Subgroups — #26033269

Order 11392.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152317	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152268	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151986	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151969	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151811	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151810	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151796	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151795	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151699	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	—	2	1	no	—
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×#13083559	C₂²	1	1	yes	characteristic
22	#15668010	C₂²	1	1	yes	characteristic
23	#15668008	C₂²	1	1	yes	characteristic
24	#15668008	C₂²	1	1	yes	characteristic
25	#15668005	—	2	1	no	—
26	#15668005	—	2	1	no	—
27	#15667999	C₂²	1	1	yes	characteristic
28	#15667989	—	2	1	no	—
29	#15667984	—	2	1	no	—
30	#15667984	—	2	1	no	—
31	#15667983	—	2	1	no	—
32	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
38	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
39	#15667968	—	2	1	no	—
40	#15667946	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667933	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	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
56	#127×C₈₉	C₂²	1	1	yes	characteristic
57	#124×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	4	4	no	—
60	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	#13083563	—	2	2	no	—
72	#13083562	—	4	1	no	—
73	#13083562	—	4	1	no	—
74	#13083561	—	4	1	no	—
75	#13083559	—	4	1	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	D₄	1	1	yes	characteristic
86	#13083553	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₉	—	2	1	no	—
99	C₂×D₄×C₈₉	—	2	1	no	—
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
103	SD₁₆×C₈₉	—	2	2	no	—
104	SD₁₆×C₈₉	—	4	1	no	—
105	D₈×C₈₉	—	2	2	no	—
106	D₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	—	2	1	no	—
114	C₂³×C₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	—	2	1	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	2	1	no	—
122	D₄×C₈₉	—	2	1	no	—
123	D₄×C₈₉	—	2	1	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	8	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂²⋊D₈₉	—	4	2	no	—
132	C₂²⋊D₈₉	—	8	1	no	—
133	C₂²⋊D₈₉	—	8	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	—	2	1	no	—
146	C₄×D₈₉	—	4	2	no	—
147	Dic₁₇₈	—	2	2	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	2	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	#41	1	2	yes	complemented
152	#10623494	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	—	4	1	no	—
178	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₈₉	—	8	1	no	—
180	C₂×C₈₉	—	8	1	no	—
181	C₂×C₈₉	—	8	1	no	—
182	C₂×C₈₉	#550	1	1	yes	characteristic
183	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₈₉	#577	1	1	yes	characteristic
185	D₈₉	—	8	1	no	—
186	#3397	—	89	1	no	maximal, Sylow
187	C₈₉	#3397	1	1	yes	characteristic, complemented, Sylow
188	#556	—	89	1	no	—
189	#550	—	89	1	no	—
190	#496	—	89	1	no	—
191	#496	D₈₉	1	1	yes	characteristic, complemented
192	#492	—	89	1	no	—
193	#463	—	89	1	no	—
194	#463	—	89	1	no	—
195	#461	—	89	1	no	—
196	#460	—	89	1	no	—
197	#460	—	89	1	no	—
198	#459	—	89	1	no	—
199	#446	—	89	1	no	—
200	#440	—	89	1	no	—
201	#440	—	89	1	no	—
202	C₄×D₈	—	89	1	no	—
203	C₂×#41	—	178	1	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×SD₁₆	—	89	1	no	—
206	C₂×SD₁₆	—	89	1	no	—
207	C₂×SD₁₆	—	89	1	no	—
208	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
209	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₂×D₈	—	89	1	no	—
211	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
212	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
213	C₄⋊Q₈	—	89	1	no	—
214	#127	—	89	1	no	—
215	#127	—	89	1	no	—
216	#127	C₂×D₈₉	1	1	yes	characteristic
