Subgroups — #26030727

Order 11392.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 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×#15668039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15668035	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#19152173	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152166	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151991	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151987	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151984	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151983	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151980	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151972	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151969	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151555	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083575	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
20	#15668044	—	2	4	no	—
21	#15668039	—	2	4	no	—
22	#15668035	—	2	4	no	—
23	C₂×#13083564	C₂²	1	1	yes	characteristic, complemented
24	C₂×#13083563	C₂²	1	1	yes	characteristic
25	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
26	C₂×#13083562	C₂²	1	1	yes	characteristic
27	C₂×#13083561	C₂²	1	1	yes	characteristic
28	C₂×#13083561	C₂²	1	1	yes	characteristic
29	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
30	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
31	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
32	SD₁₆×D₈₉	—	2	4	no	—
33	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
34	C₂×D₇₁₂	C₂²	1	1	yes	characteristic, complemented
35	C₂×#13083559	C₂²	1	1	yes	characteristic
36	C₂×#13083557	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
38	#15668010	C₂²	1	1	yes	characteristic, complemented
39	#15668009	C₂²	1	1	yes	characteristic
40	#15668008	C₂²	1	1	yes	characteristic
41	#15668001	C₂²	1	1	yes	characteristic
42	#15668001	C₂²	1	1	yes	characteristic, complemented
43	#15668000	C₂²	1	1	yes	characteristic, complemented
44	#15667997	—	2	2	no	—
45	#15667990	—	2	2	no	—
46	#15667985	—	2	2	no	—
47	#15667982	—	2	2	no	—
48	#15667969	C₂²	1	1	yes	characteristic
49	#15667922	C₂²	1	1	yes	characteristic, complemented
50	C₄×#10623494	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
54	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#124×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	#13083575	—	2	2	no	—
61	#13083575	—	2	2	no	—
62	#13083575	—	2	2	no	—
63	#13083575	—	2	2	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	Q₈×D₈₉	—	2	2	no	—
67	Q₈×D₈₉	—	2	2	no	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	—	2	1	no	—
70	C₂×D₃₅₆	—	2	2	no	—
71	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
72	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
73	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊D₈₉	—	2	1	no	—
75	C₂×C₂²⋊D₈₉	—	2	2	no	—
76	D₄×D₈₉	—	2	2	no	—
77	D₄×D₈₉	—	2	2	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×C₄×D₈₉	—	2	1	no	—
81	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
82	#13083564	—	2	2	no	—
83	#13083563	—	2	2	no	—
84	#13083563	—	2	2	no	—
85	#13083562	—	2	2	no	—
86	#13083561	—	2	2	no	—
87	#13083561	—	2	2	no	—
88	#13083561	—	2	2	no	—
89	#13083561	—	2	2	no	—
90	#13083561	—	2	2	no	—
91	D₇₁₂	—	2	2	no	—
92	D₇₁₂	—	2	2	no	—
93	#13083559	—	2	2	no	—
94	#13083557	D₄	1	4	yes	complemented
95	C₂×#10623494	C₂³	1	1	yes	characteristic
96	C₂×#10623494	C₂³	1	1	yes	characteristic
97	C₈×D₈₉	D₄	1	4	yes	complemented
98	#13083553	—	2	2	no	—
99	#13083553	D₄	1	2	yes	—
100	#13083551	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	#13083551	—	2	2	no	—
103	#13083551	D₄	1	2	yes	—
104	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
109	Q₁₆×C₈₉	—	2	2	no	—
110	Q₁₆×C₈₉	—	2	2	no	—
111	SD₁₆×C₈₉	—	2	2	no	—
112	SD₁₆×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₂²⋊C₄×C₈₉	—	2	2	no	—
