Subgroups — #26029068

Order 11392.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 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
2	C₄×#13083575	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151696	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151685	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151683	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151637	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151598	C₂	1	1	yes	maximal, characteristic
14	#19151550	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	#15668044	C₄	1	8	yes	complemented
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083563	C₂²	1	1	yes	characteristic
21	C₂×#13083561	C₂²	1	1	yes	characteristic
22	C₂×#13083557	C₂²	1	1	yes	characteristic
23	C₂×#13083557	C₂²	1	1	yes	characteristic
24	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₇₈	—	2	1	no	—
27	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
28	#15667977	—	2	1	no	—
29	#15667977	C₂²	1	1	yes	characteristic
30	C₄×D₃₅₆	—	2	1	no	—
31	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
32	#15667974	—	2	1	no	—
33	#15667974	C₂²	1	1	yes	characteristic
34	#15667970	—	2	1	no	—
35	#15667970	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	—	2	1	no	—
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	C₄²×D₈₉	—	2	1	no	—
39	C₄²×D₈₉	—	2	1	no	—
40	C₄²×D₈₉	C₂²	1	1	yes	characteristic
41	#15667946	C₂²	1	1	yes	characteristic
42	#15667945	C₂²	1	1	yes	characteristic
43	#15667943	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667923	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#106×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#97×C₈₉	C₂²	1	1	yes	characteristic
60	#13083575	—	2	2	no	—
61	#13083575	C₂×C₄	1	4	yes	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	Q₈×D₈₉	—	2	2	no	—
65	Q₈×D₈₉	C₂×C₄	1	4	yes	complemented
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
76	#13083564	C₂×C₄	1	4	yes	complemented
77	#13083563	C₂×C₄	1	4	yes	complemented
78	#13083561	C₂×C₄	1	4	yes	complemented
79	#13083557	—	2	2	no	—
80	#13083557	C₂×C₄	1	4	yes	complemented
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₂×#10623494	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	D₄	1	1	yes	characteristic
99	C₄×Dic₈₉	D₄	1	1	yes	characteristic
100	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	Q₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²×D₈₉	—	2	1	no	—
114	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
118	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₈₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	D₃₅₆	—	2	4	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	C₂²×C₄	1	2	yes	—
135	C₄×D₈₉	—	2	4	no	—
136	C₄×D₈₉	—	2	4	no	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	4	1	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	4	2	no	—
141	C₄×D₈₉	—	4	2	no	—
142	C₄×D₈₉	C₂²×C₄	1	2	yes	—
143	C₄×D₈₉	#41	1	2	yes	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	4	no	—
146	Dic₁₇₈	C₂²×C₄	1	2	yes	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	C₂²×C₄	1	2	yes	—
149	#10623494	—	2	1	no	—
150	#10623494	C₂²×C₄	1	2	yes	—
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	2	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	C₄×D₄	1	2	yes	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
165	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	C₄×D₄	1	2	yes	—
171	Dic₈₉	#137	1	2	yes	complemented
172	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
174	C₂×C₈₉	#575	1	1	yes	characteristic
175	D₈₉	—	2	2	no	—
176	D₈₉	—	4	2	no	—
177	C₄×#137	—	89	1	no	maximal, Sylow
178	C₈₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
179	C₂×#137	—	89	1	no	—
180	C₄×#41	—	89	1	no	—
181	C₂×C₄×Q₈	—	89	1	no	—
182	#441	—	89	1	no	—
183	#441	D₈₉	1	1	yes	characteristic, complemented
184	#440	—	89	1	no	—
185	#440	—	89	1	no	—
186	C₄×Q₁₆	—	89	1	no	—
187	C₄×Q₁₆	—	89	1	no	—
188	C₄×SD₁₆	—	89	1	no	—
189	C₄×SD₁₆	—	89	1	no	—
190	#428	—	89	1	no	—
191	#419	—	89	1	no	—
192	#417	—	89	1	no	—
193	C₄×M₁₆	—	89	1	no	—
194	C₂×#41	—	89	1	no	—
195	C₂²×Q₈	—	89	1	no	—
196	#137	—	89	8	no	—
