Subgroups — #26031824

Order 11392.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668044	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152260	C₂	1	1	yes	maximal, characteristic
5	#19151992	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151857	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151828	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151820	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151808	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151763	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151671	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15668044	—	2	4	no	—
21	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
23	C₂×#13083564	C₂²	1	1	yes	characteristic
24	C₂×#13083564	C₂²	1	1	yes	characteristic
25	C₂×#13083564	C₂²	1	1	yes	characteristic
26	Q₁₆×D₈₉	—	2	4	no	—
27	C₂×#13083563	C₂²	1	1	yes	characteristic
28	C₂×#13083563	C₂²	1	1	yes	characteristic
29	C₂×#13083561	C₂²	1	1	yes	characteristic
30	C₂×#13083557	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#15668012	—	2	1	no	—
33	#15668012	C₂²	1	1	yes	characteristic
34	#15668000	—	2	1	no	—
35	#15668000	C₂²	1	1	yes	characteristic
36	#15667992	—	2	1	no	—
37	#15667992	—	2	1	no	—
38	#15667992	—	2	1	no	—
39	#15667992	C₂²	1	1	yes	characteristic
40	#15667992	C₂²	1	1	yes	characteristic
41	#15667992	C₂²	1	1	yes	characteristic
42	#15667987	—	2	1	no	—
43	#15667987	C₂²	1	1	yes	characteristic
44	C₄×D₃₅₆	—	2	1	no	—
45	C₄×D₃₅₆	—	2	1	no	—
46	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
47	C₄⋊#10623494	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667935	C₂²	1	1	yes	characteristic
50	#15667935	C₂²	1	1	yes	characteristic
51	#15667932	C₂²	1	1	yes	characteristic
52	#15667924	C₂²	1	1	yes	characteristic
53	#15667924	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	#106×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#103×C₈₉	C₂²	1	1	yes	characteristic
60	#97×C₈₉	C₂²	1	1	yes	characteristic
61	#13083575	—	2	2	no	—
62	#13083575	—	4	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	—	2	1	no	—
66	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
69	Q₈×D₈₉	—	2	2	no	—
70	Q₈×D₈₉	—	2	2	no	—
71	Q₈×D₈₉	—	2	2	no	—
72	Q₈×D₈₉	—	4	1	no	—
73	Q₈×D₈₉	—	4	1	no	—
74	Q₈×D₈₉	—	4	1	no	—
75	C₂×D₃₅₆	—	2	1	no	—
76	C₂×D₃₅₆	—	2	1	no	—
77	C₂×D₃₅₆	D₄	1	1	yes	characteristic
78	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×C₄×D₈₉	—	2	1	no	—
81	C₂×C₄×D₈₉	—	2	1	no	—
82	C₂×C₄×D₈₉	—	2	1	no	—
83	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
85	Dic₃₅₆	—	2	2	no	—
86	Dic₃₅₆	—	4	1	no	—
87	#13083564	—	2	2	no	—
88	#13083564	—	2	2	no	—
89	#13083564	—	2	2	no	—
90	#13083563	—	2	2	no	—
91	#13083563	—	4	1	no	—
92	#13083561	—	2	2	no	—
93	#13083557	—	2	2	no	—
94	C₂×#10623494	C₂³	1	1	yes	characteristic
95	C₂×#10623494	C₂³	1	1	yes	characteristic
96	C₈×D₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	2	1	no	—
102	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
103	#13083551	—	2	1	no	—
104	#13083551	—	2	1	no	—
105	#13083551	—	2	1	no	—
106	#13083551	—	2	1	no	—
107	#13083551	—	2	1	no	—
108	#13083551	—	2	1	no	—
109	#13083551	D₄	1	1	yes	characteristic
110	#13083551	D₄	1	1	yes	characteristic
111	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
113	Q₁₆×C₈₉	D₄	1	4	yes	complemented
114	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
120	C₄²×C₈₉	—	2	1	no	—
121	C₄²×C₈₉	—	2	1	no	—
122	C₄²×C₈₉	C₂³	1	1	yes	characteristic
123	C₂²×D₈₉	—	2	1	no	—
