Subgroups — #36545981

Order 22784.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032540	C₂	1	2	yes	maximal, complemented
3	#26019861	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019829	C₂	1	2	yes	maximal, complemented
5	#26019653	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019641	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019480	C₂	1	2	yes	maximal, complemented
8	#26019416	C₂	1	2	yes	maximal, complemented
9	#26019137	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#13083562	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3341	C₂	1	1	yes	maximal, characteristic
12	D₄×Dic₁₇₈	C₂²	1	2	yes	—
13	#19152003	C₂²	1	1	yes	characteristic
14	#19152002	—	2	1	no	—
15	#19151950	C₂²	1	2	yes	—
16	#19151948	C₂²	1	2	yes	—
17	#19151939	—	2	2	no	—
18	#19151890	C₂²	1	2	yes	—
19	#19151878	C₂²	1	2	yes	—
20	#19151876	—	2	2	no	—
21	#19151842	—	2	2	no	—
22	#19151803	C₂²	1	2	yes	—
23	#19151736	—	2	2	no	—
24	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
25	C₄×#13083562	C₂²	1	1	yes	characteristic
26	D₈×Dic₈₉	—	2	1	no	—
27	#19151676	C₂²	1	2	yes	—
28	#19151606	—	2	2	no	—
29	#19151351	C₂²	1	1	yes	characteristic
30	#19151348	C₂²	1	1	yes	characteristic
31	#19151347	C₂²	1	2	yes	—
32	#19151343	C₂²	1	2	yes	—
33	#19151328	C₂²	1	1	yes	characteristic
34	D₄⋊#10623494	C₂²	1	2	yes	—
35	D₄⋊#10623494	C₂²	1	1	yes	characteristic
36	C₈×#10623494	C₂²	1	1	yes	characteristic
37	#555×C₈₉	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#482×C₈₉	—	2	2	no	—
40	#476×C₈₉	C₂²	1	2	yes	—
41	#466×C₈₉	—	2	1	no	—
42	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
43	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
44	#427×C₈₉	—	2	1	no	—
45	C₂²×Dic₁₇₈	—	2	2	no	—
46	C₂×#13083562	—	2	2	no	—
47	C₂×#13083562	D₄	1	1	yes	characteristic
48	C₂×#13083558	—	2	2	no	—
49	#15668012	C₂³	1	2	yes	—
50	#15668011	—	2	2	no	—
51	#15667991	—	2	2	no	—
52	#15667989	—	4	2	no	—
53	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
54	D₄×Dic₈₉	—	2	2	no	—
55	#15667946	—	2	1	no	—
56	#15667946	—	2	1	no	—
57	#15667945	—	2	2	no	—
58	#15667945	—	2	2	no	—
59	#15667943	D₄	1	1	yes	characteristic
60	C₄⋊#10623494	C₂³	1	1	yes	characteristic
61	C₄⋊#10623494	C₂³	1	2	yes	—
62	#15667941	—	2	1	no	—
63	#15667936	—	4	2	no	—
64	#15667935	—	2	2	no	—
65	#15667935	—	2	2	no	—
66	#15667935	—	2	2	no	—
67	#15667935	—	2	2	no	—
68	#15667935	D₄	1	1	yes	characteristic
69	#15667933	—	2	2	no	—
70	#15667933	D₄	1	1	yes	characteristic
71	C₂²⋊#10623494	—	2	2	no	—
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₈×Dic₈₉	—	2	1	no	—
75	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	—	2	1	no	—
77	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
78	#124×C₈₉	—	4	2	no	—
79	#121×C₈₉	—	2	2	no	—
80	#121×C₈₉	—	2	2	no	—
81	C₄×D₄×C₈₉	C₂³	1	2	yes	—
82	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
83	#117×C₈₉	—	2	2	no	—
84	#117×C₈₉	—	2	2	no	—
85	#107×C₈₉	—	2	1	no	—
86	#107×C₈₉	—	2	1	no	—
87	#107×C₈₉	D₄	1	1	yes	characteristic
88	#106×C₈₉	—	2	2	no	—
89	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
90	#102×C₈₉	—	2	2	no	—
91	#102×C₈₉	D₄	1	2	yes	—
92	C₂²⋊C₈×C₈₉	—	2	1	no	—
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	—	2	2	no	—
95	C₂×Dic₁₇₈	—	4	4	no	—
96	C₂×Dic₁₇₈	—	4	2	no	—
97	C₂×Dic₁₇₈	—	8	2	no	—
98	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×Dic₈₉	—	4	2	no	—
100	#13083562	—	2	2	no	—
101	#13083562	—	8	2	no	—
102	#13083558	—	8	2	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	1	no	—
105	C₂×#10623494	—	2	1	no	—
106	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₄⋊Dic₈₉	—	2	2	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
