Subgroups — #26029364

Order 11392.

388 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	C₄×#13083573	C₂	1	1	yes	maximal, characteristic
2	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151674	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151673	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151670	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151669	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151666	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151653	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151652	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151648	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151621	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151620	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667941	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₄	1	1	yes	characteristic
17	C₂×#13083558	C₂²	1	1	yes	characteristic
18	C₂×#13083557	C₂²	1	1	yes	characteristic
19	C₂×#13083557	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
22	#15667977	C₄	1	1	yes	characteristic
23	D₄×Dic₈₉	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
26	#15667972	C₄	1	1	yes	characteristic
27	#15667972	C₄	1	1	yes	characteristic
28	#15667970	C₄	1	1	yes	characteristic
29	#15667968	C₄	1	1	yes	characteristic
30	#15667968	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₄²×D₈₉	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667941	C₂²	1	1	yes	characteristic
36	#15667941	C₂²	1	1	yes	characteristic
37	#15667941	C₂²	1	1	yes	characteristic
38	#15667941	C₂²	1	1	yes	characteristic
39	#15667941	C₂²	1	1	yes	characteristic
40	#15667941	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
44	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
45	#15667924	C₂²	1	1	yes	characteristic
46	#15667924	C₂²	1	1	yes	characteristic
47	#15667923	C₂²	1	1	yes	characteristic
48	#15667922	C₂²	1	1	yes	characteristic
49	#15667922	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	2	4	no	—
60	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	#13083558	—	4	1	no	—
71	#13083557	—	4	1	no	—
72	#13083557	—	4	1	no	—
73	C₂×#10623494	—	4	1	no	—
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
79	#13083553	C₂×C₄	1	1	yes	characteristic
80	#13083553	C₂×C₄	1	1	yes	characteristic
81	#13083553	C₂×C₄	1	1	yes	characteristic
82	#13083553	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
85	#13083551	C₂×C₄	1	1	yes	characteristic
86	#13083551	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
99	M₁₆×C₈₉	—	4	1	no	—
100	C₂×C₈×C₈₉	—	4	1	no	—
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	2	2	no	—
123	C₂²⋊D₈₉	—	2	2	no	—
124	C₂×Dic₈₉	—	2	2	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	C₂×Dic₈₉	M₁₆	1	4	yes	complemented
128	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₈₉	—	2	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	4	1	no	—
139	Dic₁₇₈	—	2	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
154	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	2	1	no	—
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	C₂×M₁₆	1	2	yes	—
170	Dic₈₉	C₂×M₁₆	1	2	yes	—
171	C₂×C₈₉	—	2	2	no	—
172	C₂×C₈₉	—	4	1	no	—
173	C₂×C₈₉	#520	1	1	yes	characteristic
174	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₈₉	#568	1	1	yes	characteristic
176	D₈₉	—	4	1	no	—
177	#2989	—	89	1	no	maximal, Sylow
178	C₈₉	#2989	1	1	yes	characteristic, complemented, Sylow
179	C₄×#41	—	89	1	no	—
180	#446	—	89	1	no	—
181	#446	—	89	1	no	—
182	#436	—	89	1	no	—
183	#436	—	89	1	no	—
184	#435	—	89	1	no	—
185	#435	—	89	1	no	—
186	#435	—	89	1	no	—
187	#435	D₈₉	1	1	yes	characteristic, complemented
188	#432	—	89	1	no	—
189	#432	—	89	1	no	—
190	#431	—	89	1	no	—
191	#423	—	89	1	no	—
192	#423	—	89	1	no	—
193	C₂×C₄⋊C₈	—	89	1	no	—
194	C₂×#41	—	89	1	no	—
195	C₂×M₁₆	—	89	1	no	—
196	C₂×M₁₆	—	89	1	no	—
197	C₂×M₁₆	—	89	1	no	—
198	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₂²×C₈	—	89	1	no	—
