Subgroups — #26030231

Order 11392.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152178	C₂	1	1	yes	maximal, characteristic
2	#19152079	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151936	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151925	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151923	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151911	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151639	C₂	1	1	yes	maximal, characteristic, complemented
11	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151598	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151593	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151563	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083574	C₂²	1	1	yes	characteristic
18	C₂×#13083557	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15667990	—	2	1	no	—
21	#15667988	C₂²	1	1	yes	characteristic
22	#15667987	—	2	1	no	—
23	#15667987	C₂²	1	1	yes	characteristic
24	#15667986	C₂²	1	1	yes	characteristic
25	#15667985	—	2	1	no	—
26	#15667985	—	2	1	no	—
27	#15667977	—	2	1	no	—
28	#15667974	—	2	1	no	—
29	C₄×C₂²⋊D₈₉	—	2	1	no	—
30	C₄×C₂²⋊D₈₉	—	2	1	no	—
31	#15667971	C₂²	1	1	yes	characteristic
32	#15667970	C₂²	1	1	yes	characteristic
33	#15667970	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	#15667946	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667944	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667936	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667934	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	2	2	no	—
60	#13083575	—	2	2	no	—
61	#13083574	—	2	2	no	—
62	#13083574	—	2	2	no	—
63	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	D₄	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
77	#13083557	—	2	2	no	—
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₈×D₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	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	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×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	—	2	1	no	—
103	C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₈₉	—	2	1	no	—
113	Q₈×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	—	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	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	2	2	no	—
124	C₂²⋊D₈₉	—	2	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	D₃₅₆	—	2	2	no	—
132	D₃₅₆	—	2	1	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	4	2	no	—
135	D₃₅₆	—	8	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	—	8	1	no	—
140	C₄×D₈₉	—	8	1	no	—
141	C₄×D₈₉	—	8	1	no	—
142	C₄×D₈₉	#41	1	2	yes	—
143	C₄×D₈₉	#41	1	2	yes	—
144	Dic₁₇₈	—	2	2	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	#123	1	2	yes	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	#123	1	2	yes	—
174	C₂×C₈₉	—	4	1	no	—
175	C₂×C₈₉	C₂×#123	1	1	yes	characteristic
176	C₂×C₈₉	#576	1	1	yes	characteristic
177	C₂×C₈₉	#578	1	1	yes	characteristic
178	D₈₉	—	2	2	no	—
179	D₈₉	—	4	1	no	—
180	D₈₉	—	8	1	no	—
181	#3098	—	89	1	no	maximal, Sylow
182	C₈₉	#3098	1	1	yes	characteristic, complemented, Sylow
183	#537	—	89	1	no	—
184	#520	—	89	1	no	—
185	#485	—	89	1	no	—
186	#485	—	89	1	no	—
187	#484	—	89	1	no	—
188	#484	D₈₉	1	1	yes	characteristic, complemented
189	#482	—	89	1	no	—
190	#482	—	89	1	no	—
191	#480	—	89	1	no	—
192	#480	—	89	1	no	—
193	#428	—	89	1	no	—
194	C₂×#107	—	89	1	no	—
195	#417	—	89	1	no	—
196	#416	—	89	1	no	—
197	#408	—	89	1	no	—
198	C₂×#41	—	89	1	no	—
199	C₂×#41	—	89	1	no	—
200	C₂×M₁₆	—	89	1	no	—
201	C₂²×C₈	—	89	1	no	—
