Subgroups — #36471353

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033354	C₂	1	1	yes	maximal, characteristic, complemented
2	#26028955	C₂	1	1	yes	maximal, characteristic
3	#26021872	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151380	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018339	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018337	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#349×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152109	C₄	1	1	yes	characteristic
10	#19152016	—	2	2	no	—
11	#19152014	—	2	2	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	M₃₂×D₈₉	—	2	2	no	—
15	#19151674	—	2	2	no	—
16	#19151667	—	2	2	no	—
17	#19151624	C₄	1	1	yes	characteristic
18	#19151564	C₄	1	1	yes	characteristic
19	#19151562	C₄	1	1	yes	characteristic
20	#19151553	C₂²	1	1	yes	characteristic
21	#19151380	—	2	2	no	—
22	#19151379	—	2	2	no	—
23	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
26	#349×C₈₉	—	2	2	no	—
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	#15668018	—	2	1	no	—
29	#15668018	—	2	1	no	—
30	#15668018	—	2	1	no	—
31	#15668018	—	2	1	no	—
32	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
34	C₂×#13083557	D₄	1	2	yes	—
35	C₂×C₈×D₈₉	D₄	1	2	yes	—
36	#15667995	C₈	1	2	yes	—
37	#15667991	—	2	1	no	—
38	#15667986	—	2	1	no	—
39	#15667981	C₈	1	2	yes	—
40	C₄×C₂²⋊D₈₉	—	2	2	no	—
41	#15667971	C₂×C₄	1	1	yes	characteristic
42	#15667952	—	2	1	no	—
43	#15667952	—	2	1	no	—
44	#15667952	—	2	1	no	—
45	#15667952	D₄	1	2	yes	complemented
46	#15667951	—	2	4	no	—
47	C₂×#13083549	—	2	1	no	—
48	C₂×#13083549	—	2	1	no	—
49	C₁₆×D₈₉	—	2	4	no	—
50	#15667941	—	2	2	no	—
51	C₂²⋊#10623494	—	2	2	no	—
52	#15667924	—	2	2	no	—
53	#15667922	—	2	1	no	—
54	C₈×Dic₈₉	—	2	1	no	—
55	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	—	2	1	no	—
57	C₂×M₁₆×C₈₉	—	2	1	no	—
58	C₂×M₁₆×C₈₉	—	2	1	no	—
59	C₂×M₁₆×C₈₉	—	2	1	no	—
60	C₂×M₁₆×C₈₉	—	2	1	no	—
61	C₂×M₁₆×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	M₃₂×C₈₉	—	2	1	no	—
65	M₃₂×C₈₉	—	2	1	no	—
66	M₃₂×C₈₉	—	2	1	no	—
67	M₃₂×C₈₉	D₄	1	2	yes	complemented
68	C₂×C₁₆×C₈₉	—	2	1	no	—
69	C₂×C₁₆×C₈₉	—	2	1	no	—
70	#13083574	—	2	1	no	—
71	#13083574	—	2	1	no	—
72	#13083574	—	2	1	no	—
73	#13083574	—	2	1	no	—
74	C₂×Dic₁₇₈	M₁₆	1	1	yes	characteristic
75	C₂×D₃₅₆	M₁₆	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	—	2	2	no	—
77	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
79	#13083558	—	2	1	no	—
80	#13083558	—	2	1	no	—
81	#13083558	—	2	1	no	—
82	#13083558	—	2	1	no	—
83	#13083557	—	2	2	no	—
84	#13083557	—	2	2	no	—
85	C₂×#10623494	—	2	1	no	—
86	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
87	C₈×D₈₉	—	2	2	no	—
88	C₈×D₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	#13083553	—	2	2	no	—
91	C₂²⋊Dic₈₉	—	2	2	no	—
92	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
93	#13083551	—	2	2	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	#13083549	—	2	2	no	—
96	#13083549	—	4	1	no	—
97	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂²×C₄×C₈₉	—	2	1	no	—
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₈₉	—	4	1	no	—
107	M₁₆×C₈₉	—	4	1	no	—
108	M₁₆×C₈₉	—	4	1	no	—