217	#126	—	178	1	no	—
218	#126	—	178	1	no	—
219	#124	—	89	1	no	—
220	#124	C₂×D₈₉	1	1	yes	characteristic
221	C₂²⋊Q₈	—	178	1	no	—
222	#121	—	178	1	no	—
223	#121	—	178	1	no	—
224	#121	—	178	1	no	—
225	C₄×Q₈	—	89	1	no	—
226	C₄×Q₈	—	89	1	no	—
227	C₄×Q₈	—	89	1	no	—
228	C₄×D₄	—	89	1	no	—
229	C₄×D₄	—	89	1	no	—
230	C₄×D₄	—	89	1	no	—
231	C₄×D₄	—	89	1	no	—
232	#117	—	178	1	no	—
233	#107	—	89	1	no	—
234	C₄⋊C₈	—	89	1	no	—
235	C₄⋊C₈	—	89	1	no	—
236	C₄⋊C₈	—	89	1	no	—
237	#103	—	89	1	no	—
238	#103	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	#102	—	89	1	no	—
242	#102	—	89	1	no	—
243	#97	—	89	1	no	—
244	#97	C₂×D₈₉	1	1	yes	characteristic
245	C₄×C₈	—	89	1	no	—
246	#41	—	356	4	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×Q₈	—	89	1	no	—
249	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	89	1	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂×D₄	—	178	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
259	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	—	178	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	4	1	no	—
266	SD₁₆	—	356	1	no	—
267	SD₁₆	—	356	1	no	—
268	SD₁₆	—	356	1	no	—
269	SD₁₆	—	178	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	4	1	no	—
272	D₈	—	356	1	no	—
273	C₂×C₈	—	89	1	no	—
274	C₂×C₈	—	89	1	no	—
275	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	—	89	1	no	—
278	C₄⋊C₄	—	89	1	no	—
279	C₄⋊C₄	—	89	1	no	—
280	C₄⋊C₄	—	89	1	no	—
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	178	1	no	—
283	C₄⋊C₄	—	178	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₄²	—	89	1	no	—
295	C₄²	—	89	1	no	—
296	C₄²	—	178	1	no	—
297	C₄²	—	178	1	no	—
298	C₄²	C₂²×D₈₉	1	1	yes	characteristic
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	178	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	178	2	no	—
306	Q₈	—	178	1	no	—
307	Q₈	—	178	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	1	no	—
316	D₄	—	356	2	no	—
317	D₄	—	356	2	no	—
318	D₄	—	178	1	no	—
319	D₄	—	178	1	no	—
320	D₄	—	712	1	no	—
321	D₄	—	712	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	89	1	no	—
325	C₂×C₄	—	89	1	no	—
326	C₂×C₄	—	89	1	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	356	2	no	—
330	C₂×C₄	—	356	2	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	—	178	1	no	—
334	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
335	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	178	1	no	—
339	C₈	—	178	1	no	—
340	C₈	#13083573	1	2	yes	complemented
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	356	1	no	—
351	C₂²	—	356	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	89	2	no	—
358	C₄	—	356	1	no	—
359	C₄	—	356	1	no	—
360	C₄	—	356	1	no	—
361	C₄	—	178	1	no	—
362	C₄	—	178	1	no	—
363	C₄	—	178	1	no	—
364	C₄	C₂×#13083573	1	1	yes	characteristic
365	C₄	#15668057	1	1	yes	characteristic
366	C₂	—	8	1	no	—
367	C₂	—	8	1	no	—
368	C₂	—	8	1	no	—
369	C₂	—	712	1	no	—
370	C₂	#19152268	1	1	yes	characteristic
371	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
372	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	#496×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#496	D₈₉
hypocenter	C₈₉	#3397
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #19152317	C₂	semidirect
2. #19152268	C₂	semidirect
3. #19151986	C₂	semidirect
4. #19151969	C₂	semidirect
5. #19151811	C₂	semidirect
6. #19151810	C₂	semidirect
7. #19151801	C₂	semidirect
8. #19151796	C₂	semidirect
9. #19151795	C₂	semidirect
10. #19151791	C₂	semidirect
11. #19151726	C₂	semidirect
12. #19151700	C₂	semidirect
13. #19151699	C₂	semidirect
14. D₈×Dic₈₉	C₂	semidirect
15. #496×C₈₉	C₂	semidirect
151. C₈×C₈₉	#41	semidirect
187. C₈₉	#3397	semidirect
191. #496	D₈₉	semidirect
340. C₈	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#19152317	2
#19152268	2
#19151986	2
#19151969	2
#19151811	2
#19151810	2
#19151801	2
#19151796	2
#19151795	2
#19151791	2
#19151726	2
#19151700	2
#19151699	2
D₈×Dic₈₉	2
#496×C₈₉	2
#3397	89