117	C₄²×C₈₉	C₂³	1	1	yes	characteristic
118	C₂³×C₈₉	—	2	1	no	—
119	C₂²×D₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	2	1	no	—
121	C₂²×D₈₉	—	2	1	no	—
122	C₂²×D₈₉	—	4	2	no	—
123	C₂²×D₈₉	—	8	2	no	—
124	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₈₉	—	2	1	no	—
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	—	2	1	no	—
128	Q₈×C₈₉	—	2	1	no	—
129	Q₈×C₈₉	—	2	1	no	—
130	Q₈×C₈₉	—	2	1	no	—
131	D₄×C₈₉	—	2	1	no	—
132	D₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂²⋊D₈₉	—	8	2	no	—
140	C₂×Dic₈₉	—	2	1	no	—
141	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
142	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	—	2	1	no	—
146	D₃₅₆	—	2	1	no	—
147	D₃₅₆	—	2	1	no	—
148	D₃₅₆	—	2	1	no	—
149	D₃₅₆	—	4	2	no	—
150	D₃₅₆	—	4	2	no	—
151	D₃₅₆	—	8	2	no	—
152	C₄×D₈₉	—	4	2	no	—
153	C₄×D₈₉	—	4	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	C₄×D₈₉	C₂×D₄	1	2	yes	—
156	C₄×D₈₉	C₂×D₄	1	2	yes	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	4	2	no	—
160	C₈×C₈₉	C₂×D₄	1	2	yes	—
161	C₈×C₈₉	C₂×D₄	1	2	yes	—
162	#10623494	C₂×D₄	1	2	yes	—
163	#10623494	C₂×D₄	1	2	yes	—
164	C₂²×C₈₉	—	4	1	no	—
165	C₂²×C₈₉	—	4	2	no	—
166	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
167	C₂×D₈₉	—	2	2	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₂×D₈₉	—	4	2	no	—
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	8	4	no	—
175	C₂×D₈₉	#127	1	2	yes	—
176	C₄×C₈₉	—	2	2	no	—
177	C₄×C₈₉	—	4	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
181	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
182	Dic₈₉	—	2	2	no	—
183	Dic₈₉	—	4	1	no	—
184	Dic₈₉	#127	1	2	yes	—
185	C₂×C₈₉	—	8	1	no	—
186	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
187	C₂×C₈₉	#577	1	2	yes	—
188	D₈₉	—	2	2	no	—
189	D₈₉	—	8	1	no	—
190	D₈₉	—	8	1	no	—
191	D₈₉	—	8	1	no	—
192	#3154	—	89	1	no	maximal, Sylow
193	C₈₉	#3154	1	1	yes	characteristic, complemented, Sylow
194	C₂×#137	—	89	1	no	—
195	C₂×AGL(1,ℤ/8)	—	89	1	no	—
196	C₂×#135	—	89	1	no	—
197	C₂²×SD₁₆	—	89	1	no	—
198	#536	—	89	1	no	—
199	#535	—	89	1	no	—
200	#497	—	89	1	no	—
201	#497	D₈₉	1	1	yes	characteristic, complemented
202	#496	—	89	1	no	—
203	#496	—	89	1	no	—
204	#495	—	89	1	no	—
205	#495	—	89	1	no	—
206	#492	—	89	1	no	—
207	#492	—	89	1	no	—
208	#405	—	89	1	no	—
209	C₂×#41	—	89	1	no	—
210	C₂×#41	—	89	1	no	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂²×D₄	—	89	1	no	—
213	#137	—	178	4	no	—
214	AGL(1,ℤ/8)	—	178	4	no	—
215	#135	—	178	4	no	—
216	C₂×Q₁₆	—	89	1	no	—
217	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
218	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
219	C₂×SD₁₆	—	89	1	no	—
220	C₂×SD₁₆	—	89	1	no	—
221	C₂×SD₁₆	—	89	1	no	—
222	C₂×SD₁₆	—	89	1	no	—
223	C₂×SD₁₆	—	89	1	no	—
224	C₂×SD₁₆	—	89	1	no	—
225	C₂×SD₁₆	—	89	1	no	—
226	C₂×SD₁₆	—	89	1	no	—
227	C₂×SD₁₆	—	178	4	no	—
228	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
229	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
230	C₂×D₈	—	89	1	no	—
231	C₂×D₈	—	89	1	no	—
232	C₂×D₈	—	89	1	no	—
233	C₂×M₁₆	—	89	1	no	—
234	C₂²×C₈	—	89	1	no	—
235	C₄⋊Q₈	—	89	1	no	—
236	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
237	#127	—	89	1	no	—
238	#127	—	89	1	no	—
239	#124	—	89	1	no	—
240	#124	—	89	1	no	—
241	#124	—	89	1	no	—
242	#124	C₂×D₈₉	1	1	yes	characteristic
243	#123	—	178	2	no	—
244	C₂²⋊Q₈	—	178	2	no	—
245	#121	—	178	2	no	—
246	#120	—	178	2	no	—
247	#117	—	89	1	no	—
248	#97	—	89	1	no	—
249	#97	C₂×D₈₉	1	1	yes	characteristic, complemented
250	C₄×C₈	—	89	1	no	—
251	C₄×C₈	—	89	1	no	—
252	C₂⁴	—	178	1	no	—
253	#41	—	178	2	no	—
254	#41	—	178	2	no	—
255	#41	—	178	2	no	—
256	#41	—	178	2	no	—
257	C₂×Q₈	—	89	1	no	—
258	C₂×Q₈	—	178	1	no	—
259	C₂×Q₈	—	178	2	no	—
260	C₂×Q₈	—	178	2	no	—
261	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×D₄	—	89	1	no	—
265	C₂×D₄	—	89	1	no	—
266	C₂×D₄	—	89	1	no	—
267	C₂×D₄	—	178	1	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	—	178	2	no	—