109	M₁₆×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	—	2	1	no	—
112	C₂×C₈×C₈₉	—	2	1	no	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₁₆×C₈₉	—	4	1	no	—
121	C₂³×C₈₉	—	2	1	no	—
122	C₂³×C₈₉	—	2	1	no	—
123	C₂³×C₈₉	—	2	1	no	—
124	C₂³×C₈₉	—	8	1	no	—
125	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂²×D₈₉	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
140	C₂²⋊D₈₉	—	2	2	no	—
141	C₂²⋊D₈₉	—	2	2	no	—
142	C₂²⋊D₈₉	—	8	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	C₂²⋊C₈	1	2	yes	—
145	D₃₅₆	—	2	1	no	—
146	D₃₅₆	—	2	1	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	2	2	no	—
149	Dic₁₇₈	—	2	1	no	—
150	Dic₁₇₈	—	2	1	no	—
151	C₈×C₈₉	—	4	1	no	—
152	C₈×C₈₉	#120	1	1	yes	characteristic
153	C₈×C₈₉	#121	1	1	yes	characteristic
154	C₈×C₈₉	#121	1	1	yes	characteristic
155	C₈×C₈₉	#123	1	1	yes	characteristic
156	#10623494	—	2	2	no	—
157	#10623494	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	8	1	no	—
163	C₂²×C₈₉	—	8	1	no	—
164	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
166	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
167	C₂×D₈₉	—	2	2	no	—
168	C₂×D₈₉	—	4	2	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	#386	1	1	yes	characteristic
171	C₄×C₈₉	#408	1	1	yes	characteristic
172	C₄×C₈₉	#435	1	1	yes	characteristic
173	C₄×C₈₉	#435	1	1	yes	characteristic
174	Dic₈₉	—	2	2	no	—
175	Dic₈₉	—	4	2	no	—
176	#12989	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	2	1	no	—
179	C₂×C₈₉	—	2	1	no	—
180	C₂×C₈₉	—	8	1	no	—
181	C₂×C₈₉	#1869	1	1	yes	characteristic
182	D₈₉	—	4	2	no	—
183	#3407	—	89	1	no	—
184	#2930	—	89	1	no	—
185	#2115	—	89	1	no	—
186	C₂×#349	—	89	1	no	—
187	C₂×#349	D₈₉	1	1	yes	characteristic, complemented
188	#1378	—	89	1	no	—
189	#1378	—	89	1	no	—
190	C₈₉	#12989	1	1	yes	characteristic, complemented, Sylow
191	C₂×#131	—	89	1	no	—
192	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
193	#525	—	89	1	no	—
194	#504	—	178	2	no	—
195	#504	—	178	2	no	—
196	C₂×M₃₂	—	89	1	no	—
197	C₂×M₃₂	—	89	1	no	—
198	C₂×M₃₂	—	178	2	no	—
199	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
200	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
201	#436	—	178	2	no	—
202	#435	—	178	2	no	—
203	#424	—	89	1	no	—
204	#408	—	89	1	no	—
205	#407	—	89	1	no	—
206	#405	—	89	1	no	—
207	#349	—	2	2	no	—
208	#349	—	178	2	no	—
209	#349	—	178	2	no	—
210	C₂×#41	—	89	1	no	—
211	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
212	#131	—	178	1	no	—
213	#131	—	178	1	no	—
214	#131	—	178	1	no	—
215	#131	—	178	1	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	89	1	no	—
223	C₂×M₁₆	—	89	2	no	—
224	C₂×M₁₆	—	89	1	no	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	#123	—	89	2	no	—
229	C₂²⋊Q₈	—	178	1	no	—
230	#121	—	178	1	no	—
231	#120	—	89	2	no	—
232	C₄×D₄	—	178	2	no	—
233	#117	—	89	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	89	2	no	—
238	M₃₂	—	178	4	no	—
239	M₃₂	—	178	1	no	—
240	M₃₂	—	178	1	no	—
241	M₃₂	—	178	1	no	—
242	M₃₂	C₂²⋊D₈₉	1	2	yes	complemented
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	178	4	no	—
246	C₂×C₁₆	—	178	1	no	—
247	C₂×C₁₆	—	178	1	no	—
248	C₄⋊C₈	—	178	2	no	—
249	C₂²⋊C₈	—	178	2	no	—
250	C₂²⋊C₈	—	178	2	no	—
251	#97	—	178	1	no	—
