Subgroups — #26029244

Order 11392.

372 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	#19152126	C₂	1	1	yes	maximal, characteristic
2	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151725	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151672	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151672	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151668	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151666	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151663	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151661	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151660	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151554	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151552	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083558	C₂²	1	1	yes	characteristic
17	C₂×#13083557	C₂²	1	1	yes	characteristic
18	C₂²×#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	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	C₄	1	1	yes	characteristic
23	#15668005	C₄	1	1	yes	characteristic
24	#15668002	C₄	1	1	yes	characteristic
25	#15667998	C₄	1	1	yes	characteristic
26	#15667997	C₄	1	1	yes	characteristic
27	#15667994	C₄	1	1	yes	characteristic
28	#15667990	C₄	1	1	yes	characteristic
29	#15667985	C₄	1	1	yes	characteristic
30	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
34	#15667969	C₂²	1	1	yes	characteristic
35	C₄²×D₈₉	C₂²	1	1	yes	characteristic
36	#15667941	C₂²	1	1	yes	characteristic
37	#15667941	C₂²	1	1	yes	characteristic
38	#15667941	C₂²	1	1	yes	characteristic
39	#15667941	C₂²	1	1	yes	characteristic
40	#15667924	C₂²	1	1	yes	characteristic
41	#15667924	C₂²	1	1	yes	characteristic
42	#15667924	C₂²	1	1	yes	characteristic
43	#15667924	C₂²	1	1	yes	characteristic
44	#15667923	C₂²	1	1	yes	characteristic
45	#15667923	C₂²	1	1	yes	characteristic
46	#15667922	C₂²	1	1	yes	characteristic
47	#15667922	C₂²	1	1	yes	characteristic
48	#15667922	C₂²	1	1	yes	characteristic
49	C₄×#10623494	C₂²	1	1	yes	characteristic
50	C₄×#10623494	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#117×C₈₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#97×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
60	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
67	#13083558	—	2	2	no	—
68	#13083557	—	2	2	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₈×D₈₉	—	2	2	no	—
75	C₈×D₈₉	—	2	2	no	—
76	C₈×D₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
79	#13083553	C₂×C₄	1	1	yes	characteristic
80	#13083553	C₂×C₄	1	1	yes	characteristic
81	#13083553	C₂×C₄	1	1	yes	characteristic
82	#13083553	C₂×C₄	1	1	yes	characteristic
83	#13083553	C₂×C₄	1	1	yes	characteristic
84	#13083553	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
87	#13083551	C₂×C₄	1	1	yes	characteristic
88	#13083551	C₂×C₄	1	1	yes	characteristic
89	#13083551	C₂×C₄	1	1	yes	characteristic
90	#13083551	C₂×C₄	1	1	yes	characteristic
91	#13083551	C₂×C₄	1	1	yes	characteristic
92	#13083551	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×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	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₈₉	—	2	2	no	—
120	C₂²⋊D₈₉	—	4	1	no	—
121	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
127	D₃₅₆	—	2	2	no	—
128	C₄×D₈₉	—	2	2	no	—
129	C₄×D₈₉	—	2	1	no	—
130	C₄×D₈₉	—	2	1	no	—
131	C₄×D₈₉	—	2	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	4	1	no	—
134	Dic₁₇₈	—	4	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	#10623494	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	#10623494	#41	1	2	yes	—
142	#10623494	#41	1	2	yes	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₈₉	—	2	1	no	—
148	C₂×D₈₉	—	2	1	no	—
149	C₂×D₈₉	—	2	1	no	—
150	C₂×D₈₉	—	2	1	no	—
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	#131	1	2	yes	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	C₂×C₈₉	—	4	1	no	—
167	C₂×C₈₉	C₄×#41	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
169	C₂×C₈₉	#568	1	1	yes	characteristic
170	D₈₉	—	2	2	no	—
171	D₈₉	—	4	1	no	—
172	#2968	—	89	1	no	maximal, Sylow
173	C₈₉	#2968	1	1	yes	characteristic, complemented, Sylow
174	#529	—	89	1	no	—
175	#446	—	89	1	no	—
176	#446	—	89	1	no	—
177	#435	—	89	1	no	—
178	#435	—	89	1	no	—