202	C₂²⋊Q₈	—	89	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
205	#121	—	89	1	no	—
206	#121	—	89	1	no	—
207	#121	—	178	1	no	—
208	#121	—	178	1	no	—
209	#121	—	178	1	no	—
210	C₄×Q₈	—	178	1	no	—
211	C₄×D₄	—	178	1	no	—
212	C₄×D₄	—	178	1	no	—
213	C₄×D₄	—	178	1	no	—
214	#117	—	89	1	no	—
215	#117	—	89	1	no	—
216	#117	—	89	1	no	—
217	C₂×C₄⋊C₄	—	89	1	no	—
218	C₂×C₄⋊C₄	—	89	1	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
221	#107	—	89	1	no	—
222	#107	—	89	1	no	—
223	#107	—	89	1	no	—
224	#107	—	89	1	no	—
225	#107	C₂×D₈₉	1	1	yes	characteristic
226	#107	C₂×D₈₉	1	1	yes	characteristic
227	#106	—	89	1	no	—
228	#106	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	C₂×D₈₉	1	1	yes	characteristic
232	#103	C₂×D₈₉	1	1	yes	characteristic
233	#102	—	89	1	no	—
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	C₂²⋊C₈	—	89	1	no	—
239	C₂²⋊C₈	—	89	1	no	—
240	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
241	#41	—	178	2	no	—
242	#41	—	178	2	no	—
243	#41	—	178	2	no	—
244	#41	—	178	2	no	—
245	C₂×Q₈	—	89	1	no	—
246	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	89	1	no	—
249	C₂×D₄	—	89	1	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	178	1	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
262	M₁₆	—	178	2	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₂²⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	—	178	1	no	—
286	C₂²⋊C₄	—	178	1	no	—
287	C₄²	—	178	1	no	—
288	C₄²	—	178	1	no	—
289	C₄²	—	178	1	no	—
290	C₂³	—	89	1	no	—
291	C₂³	—	89	1	no	—
292	C₂³	—	178	1	no	—
293	C₂³	D₄×D₈₉	1	1	yes	characteristic
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	178	2	no	—
297	D₄	—	356	2	no	—
298	D₄	—	178	2	no	—
299	D₄	—	178	2	no	—
300	D₄	—	178	2	no	—
301	D₄	—	178	1	no	—
302	D₄	—	178	1	no	—
303	D₄	—	712	1	no	—
304	D₄	—	712	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	356	2	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	712	1	no	—
323	C₂×C₄	—	712	1	no	—
324	C₂×C₄	—	712	1	no	—
325	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	178	1	no	—
330	C₈	—	178	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	89	2	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	356	1	no	—
338	C₂²	—	178	1	no	—
339	C₂²	—	178	1	no	—
340	C₂²	—	178	1	no	—
341	C₂²	—	178	1	no	—
342	C₂²	—	178	1	no	—
343	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	89	2	no	—
350	C₄	—	356	1	no	—
351	C₄	—	356	1	no	—
352	C₄	—	356	1	no	—
353	C₄	—	178	1	no	—
354	C₄	#41×D₈₉	1	1	yes	characteristic
355	C₄	#41×D₈₉	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	356	1	no	—
358	C₂	—	178	2	no	—
359	C₂	—	712	1	no	—
360	C₂	#123×D₈₉	1	1	yes	characteristic
361	C₂	#19152402	1	1	yes	characteristic
362	C₂	#19152411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#484×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152178	C₂
hypercenter	#484	D₈₉
hypocenter	C₈₉	#3098
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
2. #19152079	C₂	semidirect
3. #19151937	C₂	semidirect
4. #19151936	C₂	semidirect
5. #19151932	C₂	semidirect
6. #19151925	C₂	semidirect
7. #19151923	C₂	semidirect
8. #19151913	C₂	semidirect
9. #19151911	C₂	semidirect
10. #19151639	C₂	semidirect
11. #107×D₈₉	C₂	semidirect
12. #19151598	C₂	semidirect
13. #19151593	C₂	semidirect
14. #19151563	C₂	semidirect
15. #484×C₈₉	C₂	semidirect
182. C₈₉	#3098	semidirect
188. #484	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152178	2
#19152079	2
#19151937	2
#19151936	2
#19151932	2
#19151925	2
#19151923	2
#19151913	2
#19151911	2
#19151639	2
#107×D₈₉	2
#19151598	2
#19151593	2
#19151563	2
#484×C₈₉	2
#3098	89