111	#13083553	—	2	1	no	—
112	#13083553	—	4	2	no	—
113	C₂²⋊Dic₈₉	—	4	2	no	—
114	C₄×Dic₈₉	—	2	1	no	—
115	C₂×Q₈×C₈₉	—	2	2	no	—
116	C₂×D₄×C₈₉	—	4	2	no	—
117	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₈₉	—	2	2	no	—
120	C₂²×C₄×C₈₉	—	2	1	no	—
121	D₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	1	no	—
123	C₂×C₈×C₈₉	—	4	2	no	—
124	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
130	C₂²⋊C₄×C₈₉	—	2	2	no	—
131	C₂²⋊C₄×C₈₉	—	2	1	no	—
132	C₂²⋊C₄×C₈₉	—	4	2	no	—
133	C₂²⋊C₄×C₈₉	—	4	2	no	—
134	C₄²×C₈₉	—	4	2	no	—
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	—	2	1	no	—
137	C₂³×C₈₉	—	2	2	no	—
138	Q₈×C₈₉	—	8	2	no	—
139	D₄×C₈₉	—	2	1	no	—
140	D₄×C₈₉	—	2	2	no	—
141	D₄×C₈₉	—	2	1	no	—
142	D₄×C₈₉	—	2	2	no	—
143	D₄×C₈₉	—	16	2	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	4	1	no	—
150	C₂×C₄×C₈₉	—	4	2	no	—
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×Dic₈₉	—	2	1	no	—
155	C₂×Dic₈₉	—	2	1	no	—
156	C₂×Dic₈₉	—	4	2	no	—
157	C₂×Dic₈₉	—	8	4	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	4	2	no	—
160	Dic₁₇₈	—	4	2	no	—
161	Dic₁₇₈	—	8	4	no	—
162	Dic₁₇₈	#135	1	2	yes	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	4	1	no	—
165	C₈×C₈₉	#137	1	2	yes	—
166	#10623494	—	2	1	no	—
167	#10623494	—	2	1	no	—
168	#10623494	—	4	1	no	—
169	#10623494	—	4	2	no	—
170	#10623494	—	4	1	no	—
171	C₂²×C₈₉	—	4	1	no	—
172	C₂²×C₈₉	—	4	2	no	—
173	C₂²×C₈₉	—	4	2	no	—
174	C₂²×C₈₉	—	4	1	no	—
175	C₂²×C₈₉	—	4	1	no	—
176	C₂²×C₈₉	—	4	2	no	—
177	C₂²×C₈₉	D₄²	1	1	yes	characteristic
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	—	4	2	no	—
181	C₄×C₈₉	—	8	2	no	—
182	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
183	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
184	C₄×C₈₉	#576	1	1	yes	characteristic
185	C₄×C₈₉	#578	1	1	yes	characteristic
186	Dic₈₉	—	2	2	no	—
187	Dic₈₉	—	4	1	no	—
188	Dic₈₉	—	8	2	no	—
189	#19905	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	—	8	1	no	—
191	C₂×C₈₉	—	8	2	no	—
192	C₂×C₈₉	#3276	1	1	yes	characteristic
193	C₂×C₈₉	#3280	1	1	yes	characteristic
194	C₂×C₈₉	#3290	1	1	yes	characteristic
195	Q₈×D₈	—	89	1	no	—
196	#3341	D₈₉	1	1	yes	characteristic
197	#3329	—	89	2	no	—
198	#1710	—	89	1	no	—
199	#1703	—	89	2	no	—
200	#1676	—	89	1	no	—
201	#1673	—	89	1	no	—
202	#1646	—	89	2	no	—
203	#1637	—	89	2	no	—
204	#1589	—	89	1	no	—
205	C₈×SD₁₆	—	89	1	no	—
206	C₈₉	#19905	1	1	yes	characteristic, complemented, Sylow
207	#555	C₂×D₈₉	1	2	yes	—
208	D₄×Q₈	—	89	2	no	—
209	#500	—	89	1	no	—
210	#500	—	178	1	no	—
211	#500	C₂×D₈₉	1	1	yes	characteristic
212	#487	—	89	2	no	—
213	#487	—	89	2	no	—
214	#485	—	178	2	no	—
215	#482	—	2	2	no	—
216	#477	—	89	2	no	—
217	#476	C₂×D₈₉	1	2	yes	—
218	#474	—	89	2	no	—
219	#474	—	178	2	no	—
220	#468	—	178	2	no	—
221	#466	—	2	1	no	—
222	#461	—	89	2	no	—
223	#448	—	178	2	no	—
224	C₈×Q₈	—	89	1	no	—
225	C₄×SD₁₆	—	89	1	no	—
226	C₄×D₈	—	178	1	no	—
227	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
228	#436	—	89	2	no	—
229	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
230	#427	—	2	1	no	—
231	#419	—	178	2	no	—
232	#335	—	89	1	no	—
233	#334	—	89	1	no	—
234	#333	—	89	2	no	—
235	#332	—	89	2	no	—
236	Q₈⋊C₈	—	89	1	no	—
237	D₄⋊C₈	—	89	1	no	—
238	D₄⋊C₈	—	89	2	no	—
239	C₈²	—	89	1	no	—
240	C₂²×Q₈	—	178	2	no	—
241	C₂×SD₁₆	—	89	1	no	—
242	C₂×SD₁₆	—	178	2	no	—
243	C₂×D₈	C₂²⋊D₈₉	1	1	yes	characteristic
244	C₂×M₁₆	—	178	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	89	2	no	—
247	C₄⋊Q₈	—	178	2	no	—
248	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
249	#124	—	4	2	no	—
250	C₂²⋊Q₈	—	356	2	no	—
251	C₂²⋊Q₈	—	178	2	no	—
252	#121	—	2	2	no	—