124	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	—	2	1	no	—
128	Q₈×C₈₉	C₂×D₄	1	2	yes	—
129	Q₈×C₈₉	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	D₃₅₆	—	2	2	no	—
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	—	4	2	no	—
146	D₃₅₆	—	4	2	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	2	1	no	—
149	C₄×D₈₉	—	2	1	no	—
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	4	2	no	—
152	C₄×D₈₉	—	4	2	no	—
153	C₄×D₈₉	—	4	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	Dic₁₇₈	—	2	1	no	—
156	Dic₁₇₈	—	2	1	no	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	2	1	no	—
160	Dic₁₇₈	—	2	1	no	—
161	Dic₁₇₈	—	4	2	no	—
162	Dic₁₇₈	—	4	2	no	—
163	Dic₁₇₈	—	4	2	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	C₂×D₄	1	2	yes	—
166	#10623494	—	2	1	no	—
167	#10623494	—	2	1	no	—
168	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₈₉	—	2	1	no	—
170	C₂×D₈₉	—	2	1	no	—
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	1	no	—
173	C₂×D₈₉	—	2	1	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	4	1	no	—
177	C₂×D₈₉	#137	1	2	yes	complemented
178	C₄×C₈₉	—	2	2	no	—
179	C₄×C₈₉	—	2	2	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	4	1	no	—
183	C₄×C₈₉	—	4	1	no	—
184	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
185	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
186	Dic₈₉	—	2	1	no	—
187	Dic₈₉	—	2	1	no	—
188	Dic₈₉	—	4	1	no	—
189	Dic₈₉	—	4	1	no	—
190	Dic₈₉	—	4	1	no	—
191	C₂×C₈₉	D₄²	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
193	C₂×C₈₉	#578	1	1	yes	characteristic
194	D₈₉	—	2	2	no	—
195	D₈₉	—	4	1	no	—
196	D₈₉	—	8	1	no	—
197	#3270	—	89	1	no	maximal, Sylow
198	C₈₉	#3270	1	1	yes	characteristic, complemented, Sylow
199	C₂×#137	—	89	1	no	—
200	C₂²×Q₁₆	—	89	1	no	—
201	D₄×Q₈	—	89	1	no	—
202	#548	—	89	1	no	—
203	#497	—	89	1	no	—
204	#470	—	89	1	no	—
205	#465	—	89	1	no	—
206	#464	—	89	1	no	—
207	#462	—	89	1	no	—
208	#452	—	89	1	no	—
209	#451	—	89	1	no	—
210	#451	—	89	1	no	—
211	#448	—	89	1	no	—
212	#441	D₈₉	1	1	yes	characteristic, complemented
213	#435	—	89	1	no	—
214	C₂×#41	—	89	1	no	—
215	C₂²×Q₈	—	89	1	no	—
216	C₂²×Q₈	—	89	1	no	—
217	C₂²×Q₈	—	89	1	no	—
218	#137	—	178	4	no	—
219	C₂×Q₁₆	—	89	1	no	—
220	C₂×Q₁₆	—	89	1	no	—
221	C₂×Q₁₆	—	89	1	no	—
222	C₂×Q₁₆	—	89	1	no	—
223	C₂×Q₁₆	—	89	1	no	—
224	C₂×Q₁₆	—	178	4	no	—
225	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
226	C₂×SD₁₆	—	89	1	no	—
227	C₂×SD₁₆	—	89	1	no	—
228	C₂×SD₁₆	—	89	1	no	—
229	C₂×M₁₆	—	89	1	no	—
230	C₂²×C₈	—	89	1	no	—
231	C₄⋊Q₈	—	89	1	no	—
232	C₄⋊Q₈	—	178	1	no	—
233	#124	—	89	1	no	—
234	#124	—	178	1	no	—
235	C₂²⋊Q₈	—	89	1	no	—
236	C₂²⋊Q₈	—	89	1	no	—
237	C₂²⋊Q₈	—	89	1	no	—
238	C₂²⋊Q₈	—	178	1	no	—
239	C₂²⋊Q₈	—	178	1	no	—
240	C₂²⋊Q₈	—	178	1	no	—
241	#121	—	89	1	no	—
242	#121	—	178	1	no	—
243	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
244	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
245	C₄×D₄	—	89	1	no	—
246	C₄×D₄	—	178	1	no	—
247	C₄×D₄	—	178	1	no	—
248	#106	C₂×D₈₉	1	1	yes	characteristic
249	C₄⋊C₈	—	89	1	no	—
250	#103	—	89	1	no	—
251	#103	—	89	1	no	—
252	#103	—	89	1	no	—
253	#103	C₂×D₈₉	1	1	yes	characteristic
254	#103	C₂×D₈₉	1	1	yes	characteristic
255	#102	—	89	1	no	—
256	C₂²⋊C₈	—	89	1	no	—
257	C₂²⋊C₈	—	89	1	no	—
258	#97	C₂×D₈₉	1	1	yes	characteristic
259	#41	—	356	1	no	—
260	#41	—	178	2	no	—
261	C₂×Q₈	—	89	1	no	—
262	C₂×Q₈	—	89	1	no	—
263	C₂×Q₈	—	89	1	no	—
264	C₂×Q₈	—	356	1	no	—
265	C₂×Q₈	—	356	1	no	—
266	C₂×Q₈	—	356	1	no	—
267	C₂×Q₈	—	178	1	no	—
268	C₂×Q₈	—	178	1	no	—
