Subgroups — #26030303

Order 11392.

454 conjugacy classes of subgroups.

The subgroup lattice has 370 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	C₄×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151918	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151896	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151815	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151747	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151733	C₂	1	1	yes	maximal, characteristic
11	C₄×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13083559	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	—	2	1	no	—
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	#15668007	—	2	1	no	—
22	#15668003	C₂²	1	1	yes	characteristic, complemented
23	#15668001	C₂²	1	1	yes	characteristic, complemented
24	#15667997	—	2	1	no	—
25	#15667988	C₂²	1	1	yes	characteristic
26	#15667987	C₂²	1	1	yes	characteristic
27	C₄×D₃₅₆	—	2	1	no	—
28	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
29	C₄×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
30	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	#15667974	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₉	—	2	1	no	—
33	#15667972	—	2	1	no	—
34	#15667969	—	2	1	no	—
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×D₈₉	—	2	1	no	—
37	C₄²×D₈₉	C₂²	1	1	yes	characteristic
38	#15667946	C₂²	1	1	yes	characteristic, complemented
39	#15667945	C₂²	1	1	yes	characteristic, complemented
40	#15667943	C₂²	1	1	yes	characteristic, complemented
41	#15667941	C₂²	1	1	yes	characteristic, complemented
42	#15667941	C₂²	1	1	yes	characteristic, complemented
43	#15667936	C₂²	1	1	yes	characteristic, complemented
44	#15667934	C₂²	1	1	yes	characteristic
45	#15667934	C₂²	1	1	yes	characteristic, complemented
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic, complemented
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₄×#10623494	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	D₄×D₈₉	—	2	2	no	—
65	D₄×D₈₉	—	2	2	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	4	4	no	—
72	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	#13083561	—	2	2	no	—
76	D₇₁₂	—	2	2	no	—
77	#13083559	—	2	2	no	—
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	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	#13083551	—	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×D₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	SD₁₆×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂³×C₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	4	no	—
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	4	2	no	—
114	C₂²×D₈₉	—	4	2	no	—
115	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	—	2	1	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	#41	1	2	yes	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₈₉	—	2	4	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	4	2	no	—
138	D₃₅₆	#41	1	2	yes	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	2	1	no	—
142	C₄×D₈₉	—	2	1	no	—
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	—	4	2	no	—
146	C₄×D₈₉	—	4	2	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	2	no	—
158	C₂×D₈₉	—	4	8	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	#135	1	2	yes	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
170	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₈₉	C₂×#41	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	C₂×C₈₉	—	4	2	no	—
178	C₂×C₈₉	#525	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
180	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	D₈₉	—	2	2	no	—
182	D₈₉	—	4	2	no	—
183	D₈₉	—	8	1	no	—
184	#3105	—	89	1	no	maximal, Sylow
185	C₈₉	#3105	1	1	yes	characteristic, complemented, Sylow
186	#529	—	89	1	no	—
187	C₂×C₄×D₄	—	89	1	no	—
188	#484	—	89	1	no	—
189	#481	—	89	1	no	—
190	#478	—	89	1	no	—
191	#478	—	89	1	no	—
192	#463	—	89	1	no	—
193	#463	—	89	1	no	—
194	#450	—	89	1	no	—
195	#447	—	89	1	no	—
196	C₄×SD₁₆	—	89	1	no	—
197	C₄×SD₁₆	D₈₉	1	1	yes	characteristic, complemented
198	C₄×D₈	—	89	1	no	—
199	C₄×D₈	—	89	1	no	—
200	#431	—	89	1	no	—
201	C₂²×D₄	—	89	1	no	—
202	C₂³×C₄	—	178	1	no	—
