Subgroups — #26032326

Order 11392.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152306	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152236	C₂	1	1	yes	maximal, characteristic
3	#19151959	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151912	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151876	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151846	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151836	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151822	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151747	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151699	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151672	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151599	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083561	C₂²	1	1	yes	characteristic
18	C₂×#13083558	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic, complemented
21	#15668002	—	2	1	no	—
22	#15668001	C₂²	1	1	yes	characteristic
23	#15667995	—	2	1	no	—
24	#15667992	—	2	1	no	—
25	#15667989	—	2	1	no	—
26	#15667988	C₂²	1	1	yes	characteristic
27	#15667988	C₂²	1	1	yes	characteristic
28	#15667986	C₂²	1	1	yes	characteristic
29	#15667984	—	2	1	no	—
30	#15667980	—	2	1	no	—
31	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
34	#15667972	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic, complemented
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₂×#13083551	—	2	1	no	—
38	#15667946	C₂²	1	1	yes	characteristic, complemented
39	#15667946	C₂²	1	1	yes	characteristic, complemented
40	#15667945	C₂²	1	1	yes	characteristic, complemented
41	#15667941	C₂²	1	1	yes	characteristic, complemented
42	#15667936	C₂²	1	1	yes	characteristic, complemented
43	#15667936	C₂²	1	1	yes	characteristic, complemented
44	#15667936	C₂²	1	1	yes	characteristic, complemented
45	#15667934	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	#15667932	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667922	C₂²	1	1	yes	characteristic, complemented
52	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
54	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×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	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
65	D₄×D₈₉	—	2	2	no	—
66	D₄×D₈₉	—	4	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	#13083561	—	4	1	no	—
72	#13083558	—	4	1	no	—
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₈×D₈₉	—	4	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	#41×C₈₉	—	2	2	no	—
97	#41×C₈₉	—	4	1	no	—
98	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	—	2	1	no	—
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	4	1	no	—
105	SD₁₆×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₉	—	2	1	no	—
114	C₂²×D₈₉	—	4	2	no	—
115	C₂²×D₈₉	—	4	2	no	—
116	C₂²×D₈₉	—	8	1	no	—
117	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₈₉	—	2	1	no	—
119	Q₈×C₈₉	—	2	1	no	—
120	Q₈×C₈₉	—	2	1	no	—
121	Q₈×C₈₉	—	2	1	no	—
122	D₄×C₈₉	—	2	1	no	—
123	D₄×C₈₉	—	4	2	no	—
124	D₄×C₈₉	#41	1	2	yes	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	2	2	no	—
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂²⋊D₈₉	—	8	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	8	1	no	—
140	C₂×Dic₈₉	—	8	1	no	—
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	—	2	1	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	8	1	no	—
149	Dic₁₇₈	—	8	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	#10623494	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	C₂²×C₈₉	—	2	2	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	8	4	no	—
167	C₂×D₈₉	#137	1	2	yes	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
173	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	2	1	no	—
176	Dic₈₉	—	2	1	no	—
177	Dic₈₉	—	4	1	no	—
178	Dic₈₉	—	4	1	no	—
179	Dic₈₉	—	4	1	no	—
180	C₂×C₈₉	—	4	2	no	—
181	C₂×C₈₉	—	4	1	no	—
182	C₂×C₈₉	#546	1	1	yes	characteristic
183	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
184	C₂×C₈₉	#576	1	1	yes	characteristic
185	D₈₉	—	2	2	no	—
186	D₈₉	—	8	1	no	—
187	#3314	—	89	1	no	maximal, Sylow
188	C₈₉	#3314	1	1	yes	characteristic, complemented, Sylow
189	#554	—	89	1	no	—
190	#546	—	89	1	no	—
191	#489	—	89	1	no	—
192	#484	—	89	1	no	—
193	#480	—	89	1	no	—
194	#474	—	89	1	no	—
195	#468	—	89	1	no	—
196	#467	—	89	1	no	—
197	#464	—	89	1	no	—
198	#452	D₈₉	1	1	yes	characteristic, complemented
199	#450	—	89	1	no	—
200	#440	—	89	1	no	—
201	#435	—	89	1	no	—
202	#417	—	89	1	no	—
203	C₂×#102	—	89	1	no	—
204	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
205	C₂²×D₄	—	89	1	no	—
206	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	89	1	no	—
208	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
209	C₂×M₁₆	—	89	1	no	—
210	C₂²×C₈	—	89	1	no	—
211	C₄⋊Q₈	—	89	1	no	—