269	C₂×Q₈	—	178	1	no	—
270	C₂×Q₈	—	178	2	no	—
271	C₂×Q₈	—	178	2	no	—
272	C₂×Q₈	—	178	2	no	—
273	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₂×D₄	—	89	1	no	—
276	C₂×D₄	—	89	1	no	—
277	C₂×D₄	—	178	1	no	—
278	C₂×D₄	—	178	1	no	—
279	C₂²×C₄	—	89	1	no	—
280	C₂²×C₄	—	89	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	Q₁₆	—	356	1	no	—
286	Q₁₆	—	178	2	no	—
287	Q₁₆	—	178	2	no	—
288	Q₁₆	—	178	2	no	—
289	Q₁₆	—	178	2	no	—
290	Q₁₆	D₃₅₆	1	4	yes	complemented
291	SD₁₆	—	356	1	no	—
292	SD₁₆	—	178	2	no	—
293	SD₁₆	—	178	2	no	—
294	M₁₆	—	178	2	no	—
295	C₂×C₈	—	89	1	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	178	2	no	—
298	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	89	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₄⋊C₄	—	178	1	no	—
307	C₄⋊C₄	—	178	1	no	—
308	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
310	C₂²⋊C₄	—	89	1	no	—
311	C₂²⋊C₄	—	89	1	no	—
312	C₂²⋊C₄	—	178	1	no	—
313	C₂²⋊C₄	—	178	1	no	—
314	C₂²⋊C₄	—	178	1	no	—
315	C₂²⋊C₄	—	178	1	no	—
316	C₂²⋊C₄	—	178	1	no	—
317	C₂²⋊C₄	—	178	1	no	—
318	C₄²	—	2	1	no	—
319	C₄²	—	2	1	no	—
320	C₄²	C₂²×D₈₉	1	1	yes	characteristic
321	C₂³	—	89	1	no	—
322	C₂³	—	89	1	no	—
323	C₂³	—	178	1	no	—
324	Q₈	—	2	1	no	—
325	Q₈	—	2	1	no	—
326	Q₈	—	356	2	no	—
327	Q₈	—	356	2	no	—
328	Q₈	—	356	2	no	—
329	Q₈	—	178	1	no	—
330	Q₈	—	178	1	no	—
331	Q₈	—	178	1	no	—
332	Q₈	—	178	1	no	—
333	Q₈	—	178	1	no	—
334	Q₈	—	178	1	no	—
335	Q₈	C₂×D₃₅₆	1	2	yes	—
336	Q₈	C₂×D₃₅₆	1	2	yes	—
337	D₄	—	356	2	no	—
338	D₄	—	356	2	no	—
339	D₄	—	178	2	no	—
340	D₄	—	178	1	no	—
341	D₄	—	178	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	89	1	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	—	178	1	no	—
356	C₂×C₄	—	178	1	no	—
357	C₂×C₄	—	178	1	no	—
358	C₂×C₄	—	178	1	no	—
359	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
360	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	178	1	no	—
364	C₈	—	178	1	no	—
365	C₈	C₂×D₃₅₆	1	2	yes	—
366	C₂²	—	89	2	no	—
367	C₂²	—	356	1	no	—
368	C₂²	—	356	1	no	—
369	C₂²	—	356	1	no	—
370	C₂²	—	178	1	no	—
371	C₂²	—	178	1	no	—
372	C₂²	—	178	1	no	—
373	C₂²	—	178	1	no	—
374	C₂²	—	178	1	no	—
375	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	1	no	—
382	C₄	—	356	1	no	—
383	C₄	—	356	1	no	—
384	C₄	—	356	1	no	—
385	C₄	—	178	1	no	—
386	C₄	—	178	1	no	—
387	C₄	C₂²×D₃₅₆	1	1	yes	characteristic
388	C₄	#15668058	1	1	yes	characteristic
389	C₂	—	356	1	no	—
390	C₂	—	178	2	no	—
391	C₂	—	712	1	no	—
392	C₂	D₄×D₃₅₆	1	1	yes	characteristic
393	C₂	#137×D₈₉	1	1	yes	characteristic
394	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#441×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152260	C₂
hypercenter	#441	D₈₉
hypocenter	C₈₉	#3270
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. C₂×#15668044	C₂	semidirect
2. C₂×Q₁₆×D₈₉	C₂	semidirect
3. Q₈×D₃₅₆	C₂	semidirect
5. #19151992	C₂	semidirect
6. #19151857	C₂	semidirect
7. #19151828	C₂	semidirect
8. #19151820	C₂	semidirect
9. #19151808	C₂	semidirect
10. #19151763	C₂	semidirect
11. #19151755	C₂	semidirect
12. #19151755	C₂	semidirect
13. #19151738	C₂	semidirect
14. #19151671	C₂	semidirect
15. #441×C₈₉	C₂	semidirect
113. Q₁₆×C₈₉	D₄	semidirect
177. C₂×D₈₉	#137	semidirect
198. C₈₉	#3270	semidirect
212. #441	D₈₉	semidirect
290. Q₁₆	D₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668044	2
C₂×Q₁₆×D₈₉	2
Q₈×D₃₅₆	2
#19152260	2
#19151992	2
#19151857	2
#19151828	2
#19151820	2
#19151808	2
#19151763	2
#19151755	2
#19151738	2
#19151671	2
#441×C₈₉	2
#3270	89