270	C₂×D₄	—	178	2	no	—
271	C₂×D₄	—	178	1	no	—
272	C₂×D₄	—	178	2	no	—
273	C₂×D₄	—	178	2	no	—
274	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	—	89	1	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	C₂²×C₄	—	178	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	178	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	178	2	no	—
285	SD₁₆	—	178	2	no	—
286	SD₁₆	—	178	2	no	—
287	SD₁₆	—	178	2	no	—
288	SD₁₆	—	178	2	no	—
289	SD₁₆	—	178	2	no	—
290	SD₁₆	—	178	2	no	—
291	SD₁₆	—	178	2	no	—
292	D₈	—	178	2	no	—
293	D₈	—	178	2	no	—
294	D₈	—	178	2	no	—
295	M₁₆	—	89	4	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	89	1	no	—
298	C₂×C₈	—	89	4	no	—
299	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	89	2	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	89	2	no	—
306	C₂²⋊C₄	—	178	2	no	—
307	C₂²⋊C₄	—	178	2	no	—
308	C₂²⋊C₄	—	178	2	no	—
309	C₄²	—	89	1	no	—
310	C₄²	C₂²×D₈₉	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	89	1	no	—
313	C₂³	—	356	2	no	—
314	C₂³	—	178	1	no	—
315	C₂³	—	178	1	no	—
316	C₂³	—	178	1	no	—
317	C₂³	—	712	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	356	2	no	—
325	Q₈	—	178	1	no	—
326	Q₈	—	178	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	356	2	no	—
330	D₄	—	356	2	no	—
331	D₄	—	356	2	no	—
332	D₄	—	178	1	no	—
333	D₄	—	178	1	no	—
334	D₄	—	178	1	no	—
335	D₄	—	178	1	no	—
336	D₄	—	178	1	no	—
337	D₄	—	178	1	no	—
338	D₄	—	712	2	no	—
339	D₄	—	712	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	2	no	—
345	C₂×C₄	—	89	2	no	—
346	C₂×C₄	—	89	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	D₄×D₈₉	1	2	yes	—
352	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
353	C₈	—	89	2	no	—
354	C₈	—	89	2	no	—
355	C₈	D₄×D₈₉	1	2	yes	—
356	C₈	D₄×D₈₉	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	89	2	no	—
360	C₂²	—	356	1	no	—
361	C₂²	—	356	1	no	—
362	C₂²	—	356	1	no	—
363	C₂²	—	356	2	no	—
364	C₂²	—	356	2	no	—
365	C₂²	—	356	2	no	—
366	C₂²	—	178	2	no	—
367	C₂²	—	712	4	no	—
368	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	89	2	no	—
374	C₄	—	356	1	no	—
375	C₄	—	178	2	no	—
376	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
377	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	178	2	no	—
380	C₂	—	712	1	no	—
381	C₂	—	712	1	no	—
382	C₂	—	712	1	no	—
383	C₂	#127×D₈₉	1	1	yes	characteristic
384	C₂	#19152407	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#497×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#497	D₈₉
hypocenter	C₈₉	#3154
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#15668044	C₂	semidirect
2. C₂×#15668039	C₂	semidirect
3. C₂×#15668035	C₂	semidirect
4. C₂×SD₁₆×D₈₉	C₂	semidirect
5. #19152173	C₂	semidirect
6. #19152166	C₂	semidirect
7. #19151991	C₂	semidirect
8. #19151987	C₂	semidirect
9. #19151984	C₂	semidirect
10. #19151983	C₂	semidirect
11. #19151980	C₂	semidirect
12. #19151972	C₂	semidirect
13. #19151969	C₂	semidirect
14. #19151555	C₂	semidirect
15. #497×C₈₉	C₂	semidirect
23. C₂×#13083564	C₂²	semidirect
25. C₂×#13083563	C₂²	semidirect
29. C₂×#13083561	C₂²	semidirect
30. C₂×#13083561	C₂²	semidirect
31. C₂×#13083561	C₂²	semidirect
34. C₂×D₇₁₂	C₂²	semidirect
38. #15668010	C₂²	semidirect
42. #15668001	C₂²	semidirect
43. #15668000	C₂²	semidirect
49. #15667922	C₂²	semidirect
50. C₄×#10623494	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
52. C₂×Q₁₆×C₈₉	C₂²	semidirect
53. C₂×Q₁₆×C₈₉	C₂²	semidirect
56. C₄⋊Q₈×C₈₉	C₂²	semidirect
58. #97×C₈₉	C₂²	semidirect
94. #13083557	D₄	semidirect
97. C₈×D₈₉	D₄	semidirect
193. C₈₉	#3154	semidirect
201. #497	D₈₉	semidirect
217. C₂×Q₁₆	C₂×D₈₉	semidirect
218. C₂×Q₁₆	C₂×D₈₉	semidirect
236. C₄⋊Q₈	C₂×D₈₉	semidirect
249. #97	C₂×D₈₉	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×#15668039	2
C₂×#15668035	2
C₂×SD₁₆×D₈₉	2
#19152173	2
#19152166	2
#19151991	2
#19151987	2
#19151984	2
#19151983	2
#19151980	2
#19151972	2
#19151969	2
#19151555	2
#497×C₈₉	2
#3154	89