yes	characteristic
230	#97	—	89	1	no	—
231	#97	C₂×D₈₉	1	1	yes	characteristic
232	C₄×C₈	—	89	1	no	—
233	C₂×Q₈	—	89	1	no	—
234	C₂×Q₈	—	89	1	no	—
235	C₂×Q₈	—	89	1	no	—
236	C₂×Q₈	—	89	1	no	—
237	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×D₄	—	89	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂×D₄	—	178	1	no	—
241	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	178	1	no	—
244	Q₁₆	—	356	1	no	—
245	Q₁₆	—	356	1	no	—
246	Q₁₆	—	178	2	no	—
247	SD₁₆	—	2	2	no	—
248	SD₁₆	—	356	1	no	—
249	SD₁₆	—	356	1	no	—
250	SD₁₆	—	356	1	no	—
251	SD₁₆	—	356	1	no	—
252	SD₁₆	—	178	2	no	—
253	C₂×C₈	—	89	1	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	89	1	no	—
280	C₄²	—	89	1	no	—
281	C₄²	—	178	1	no	—
282	C₄²	—	178	1	no	—
283	C₄²	—	178	1	no	—
284	C₄²	C₂²×D₈₉	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	—	178	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	178	2	no	—
290	Q₈	—	178	1	no	—
291	Q₈	—	178	1	no	—
292	Q₈	—	178	1	no	—
293	Q₈	—	178	1	no	—
294	Q₈	—	178	1	no	—
295	Q₈	—	178	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	356	2	no	—
299	D₄	—	178	1	no	—
300	D₄	—	178	1	no	—
301	D₄	—	712	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	89	1	no	—
307	C₂×C₄	—	89	1	no	—
308	C₂×C₄	—	89	1	no	—
309	C₂×C₄	—	89	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	712	1	no	—
314	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₈	#13083574	1	2	yes	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	356	1	no	—
325	C₂²	—	356	1	no	—
326	C₂²	—	356	1	no	—
327	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	89	2	no	—
334	C₄	—	356	1	no	—
335	C₄	—	356	1	no	—
336	C₄	—	356	1	no	—
337	C₄	—	178	1	no	—
338	C₄	—	178	1	no	—
339	C₄	—	178	1	no	—
340	C₄	C₂×#13083574	1	1	yes	characteristic
341	C₄	#15668058	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	—	712	1	no	—
344	C₂	#19152269	1	1	yes	characteristic
345	C₂	#137×D₈₉	1	1	yes	characteristic
346	C₂	#19152408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#440×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15668008	C₂²
hypercenter	#440	D₈₉
hypocenter	C₈₉	#3400
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #19152326	C₂	semidirect
3. #19151992	C₂	semidirect
4. #19151989	C₂	semidirect
6. #19151820	C₂	semidirect
7. #19151819	C₂	semidirect
8. #19151807	C₂	semidirect
9. #19151803	C₂	semidirect
10. #19151802	C₂	semidirect
12. #19151726	C₂	semidirect
13. #19151697	C₂	semidirect
14. #19151692	C₂	semidirect
15. #440×C₈₉	C₂	semidirect
174. C₈₉	#3400	semidirect
188. #440	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152331	2
#19152326	2
#19151992	2
#19151989	2
#19151969	2
#19151820	2
#19151819	2
#19151807	2
#19151803	2
#19151802	2
#19151797	2
#19151726	2
#19151697	2
#19151692	2
#440×C₈₉	2
#3400	89