Subgroups — #135×#10623494 #36523807

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10623494	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#10623494	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#10623494	C₂	1	4	yes	maximal, complemented
4	D₈×#10623494	C₂	1	2	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018634	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic
9	Q₈×#10623494	C₂²	1	4	yes	—
10	D₄×#10623494	—	2	4	no	—
11	D₄×#10623494	C₂²	1	4	yes	—
12	#19151651	—	2	2	no	—
13	#19151651	C₂²	1	2	yes	—
14	C₂×C₄×#10623494	—	2	2	no	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	#19151351	C₂²	1	2	yes	—
18	#19151349	C₂²	1	2	yes	—
19	Q₈⋊#10623494	C₂²	1	4	yes	—
20	D₄⋊#10623494	C₂²	1	4	yes	—
21	C₈×#10623494	C₂²	1	2	yes	—
22	C₈×#10623494	C₂²	1	2	yes	—
23	C₂×#135×C₈₉	C₄	1	1	yes	characteristic
24	C₄×#41×C₈₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₈₉	C₄	1	2	yes	—
28	C₄×SD₁₆×C₈₉	C₂²	1	2	yes	—
29	C₄×D₈×C₈₉	C₄	1	1	yes	characteristic
30	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
31	#427×C₈₉	C₄	1	1	yes	characteristic
32	#416×C₈₉	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂²×#10623494	—	2	4	no	—
35	C₂²×#10623494	—	2	1	no	—
36	C₂²×#10623494	D₄	1	2	yes	—
37	C₄⋊#10623494	—	2	4	no	—
38	C₄⋊#10623494	C₂³	1	4	yes	—
39	C₂²⋊#10623494	—	2	4	no	—
40	C₂²⋊#10623494	—	2	2	no	—
41	C₄×#10623494	—	2	4	no	—
42	C₄×#10623494	—	2	2	no	—
43	C₄×#10623494	—	2	1	no	—
44	C₄×#10623494	D₄	1	1	yes	characteristic
45	C₄×#10623494	D₄	1	1	yes	characteristic
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₄×#10623494	C₂³	1	2	yes	—
48	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
49	#135×C₈₉	C₈	1	8	yes	complemented
50	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
57	C₄×D₄×C₈₉	—	2	2	no	—
58	C₄×D₄×C₈₉	—	2	2	no	—
59	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₈₉	C₂³	1	2	yes	—
61	#117×C₈₉	—	2	2	no	—
62	#117×C₈₉	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
66	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
67	#106×C₈₉	C₂×C₄	1	2	yes	—
68	#103×C₈₉	C₂×C₄	1	2	yes	—
69	#103×C₈₉	C₂×C₄	1	2	yes	—
70	#102×C₈₉	C₂×C₄	1	2	yes	—
71	#102×C₈₉	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×#10623494	—	2	4	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	4	no	—
79	C₂×#10623494	—	2	4	no	—
80	C₂×#10623494	—	4	8	no	—
81	C₂×#10623494	C₂×D₄	1	2	yes	—
82	C₂×#10623494	C₂×D₄	1	2	yes	—
83	#41×C₈₉	—	2	4	no	—
84	#41×C₈₉	C₂×C₈	1	8	yes	complemented
85	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₈₉	—	2	2	no	—
87	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	—	2	2	no	—
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
94	Q₁₆×C₈₉	C₂×C₈	1	4	yes	complemented
95	SD₁₆×C₈₉	C₂×C₈	1	8	yes	complemented
96	D₈×C₈₉	C₂×C₈	1	4	yes	complemented
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	complemented
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	2	2	no	—
105	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
106	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂²⋊C₄×C₈₉	—	2	2	no	—
108	C₂²⋊C₄×C₈₉	—	2	2	no	—
109	C₂²⋊C₄×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₉	—	2	2	no	—