252	C₄×C₈	—	178	1	no	—
253	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
254	#41	—	178	1	no	—
255	#41	—	178	1	no	—
256	#41	—	178	1	no	—
257	#41	—	178	1	no	—
258	C₂×Q₈	—	89	1	no	—
259	C₂×D₄	—	89	2	no	—
260	C₂×D₄	—	89	1	no	—
261	C₂×D₄	—	178	2	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	89	2	no	—
269	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	178	1	no	—
276	M₁₆	—	178	1	no	—
277	M₁₆	—	178	1	no	—
278	M₁₆	—	178	1	no	—
279	M₁₆	—	178	2	no	—
280	M₁₆	—	178	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	89	2	no	—
285	C₂×C₈	—	178	1	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	—	178	2	no	—
288	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
290	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
292	C₂×C₈	#13083573	1	1	yes	characteristic
293	C₂×C₈	#13083574	1	1	yes	characteristic
294	C₄⋊C₄	—	178	2	no	—
295	C₄⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	89	2	no	—
297	C₂²⋊C₄	—	178	2	no	—
298	C₂²⋊C₄	—	178	2	no	—
299	C₄²	—	178	1	no	—
300	C₁₆	—	2	2	no	—
301	C₁₆	—	4	1	no	—
302	C₁₆	—	356	1	no	—
303	C₁₆	—	178	2	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	8	1	no	—
308	C₂³	—	89	2	no	—
309	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
310	Q₈	—	178	1	no	—
311	Q₈	—	178	1	no	—
312	D₄	—	178	2	no	—
313	D₄	—	178	2	no	—
314	D₄	—	178	1	no	—
315	D₄	—	178	1	no	—
316	D₄	—	712	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	89	2	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	#15667972	1	1	yes	characteristic
330	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
332	C₂×C₄	#15668018	1	1	yes	characteristic
333	C₂×C₄	#15668020	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	356	1	no	—
336	C₈	—	178	2	no	—
337	C₈	#15667980	1	1	yes	characteristic
338	C₈	#15667983	1	1	yes	characteristic
339	C₈	#15667985	1	1	yes	characteristic
340	C₈	#15667994	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	356	2	no	—
348	C₂²	—	178	2	no	—
349	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
350	C₂²	#19151661	1	1	yes	characteristic
351	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
352	C₄	—	4	1	no	—
353	C₄	—	356	2	no	—
354	C₄	—	178	2	no	—
355	C₄	#19151474	1	1	yes	characteristic
356	C₄	#19151563	1	1	yes	characteristic
357	C₄	#19151667	1	1	yes	characteristic
358	C₄	#19151672	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	356	2	no	—
364	C₂	#26020541	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151474
commutator	C₂³×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#349×C₈₉	C₂
socle	C₂×C₈₉	#1869
prime radical	#19152109	C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#349	D₈₉
hypocenter	C₈₉	#12989
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
1. #26033354	C₂	semidirect
3. #26021872	C₂	semidirect
4. C₂×#19151380	C₂	semidirect
5. #26018339	C₂	semidirect
6. #26018337	C₂	semidirect
7. C₂×#349×C₈₉	C₂	semidirect
45. #15667952	D₄	semidirect
67. M₃₂×C₈₉	D₄	semidirect
187. C₂×#349	D₈₉	semidirect
190. C₈₉	#12989	semidirect
242. M₃₂	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033354	2
#26028955	2
#26021872	2
C₂×#19151380	2
#26018339	2
#26018337	2
C₂×#349×C₈₉	2
#12989	89