179	#435	—	89	1	no	—
180	#435	—	89	1	no	—
181	C₈×D₄	—	89	1	no	—
182	C₈×D₄	—	89	1	no	—
183	#433	—	89	1	no	—
184	#433	D₈₉	1	1	yes	characteristic, complemented
185	#431	—	89	1	no	—
186	#405	—	89	1	no	—
187	#405	—	89	1	no	—
188	C₂×#97	—	89	1	no	—
189	C₂×M₁₆	—	89	1	no	—
190	C₂×M₁₆	—	89	1	no	—
191	C₂²×C₈	—	89	1	no	—
192	C₂²×C₈	—	89	1	no	—
193	C₂²×C₈	—	89	1	no	—
194	C₂²×C₈	—	89	1	no	—
195	#126	—	89	1	no	—
196	#126	—	89	1	no	—
197	#125	—	89	1	no	—
198	#124	—	89	1	no	—
199	#123	—	89	1	no	—
200	#123	—	89	1	no	—
201	C₂²⋊Q₈	—	89	1	no	—
202	#121	—	89	1	no	—
203	C₄×Q₈	—	89	1	no	—
204	C₄×D₄	—	89	1	no	—
205	C₄×D₄	—	89	1	no	—
206	C₄×D₄	—	89	1	no	—
207	#117	—	89	1	no	—
208	#117	C₂×D₈₉	1	1	yes	characteristic
209	C₂×C₄²	—	89	1	no	—
210	C₄⋊C₈	—	89	1	no	—
211	C₄⋊C₈	—	89	1	no	—
212	C₄⋊C₈	—	89	1	no	—
213	C₄⋊C₈	—	89	1	no	—
214	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
216	C₂²⋊C₈	—	89	1	no	—
217	C₂²⋊C₈	—	89	1	no	—
218	C₂²⋊C₈	—	89	1	no	—
219	C₂²⋊C₈	—	89	1	no	—
220	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
221	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
222	#97	—	89	1	no	—
223	#97	—	89	1	no	—
224	#97	—	89	1	no	—
225	#97	—	89	1	no	—
226	#97	—	89	1	no	—
227	#97	C₂×D₈₉	1	1	yes	characteristic
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	—	89	1	no	—
230	C₄×C₈	—	89	1	no	—
231	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
232	C₂×Q₈	—	89	1	no	—
233	C₂×D₄	—	89	1	no	—
234	C₂×D₄	—	89	1	no	—
235	C₂×D₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	89	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
241	M₁₆	—	178	2	no	—
242	M₁₆	—	178	2	no	—
243	C₂×C₈	—	89	1	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	—	89	1	no	—
246	C₂×C₈	—	89	1	no	—
247	C₂×C₈	—	178	2	no	—
248	C₂×C₈	—	178	2	no	—
249	C₂×C₈	—	178	2	no	—
250	C₂×C₈	—	178	2	no	—
251	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	89	1	no	—
266	C₂²⋊C₄	—	89	1	no	—
267	C₂²⋊C₄	—	89	1	no	—
268	C₂²⋊C₄	—	89	1	no	—
269	C₂²⋊C₄	—	89	1	no	—
270	C₂²⋊C₄	—	89	1	no	—
271	C₂²⋊C₄	—	89	1	no	—
272	C₂²⋊C₄	—	89	1	no	—
273	C₂²⋊C₄	C₄×D₈₉	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₈₉	1	1	yes	characteristic
275	C₄²	—	89	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	89	1	no	—
279	C₄²	C₂²×D₈₉	1	1	yes	characteristic
280	C₄²	C₂²×D₈₉	1	1	yes	characteristic
281	C₂³	—	89	1	no	—
282	C₂³	—	89	1	no	—
283	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
284	Q₈	—	356	1	no	—
285	D₄	—	356	1	no	—
286	D₄	—	178	2	no	—
287	D₄	—	178	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	89	1	no	—
291	C₂×C₄	—	89	1	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	356	1	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	89	2	no	—
313	C₈	—	89	2	no	—
314	C₈	—	178	1	no	—
315	C₈	—	178	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	89	2	no	—
320	C₂²	—	178	1	no	—
321	C₂²	—	178	1	no	—
322	C₂²	—	178	1	no	—
323	C₂²	—	178	1	no	—
324	C₂²	—	178	1	no	—
325	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	178	1	no	—
332	C₄	—	178	1	no	—
333	C₄	—	178	1	no	—
334	C₄	—	178	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	1	no	—
337	C₄	#41×D₈₉	1	1	yes	characteristic
338	C₄	#41×D₈₉	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	356	1	no	—
341	C₂	—	178	2	no	—
342	C₂	#19152066	1	1	yes	characteristic
343	C₂	#131×D₈₉	1	1	yes	characteristic
344	C₂	#19152363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#433×C₈₉	C₂
prime radical	#15667985	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#433	D₈₉
hypocenter	C₈₉	#2968
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #19151727	C₂	semidirect
3. #19151725	C₂	semidirect
4. #19151672	C₂	semidirect
5. #19151672	C₂	semidirect
6. #19151668	C₂	semidirect
7. #19151666	C₂	semidirect
8. #19151663	C₂	semidirect
9. #19151661	C₂	semidirect
10. #19151660	C₂	semidirect
11. #19151649	C₂	semidirect
12. #19151554	C₂	semidirect
13. #19151552	C₂	semidirect
14. #97×D₈₉	C₂	semidirect
15. #433×C₈₉	C₂	semidirect
173. C₈₉	#2968	semidirect
184. #433	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152126	2
#19151727	2
#19151725	2
#19151672	2
#19151668	2
#19151666	2
#19151663	2
#19151661	2
#19151660	2
#19151649	2
#19151554	2
#19151552	2
#97×D₈₉	2
#433×C₈₉	2
#2968	89