203	C₂×SD₁₆	—	89	1	no	—
204	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₂×D₈	—	89	1	no	—
206	C₂×D₈	—	89	1	no	—
207	#126	—	178	1	no	—
208	#125	—	89	1	no	—
209	#124	—	89	1	no	—
210	#123	—	178	1	no	—
211	C₂²⋊Q₈	—	89	1	no	—
212	#121	—	89	1	no	—
213	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	178	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₄×D₄	—	178	1	no	—
221	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
222	#117	—	178	1	no	—
223	C₂×C₄⋊C₄	—	89	1	no	—
224	C₂×C₂²⋊C₄	—	178	1	no	—
225	C₂×C₄²	—	89	1	no	—
226	#107	—	89	1	no	—
227	#107	—	89	1	no	—
228	#106	—	89	1	no	—
229	#106	C₂×D₈₉	1	1	yes	characteristic, complemented
230	C₄⋊C₈	—	89	1	no	—
231	C₄⋊C₈	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	C₂×D₈₉	1	1	yes	characteristic
240	C₂²⋊C₈	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₄×C₈	—	89	1	no	—
243	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
244	C₂⁴	—	178	1	no	—
245	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×D₄	—	89	1	no	—
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	2	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	356	4	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	—	178	1	no	—
263	SD₁₆	—	2	2	no	—
264	SD₁₆	—	178	2	no	—
265	D₈	—	178	2	no	—
266	D₈	—	178	2	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	1	no	—
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	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	89	1	no	—
291	C₄²	—	89	1	no	—
292	C₄²	—	89	1	no	—
293	C₄²	C₂²×D₈₉	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	89	1	no	—
296	C₂³	—	356	2	no	—
297	C₂³	—	356	2	no	—
298	C₂³	—	178	4	no	—
299	C₂³	—	178	1	no	—
300	C₂³	—	178	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	89	2	no	—
305	D₄	—	356	2	no	—
306	D₄	—	178	4	no	—
307	D₄	—	178	1	no	—
308	D₄	—	178	1	no	—
309	D₄	—	178	1	no	—
310	D₄	#13083574	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	178	1	no	—
335	C₈	—	178	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	89	2	no	—
340	C₂²	—	356	8	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	356	1	no	—
344	C₂²	—	356	1	no	—
345	C₂²	—	356	1	no	—
346	C₂²	—	178	2	no	—
347	C₂²	—	178	1	no	—
348	C₂²	—	178	1	no	—
349	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	356	1	no	—
355	C₄	—	356	1	no	—
356	C₄	—	178	1	no	—
357	C₄	—	178	1	no	—
358	C₄	—	178	1	no	—
359	C₄	#15668039	1	2	yes	—
360	C₄	#41×D₈₉	1	1	yes	characteristic
361	C₄	C₂×#13083574	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	356	2	no	—
364	C₂	—	178	2	no	—
365	C₂	—	712	1	no	—
366	C₂	#19152107	1	1	yes	characteristic
367	C₂	#135×D₈₉	1	1	yes	characteristic
368	C₂	C₂×#15668039	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₄×SD₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151733	C₂
hypercenter	C₄×SD₁₆	D₈₉
hypocenter	C₈₉	#3105
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #19152135	C₂	semidirect
2. C₄×D₄×D₈₉	C₂	semidirect
3. #19151932	C₂	semidirect
4. #19151918	C₂	semidirect
5. #19151897	C₂	semidirect
6. #19151896	C₂	semidirect
7. #19151816	C₂	semidirect
8. #19151815	C₂	semidirect
9. #19151747	C₂	semidirect
11. C₄×#13083561	C₂	semidirect
12. C₄×D₇₁₂	C₂	semidirect
13. C₄×#13083559	C₂	semidirect
14. #19151649	C₂	semidirect
15. C₄×SD₁₆×C₈₉	C₂	semidirect
22. #15668003	C₂²	semidirect
23. #15668001	C₂²	semidirect
29. C₄×D₃₅₆	C₂²	semidirect
38. #15667946	C₂²	semidirect
39. #15667945	C₂²	semidirect
40. #15667943	C₂²	semidirect
41. #15667941	C₂²	semidirect
42. #15667941	C₂²	semidirect
43. #15667936	C₂²	semidirect
45. #15667934	C₂²	semidirect
48. #15667932	C₂²	semidirect
51. C₄×#10623494	C₂²	semidirect
53. C₄×Q₈×C₈₉	C₂²	semidirect
55. #106×C₈₉	C₂²	semidirect
56. #103×C₈₉	C₂²	semidirect
58. C₄×C₈×C₈₉	C₂²	semidirect
185. C₈₉	#3105	semidirect
197. C₄×SD₁₆	D₈₉	semidirect
213. C₄×Q₈	C₂×D₈₉	semidirect
229. #106	C₂×D₈₉	semidirect
233. #103	C₂×D₈₉	semidirect
243. C₄×C₈	C₂×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
C₄×D₄×D₈₉	2
#19151932	2
#19151918	2
#19151897	2
#19151896	2
#19151816	2
#19151815	2
#19151747	2
#19151733	2
C₄×#13083561	2
C₄×D₇₁₂	2
C₄×#13083559	2
#19151649	2
C₄×SD₁₆×C₈₉	2
#3105	89