200	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
201	C₄×Q₈	—	89	1	no	—
202	C₄×Q₈	—	89	1	no	—
203	C₄×D₄	—	89	1	no	—
204	C₄×D₄	—	89	1	no	—
205	C₄×D₄	—	89	1	no	—
206	C₄×D₄	—	89	1	no	—
207	C₄×D₄	—	89	1	no	—
208	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
209	#117	—	89	1	no	—
210	#117	—	89	1	no	—
211	#117	—	89	1	no	—
212	C₂×C₄²	—	89	1	no	—
213	C₂×C₄²	—	89	1	no	—
214	C₂×C₄²	—	89	1	no	—
215	C₄⋊C₈	—	89	1	no	—
216	C₄⋊C₈	—	89	1	no	—
217	C₄⋊C₈	—	89	1	no	—
218	C₄⋊C₈	—	89	1	no	—
219	C₄⋊C₈	—	89	1	no	—
220	C₄⋊C₈	—	89	1	no	—
221	C₄⋊C₈	—	89	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₄⋊C₈	—	89	1	no	—
224	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
225	C₂²⋊C₈	—	89	1	no	—
226	C₂²⋊C₈	—	89	1	no	—
227	C₂²⋊C₈	—	89	1	no	—
228	C₂²⋊C₈	—	89	1	no	—
229	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
230	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
231	#97	—	89	1	no	—
232	#97	—	89	1	no	—
233	#97	—	89	1	no	—
234	#97	C₂×D₈₉	1	1	yes	characteristic
235	C₄×C₈	—	89	1	no	—
236	C₄×C₈	—	89	1	no	—
237	#41	—	178	4	no	—
238	C₂×Q₈	—	89	1	no	—
239	C₂×D₄	—	89	1	no	—
240	C₂×D₄	—	89	1	no	—
241	C₂×D₄	C₄×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	—	89	1	no	—
243	C₂²×C₄	—	89	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	356	1	no	—
253	M₁₆	—	356	1	no	—
254	M₁₆	—	356	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	89	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	356	1	no	—
261	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
271	C₂²⋊C₄	—	89	1	no	—
272	C₂²⋊C₄	—	89	1	no	—
273	C₂²⋊C₄	—	89	1	no	—
274	C₂²⋊C₄	—	89	1	no	—
275	C₂²⋊C₄	C₄×D₈₉	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₈₉	1	1	yes	characteristic
277	C₄²	—	89	1	no	—
278	C₄²	—	89	1	no	—
279	C₄²	—	89	1	no	—
280	C₄²	—	89	1	no	—
281	C₄²	—	89	1	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	—	89	1	no	—
285	C₄²	—	89	1	no	—
286	C₄²	C₂²×D₈₉	1	1	yes	characteristic
287	C₂³	—	89	1	no	—
288	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
290	Q₈	—	178	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	178	2	no	—
293	D₄	—	178	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	89	4	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	89	1	no	—
306	C₂×C₄	—	356	1	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	178	2	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	178	1	no	—
323	C₈	—	178	1	no	—
324	C₈	—	178	1	no	—
325	C₈	—	178	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	178	1	no	—
332	C₂²	—	178	1	no	—
333	C₂²	—	178	1	no	—
334	C₂²	M₁₆×D₈₉	1	2	yes	—
335	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	89	2	no	—
341	C₄	—	89	2	no	—
342	C₄	—	178	1	no	—
343	C₄	—	178	1	no	—
344	C₄	—	178	1	no	—
345	C₄	—	178	1	no	—
346	C₄	—	178	1	no	—
347	C₄	#15668057	1	1	yes	characteristic
348	C₄	#15668059	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	356	1	no	—
352	C₂	#19152077	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₈₉	1	1	yes	characteristic
354	C₂	#19152363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#435×C₈₉	C₂
prime radical	C₂×#13083573	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#435	D₈₉
hypocenter	C₈₉	#2989
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #19151727	C₂	semidirect
3. #19151726	C₂	semidirect
4. #19151674	C₂	semidirect
5. #19151673	C₂	semidirect
6. #19151670	C₂	semidirect
7. #19151669	C₂	semidirect
8. #19151666	C₂	semidirect
9. #19151653	C₂	semidirect
10. #19151652	C₂	semidirect
11. #19151648	C₂	semidirect
12. #19151621	C₂	semidirect
13. #19151620	C₂	semidirect
14. C₂×#15667941	C₂	semidirect
15. #435×C₈₉	C₂	semidirect
127. C₂×Dic₈₉	M₁₆	semidirect
178. C₈₉	#2989	semidirect
187. #435	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13083573	2
#19151727	2
#19151726	2
#19151674	2
#19151673	2
#19151670	2
#19151669	2
#19151666	2
#19151653	2
#19151652	2
#19151648	2
#19151621	2
#19151620	2
C₂×#15667941	2
#435×C₈₉	2
#2989	89