253	#121	—	2	2	no	—
254	C₄×Q₈	—	89	1	no	—
255	C₄×D₄	—	178	2	no	—
256	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₄×D₄	C₂²×D₈₉	1	2	yes	—
258	#117	—	2	2	no	—
259	#117	—	2	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	178	2	no	—
263	#107	—	178	1	no	—
264	#107	—	178	1	no	—
265	#107	—	178	2	no	—
266	#107	C₂²⋊D₈₉	1	1	yes	characteristic
267	#106	—	2	2	no	—
268	#106	—	89	1	no	—
269	C₄⋊C₈	—	89	2	no	—
270	C₄⋊C₈	—	89	1	no	—
271	C₄⋊C₈	—	178	1	no	—
272	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
273	#103	—	89	1	no	—
274	#103	—	356	2	no	—
275	#103	—	178	2	no	—
276	#103	—	178	2	no	—
277	#103	—	178	2	no	—
278	#103	—	178	2	no	—
279	#102	—	2	2	no	—
280	#102	—	89	1	no	—
281	#102	—	178	2	no	—
282	#102	C₂²⋊D₈₉	1	2	yes	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	178	2	no	—
285	C₄×C₈	—	89	1	no	—
286	C₄×C₈	—	89	1	no	—
287	C₄×C₈	—	178	1	no	—
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	89	1	no	—
291	C₂×Q₈	—	356	2	no	—
292	C₂×Q₈	—	356	4	no	—
293	C₂×Q₈	—	178	2	no	—
294	C₂×Q₈	—	712	2	no	—
295	C₂×D₄	—	4	2	no	—
296	C₂×D₄	D₄×D₈₉	1	1	yes	characteristic
297	C₂×D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	356	2	no	—
301	SD₁₆	—	178	2	no	—
302	SD₁₆	—	712	2	no	—
303	D₈	—	2	2	no	—
304	M₁₆	—	712	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	89	1	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	1	no	—
311	C₂×C₈	—	178	1	no	—
312	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	89	1	no	—
317	C₄⋊C₄	—	356	2	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	—	178	1	no	—
321	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
322	C₄⋊C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	2	1	no	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	356	2	no	—
328	C₄²	—	4	2	no	—
329	C₄²	—	178	1	no	—
330	C₄²	C₂³×D₈₉	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	2	2	no	—
333	Q₈	—	8	2	no	—
334	Q₈	—	89	2	no	—
335	Q₈	—	356	2	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	178	1	no	—
338	Q₈	—	712	4	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	1	no	—
341	D₄	—	2	2	no	—
342	D₄	—	2	1	no	—
343	D₄	—	16	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	712	4	no	—
355	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
357	C₂×C₄	#15668057	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	356	1	no	—
361	C₈	—	356	2	no	—
362	C₈	—	356	1	no	—
363	C₈	—	178	1	no	—
364	C₈	—	178	1	no	—
365	C₈	#15668042	1	2	yes	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	4	2	no	—
372	C₂²	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	356	1	no	—
378	C₄	—	178	2	no	—
379	C₄	—	712	2	no	—
380	C₄	#135×D₈₉	1	1	yes	characteristic
381	C₄	C₂×#15668042	1	1	yes	characteristic
382	C₄	#19152400	1	1	yes	characteristic
383	C₄	#19152413	1	1	yes	characteristic
384	C₂	—	8	1	no	—
385	C₂	—	8	2	no	—
386	C₂	#26031894	1	1	yes	characteristic
387	C₂	#26031931	1	1	yes	characteristic
388	C₂	#26032034	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3341	C₂
socle	C₂²×C₈₉	D₄²
hypercenter	#3341	D₈₉
hypocenter	C₈₉	#19905
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3341

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₁₇₈	C₂	semidirect
2. #26032540	C₂	semidirect
3. #26019861	C₂	semidirect
4. #26019829	C₂	semidirect
5. #26019653	C₂	semidirect
6. #26019641	C₂	semidirect
7. #26019480	C₂	semidirect
8. #26019416	C₂	semidirect
9. #26019137	C₂	semidirect
10. C₈×#13083562	C₂	semidirect
206. C₈₉	#19905	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₁₇₈	2
#26032540	2
#26019861	2
#26019829	2
#26019653	2
#26019641	2
#26019480	2
#26019416	2
#26019137	2
C₈×#13083562	2
C₈₉×#3341	2
#19905	89