197	C₂×Q₁₆	—	89	1	no	—
198	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×SD₁₆	—	89	1	no	—
201	C₂×M₁₆	—	89	1	no	—
202	C₂×M₁₆	—	89	1	no	—
203	C₄×Q₈	—	89	1	no	—
204	C₄×Q₈	—	89	1	no	—
205	C₄×Q₈	—	89	1	no	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×Q₈	—	178	1	no	—
208	C₄×Q₈	—	178	1	no	—
209	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
211	C₄×D₄	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	178	1	no	—
214	C₄×D₄	—	178	1	no	—
215	#117	—	89	1	no	—
216	#117	—	178	1	no	—
217	C₂×C₄⋊C₄	—	89	1	no	—
218	C₂×C₄⋊C₄	—	178	1	no	—
219	C₂×C₄²	—	89	1	no	—
220	C₂×C₄²	—	178	1	no	—
221	C₂×C₄²	—	178	1	no	—
222	#107	—	89	1	no	—
223	#107	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	C₂×D₈₉	1	1	yes	characteristic
226	#103	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	C₂×D₈₉	1	1	yes	characteristic
231	#103	C₂×D₈₉	1	1	yes	characteristic
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#97	—	89	1	no	—
235	#97	C₂×D₈₉	1	1	yes	characteristic
236	C₄×C₈	—	89	1	no	—
237	C₄×C₈	—	89	1	no	—
238	#41	—	89	4	no	—
239	#41	—	178	2	no	—
240	C₂×Q₈	—	89	4	no	—
241	C₂×Q₈	—	89	1	no	—
242	C₂×Q₈	—	178	1	no	—
243	C₂×Q₈	—	178	2	no	—
244	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×D₄	—	89	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	—	89	1	no	—
250	C₂²×C₄	—	89	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	Q₁₆	—	89	4	no	—
257	Q₁₆	C₄×D₈₉	1	4	yes	complemented
258	SD₁₆	—	89	4	no	—
259	SD₁₆	—	89	4	no	—
260	M₁₆	—	89	4	no	—
261	M₁₆	—	178	2	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	—	89	1	no	—
285	C₄²	—	178	1	no	—
286	C₄²	—	178	1	no	—
287	C₄²	—	178	1	no	—
288	C₄²	—	178	1	no	—
289	C₄²	—	178	1	no	—
290	C₄²	C₂²×D₈₉	1	1	yes	characteristic
291	C₂³	—	89	1	no	—
292	C₂³	—	178	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	89	2	no	—
296	Q₈	—	178	4	no	—
297	Q₈	—	178	1	no	—
298	Q₈	C₂×C₄×D₈₉	1	2	yes	—
299	Q₈	C₂×C₄×D₈₉	1	2	yes	—
300	D₄	—	89	2	no	—
301	D₄	—	178	4	no	—
302	D₄	—	178	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	89	1	no	—
308	C₂×C₄	—	89	1	no	—
309	C₂×C₄	—	89	2	no	—
310	C₂×C₄	—	89	2	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	1	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	178	2	no	—
317	C₂×C₄	—	178	4	no	—
318	C₂×C₄	—	178	4	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
323	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	89	2	no	—
327	C₈	—	178	1	no	—
328	C₈	C₂×C₄×D₈₉	1	2	yes	—
329	C₂²	—	89	2	no	—
330	C₂²	—	356	1	no	—
331	C₂²	—	356	1	no	—
332	C₂²	—	178	2	no	—
333	C₂²	—	178	1	no	—
334	C₂²	—	178	1	no	—
335	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	89	2	no	—
343	C₄	—	89	2	no	—
344	C₄	—	356	1	no	—
345	C₄	—	356	1	no	—
346	C₄	—	178	1	no	—
347	C₄	—	178	2	no	—
348	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
349	C₄	#41×D₈₉	1	1	yes	characteristic
350	C₂	—	356	2	no	—
351	C₂	—	178	2	no	—
352	C₂	C₄×D₄×D₈₉	1	1	yes	characteristic
353	C₂	#137×D₈₉	1	1	yes	characteristic
354	C₂	#19152397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#441×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083575	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#441	D₈₉
hypocenter	C₈₉	C₄×#137
LCS	G ▷ 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
3. C₄×Q₈×D₈₉	C₂	semidirect
4. #19151703	C₂	semidirect
5. #19151700	C₂	semidirect
6. #19151696	C₂	semidirect
7. Q₁₆×Dic₈₉	C₂	semidirect
8. #19151690	C₂	semidirect
9. #19151685	C₂	semidirect
10. #19151683	C₂	semidirect
11. #19151637	C₂	semidirect
12. #19151605	C₂	semidirect
14. #19151550	C₂	semidirect
15. #441×C₈₉	C₂	semidirect
18. #15668044	C₄	semidirect
65. Q₈×D₈₉	C₂×C₄	semidirect
76. #13083564	C₂×C₄	semidirect
77. #13083563	C₂×C₄	semidirect
78. #13083561	C₂×C₄	semidirect
80. #13083557	C₂×C₄	semidirect
103. Q₁₆×C₈₉	C₂×C₄	semidirect
171. Dic₈₉	#137	semidirect
178. C₈₉	C₄×#137	semidirect
183. #441	D₈₉	semidirect
257. Q₁₆	C₄×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
C₄×#13083575	2
C₄×Q₈×D₈₉	2
#19151703	2
#19151700	2
#19151696	2
Q₁₆×Dic₈₉	2
#19151690	2
#19151685	2
#19151683	2
#19151637	2
#19151605	2
#19151598	2
#19151550	2
#441×C₈₉	2
C₄×#137	89