118	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₈₉	—	2	2	no	—
120	Q₈×C₈₉	C₂²×C₈	1	4	yes	—
121	D₄×C₈₉	—	2	8	no	—
122	D₄×C₈₉	—	2	2	no	—
123	D₄×C₈₉	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	8	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	4	4	no	—
131	C₂×C₄×C₈₉	—	4	4	no	—
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₈₉	#131	1	2	yes	—
137	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	C₂²×C₈	1	2	yes	—
143	C₈×C₈₉	C₂²×C₈	1	2	yes	—
144	#10623494	—	2	4	no	—
145	#10623494	—	2	2	no	—
146	#10623494	—	4	4	no	—
147	#10623494	#135	1	4	yes	direct product factor
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	4	no	—
151	C₂²×C₈₉	—	4	2	no	—
152	C₂²×C₈₉	—	4	2	no	—
153	C₂²×C₈₉	C₈×D₄	1	2	yes	—
154	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₈₉	—	2	4	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	4	2	no	—
159	C₄×C₈₉	—	4	2	no	—
160	C₄×C₈₉	C₈×D₄	1	2	yes	—
161	C₄×C₈₉	#443	1	2	yes	—
162	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
164	C₄×C₈₉	C₂×#135	1	2	yes	—
165	C₈×#135	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	2	2	no	—
167	C₂×C₈₉	—	4	4	no	—
168	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₈₉	C₄×#135	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
171	C₈×#41	—	89	2	no	—
172	C₄×#135	D₈₉	1	1	yes	characteristic
173	C₈×Q₁₆	—	89	2	no	—
174	C₈×SD₁₆	—	89	4	no	—
175	C₈×D₈	—	89	2	no	—
176	#1564	—	89	1	no	—
177	#1476	—	89	2	no	—
178	C₂×C₈²	—	89	1	no	—
179	C₈₉	C₈×#135	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	Dic₈₉	1	1	yes	characteristic
181	C₄×#41	C₂×D₈₉	1	2	yes	—
182	C₈×Q₈	—	89	4	no	—
183	C₄×Q₁₆	Dic₈₉	1	1	yes	characteristic
184	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
185	C₄×SD₁₆	Dic₈₉	1	2	yes	—
186	C₄×SD₁₆	C₂×D₈₉	1	2	yes	—
187	C₄×D₈	Dic₈₉	1	1	yes	characteristic
188	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
189	C₈×D₄	—	89	4	no	—
190	C₈×D₄	—	178	4	no	—
191	#431	—	89	2	no	—
192	#431	—	178	2	no	—
193	#427	Dic₈₉	1	1	yes	characteristic
194	#416	Dic₈₉	1	2	yes	—
195	C₂×C₄×C₈	—	89	1	no	—
196	C₂×C₄×C₈	—	89	1	no	—
197	C₂×C₄×C₈	—	178	2	no	—
198	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
199	#335	—	89	2	no	—
200	#334	—	89	2	no	—
201	Q₈⋊C₈	—	89	4	no	—
202	D₄⋊C₈	—	89	4	no	—
203	C₈²	—	89	2	no	—
204	C₈²	—	89	2	no	—
205	C₂×#41	C₂×Dic₈₉	1	2	yes	—
206	#135	#10623494	1	8	yes	direct product factor
207	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
208	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
209	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
210	C₂²×C₈	—	89	2	no	—
211	C₂²×C₈	—	178	4	no	—
212	C₂²×C₈	—	178	1	no	—
213	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
215	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
216	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
220	C₄×D₄	C₂²×D₈₉	1	2	yes	—
221	#117	—	2	2	no	—
222	#117	C₂×Dic₈₉	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
225	#107	C₂×Dic₈₉	1	1	yes	characteristic
226	#107	C₂×Dic₈₉	1	1	yes	characteristic
227	#106	C₂×Dic₈₉	1	2	yes	—
228	C₄⋊C₈	—	89	4	no	—
229	C₄⋊C₈	—	178	4	no	—
230	#103	C₂×Dic₈₉	1	2	yes	—