212	#125	—	178	1	no	—
213	#124	—	89	1	no	—
214	#123	—	178	1	no	—
215	C₂²⋊Q₈	—	89	1	no	—
216	C₂²⋊Q₈	—	89	1	no	—
217	C₂²⋊Q₈	—	178	1	no	—
218	C₂²⋊Q₈	—	178	1	no	—
219	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
220	#121	—	89	1	no	—
221	#121	—	178	1	no	—
222	#120	—	178	1	no	—
223	C₄×Q₈	—	89	1	no	—
224	C₄×D₄	—	89	1	no	—
225	C₄×D₄	—	89	1	no	—
226	C₄×D₄	—	178	1	no	—
227	C₂×C₄⋊C₄	—	89	1	no	—
228	C₂×C₄⋊C₄	—	89	1	no	—
229	C₂×C₂²⋊C₄	—	178	1	no	—
230	#107	—	89	1	no	—
231	#107	—	89	1	no	—
232	#107	—	89	1	no	—
233	C₄⋊C₈	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#103	—	89	1	no	—
237	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	#102	—	89	1	no	—
242	#102	—	89	1	no	—
243	#102	C₂×D₈₉	1	1	yes	characteristic
244	C₂²⋊C₈	—	89	1	no	—
245	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
246	#97	—	89	1	no	—
247	C₂⁴	—	178	1	no	—
248	#41	—	2	2	no	—
249	#41	—	4	1	no	—
250	C₂×Q₈	—	178	1	no	—
251	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	89	1	no	—
255	C₂×D₄	—	89	1	no	—
256	C₂×D₄	—	356	1	no	—
257	C₂×D₄	—	178	1	no	—
258	C₂×D₄	—	178	2	no	—
259	C₂×D₄	—	178	1	no	—
260	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	89	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	C₂²×C₄	—	178	1	no	—
265	C₂²×C₄	—	178	1	no	—
266	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
267	Q₁₆	—	4	1	no	—
268	SD₁₆	—	2	2	no	—
269	SD₁₆	—	356	1	no	—
270	M₁₆	—	356	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	89	1	no	—
273	C₂×C₈	—	356	1	no	—
274	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
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	—	178	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	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	89	1	no	—
290	C₂²⋊C₄	—	89	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₄²	—	89	1	no	—
298	C₄²	—	178	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	89	1	no	—
301	C₂³	—	356	2	no	—
302	C₂³	—	356	2	no	—
303	C₂³	—	178	1	no	—
304	C₂³	—	712	1	no	—
305	C₂³	D₄×D₈₉	1	1	yes	characteristic
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	712	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	2	no	—
313	D₄	—	356	2	no	—
314	D₄	—	178	2	no	—
315	D₄	—	178	1	no	—
316	D₄	—	178	1	no	—
317	D₄	—	712	1	no	—
318	D₄	#13083573	1	2	yes	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	89	1	no	—
326	C₂×C₄	—	89	1	no	—
327	C₂×C₄	—	89	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	178	1	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	—	712	1	no	—
334	C₂×C₄	—	712	1	no	—
335	C₂×C₄	—	712	1	no	—
336	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
337	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	178	1	no	—
341	C₈	—	178	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	89	2	no	—
349	C₂²	—	356	1	no	—
350	C₂²	—	356	1	no	—
351	C₂²	—	356	1	no	—
352	C₂²	—	178	1	no	—
353	C₂²	—	178	1	no	—
354	C₂²	—	712	4	no	—
355	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	356	1	no	—
361	C₄	—	356	1	no	—
362	C₄	—	356	1	no	—
363	C₄	—	178	1	no	—
364	C₄	—	178	1	no	—
365	C₄	—	178	1	no	—
366	C₄	C₂×#13083573	1	1	yes	characteristic
367	C₄	#15668057	1	1	yes	characteristic
368	C₂	—	4	1	no	—
369	C₂	—	4	2	no	—
370	C₂	—	178	2	no	—
371	C₂	—	712	1	no	—
372	C₂	#19152235	1	1	yes	characteristic
373	C₂	#137×D₈₉	1	1	yes	characteristic
374	C₂	#19152400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#452×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152236	C₂
hypercenter	#452	D₈₉
hypocenter	C₈₉	#3314
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #19152306	C₂	semidirect
3. #19151959	C₂	semidirect
4. #19151932	C₂	semidirect
5. #19151912	C₂	semidirect
6. #19151876	C₂	semidirect
7. #19151846	C₂	semidirect
8. #19151836	C₂	semidirect
9. #19151822	C₂	semidirect
10. #19151747	C₂	semidirect
11. #19151699	C₂	semidirect
12. #19151672	C₂	semidirect
13. #19151599	C₂	semidirect
14. #102×D₈₉	C₂	semidirect
15. #452×C₈₉	C₂	semidirect
18. C₂×#13083558	C₂²	semidirect
20. #15668011	C₂²	semidirect
31. Q₈×Dic₈₉	C₂²	semidirect
35. C₂×C₄⋊Dic₈₉	C₂²	semidirect
38. #15667946	C₂²	semidirect
39. #15667946	C₂²	semidirect
40. #15667945	C₂²	semidirect
41. #15667941	C₂²	semidirect
42. #15667936	C₂²	semidirect
43. #15667936	C₂²	semidirect
44. #15667936	C₂²	semidirect
51. #15667922	C₂²	semidirect
53. C₂×Q₁₆×C₈₉	C₂²	semidirect
55. C₂²⋊Q₈×C₈₉	C₂²	semidirect
56. #103×C₈₉	C₂²	semidirect
58. C₂²⋊C₈×C₈₉	C₂²	semidirect
188. C₈₉	#3314	semidirect
198. #452	D₈₉	semidirect
206. C₂×Q₁₆	C₂×D₈₉	semidirect
219. C₂²⋊Q₈	C₂×D₈₉	semidirect
237. #103	C₂×D₈₉	semidirect
245. C₂²⋊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
#19152306	2
#19152236	2
#19151959	2
#19151932	2
#19151912	2
#19151876	2
#19151846	2
#19151836	2
#19151822	2
#19151747	2
#19151699	2
#19151672	2
#19151599	2
#102×D₈₉	2
#452×C₈₉	2
#3314	89