231	#103	C₂×Dic₈₉	1	2	yes	—
232	#102	C₂×Dic₈₉	1	2	yes	—
233	#102	C₂×Dic₈₉	1	2	yes	—
234	C₂²⋊C₈	—	178	4	no	—
235	C₂²⋊C₈	—	178	2	no	—
236	C₄×C₈	—	89	1	no	—
237	C₄×C₈	—	89	1	no	—
238	C₄×C₈	—	89	2	no	—
239	C₄×C₈	—	89	2	no	—
240	C₄×C₈	—	178	4	no	—
241	C₄×C₈	—	178	2	no	—
242	C₄×C₈	—	178	1	no	—
243	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	#41	—	2	4	no	—
248	#41	C₂×#10623494	1	8	yes	complemented
249	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
257	C₂²×C₄	#13083573	1	1	yes	characteristic
258	Q₁₆	C₂×#10623494	1	4	yes	complemented
259	SD₁₆	C₂×#10623494	1	8	yes	complemented
260	D₈	C₂×#10623494	1	4	yes	complemented
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	89	2	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	356	8	no	—
265	C₂×C₈	—	178	4	no	—
266	C₂×C₈	—	178	4	no	—
267	C₂×C₈	—	178	4	no	—
268	C₂×C₈	—	178	2	no	—
269	C₂×C₈	C₂×#10623494	1	4	yes	complemented
270	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
277	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
285	C₄²	D₄×D₈₉	1	1	yes	characteristic
286	C₄²	#13083573	1	1	yes	characteristic
287	C₄²	C₂³×D₈₉	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	C₂²×#10623494	1	4	yes	—
292	D₄	—	2	8	no	—
293	D₄	—	2	2	no	—
294	D₄	C₂²×#10623494	1	4	yes	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂²×#10623494	1	2	yes	—
307	C₂×C₄	#15668019	1	2	yes	—
308	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	89	4	no	—
314	C₈	—	356	4	no	—
315	C₈	—	178	4	no	—
316	C₈	—	178	2	no	—
317	C₈	C₂²×#10623494	1	2	yes	—
318	C₈	C₂²×#10623494	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	D₄×#10623494	1	2	yes	—
325	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	4	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	D₄×#10623494	1	2	yes	—
332	C₄	#19151715	1	2	yes	—
333	C₄	C₂³×#10623494	1	1	yes	characteristic
334	C₄	C₂×#15668019	1	1	yes	characteristic
335	C₄	#135×D₈₉	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	4	4	no	—
338	C₂	C₂×D₄×#10623494	1	1	yes	characteristic
339	C₂	#135×Dic₈₉	1	1	yes	characteristic
340	C₂	C₂×#19151715	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#135×C₈₉	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	C₈×#135
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #41×#10623494	C₂	semidirect
2. Q₁₆×#10623494	C₂	semidirect
3. SD₁₆×#10623494	C₂	semidirect
4. D₈×#10623494	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018634	C₂	semidirect
7. C₂×C₈×#10623494	C₂	semidirect
49. #135×C₈₉	C₈	semidirect
84. #41×C₈₉	C₂×C₈	semidirect
94. Q₁₆×C₈₉	C₂×C₈	semidirect
95. SD₁₆×C₈₉	C₂×C₈	semidirect
96. D₈×C₈₉	C₂×C₈	semidirect
98. C₂×C₈×C₈₉	C₂×C₈	semidirect
147. #10623494	#135	direct (N × H)
179. C₈₉	C₈×#135	semidirect
206. #135	#10623494	direct (N × H)
248. #41	C₂×#10623494	semidirect
258. Q₁₆	C₂×#10623494	semidirect
259. SD₁₆	C₂×#10623494	semidirect
260. D₈	C₂×#10623494	semidirect
269. C₂×C₈	C₂×#10623494	semidirect

Maximal subgroups

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

subgroup	index
#41×#10623494	2
Q₁₆×#10623494	2
SD₁₆×#10623494	2
D₈×#10623494	2
#26019000	2
#26018634	2
C₂×C₈×#10623494	2
C₄×#135×C₈₉	2
C₈×#135	89