Subgroups — #36524376

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030284	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028965	C₂	1	1	yes	maximal, characteristic
4	#26019132	C₂	1	4	yes	maximal, complemented
5	#26019066	C₂	1	4	yes	maximal, complemented
6	#26019027	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018682	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018652	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	—	2	1	no	—
11	#19152148	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	#19151917	C₄	1	2	yes	—
14	#19151884	C₄	1	2	yes	—
15	#19151799	C₄	1	2	yes	—
16	#19151739	C₄	1	2	yes	—
17	C₈×Dic₁₇₈	—	2	1	no	—
18	C₈×Dic₁₇₈	—	2	1	no	—
19	C₈×Dic₁₇₈	C₂²	1	2	yes	—
20	C₈×Dic₁₇₈	C₂²	1	2	yes	—
21	C₄×#13083563	C₂²	1	4	yes	—
22	#19151677	—	2	1	no	—
23	#19151677	C₂²	1	2	yes	—
24	C₈×D₃₅₆	—	2	1	no	—
25	C₈×D₃₅₆	C₂²	1	2	yes	—
26	#19151622	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
28	C₂×#15667941	—	2	1	no	—
29	#19151564	—	2	1	no	—
30	C₂×C₈×Dic₈₉	—	2	1	no	—
31	#19151351	C₂²	1	4	yes	—
32	#19151330	C₂²	1	4	yes	—
33	#19151327	C₂²	1	4	yes	—
34	#431×C₈₉	C₂²	1	1	yes	characteristic
35	#431×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	#322×C₈₉	C₂²	1	2	yes	—
38	#322×C₈₉	C₂²	1	2	yes	—
39	C₂×#13083574	—	2	1	no	—
40	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083563	C₂×C₄	1	4	yes	—
42	#15668018	—	4	4	no	—
43	C₂×#13083558	—	2	1	no	—
44	C₂×#13083557	—	2	2	no	—
45	C₂²×#10623494	—	2	1	no	—
46	C₂×C₈×D₈₉	—	2	2	no	—
47	#15668012	C₂×C₄	1	1	yes	characteristic
48	#15668009	C₂×C₄	1	1	yes	characteristic
49	#15668000	—	2	1	no	—
50	#15667986	—	2	1	no	—
51	#15667986	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	C₄×Dic₁₇₈	—	2	1	no	—
54	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₇₈	C₂³	1	2	yes	—
56	C₄×D₃₅₆	—	2	1	no	—
57	C₄×D₃₅₆	C₂³	1	2	yes	—
58	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×#13083553	—	2	1	no	—
60	C₂×C₄×Dic₈₉	—	2	1	no	—
61	#15667944	C₂×C₄	1	4	yes	—
62	C₄⋊#10623494	—	2	1	no	—
63	C₄⋊#10623494	—	2	1	no	—
64	C₄⋊#10623494	C₂³	1	2	yes	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	#15667941	—	2	2	no	—
67	#15667941	—	2	2	no	—
68	#15667937	C₂×C₄	1	4	yes	—
69	#15667934	C₂×C₄	1	4	yes	—
70	#15667924	—	2	2	no	—
71	#15667924	—	2	2	no	—
72	C₈×Dic₈₉	—	2	2	no	—
73	C₈×Dic₈₉	—	2	2	no	—
74	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	—	2	1	no	—
78	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₈₉	—	2	1	no	—
80	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
82	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
83	C₄×C₈×C₈₉	C₂³	1	2	yes	—
84	C₄×C₈×C₈₉	C₂³	1	2	yes	—
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	#13083574	—	4	2	no	—
88	#13083574	—	4	2	no	—
89	C₂×Dic₁₇₈	—	2	1	no	—
90	C₂×Dic₁₇₈	—	2	4	no	—
91	C₂×Dic₁₇₈	—	2	1	no	—
92	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
93	C₂×D₃₅₆	—	2	1	no	—
94	C₂×D₃₅₆	—	2	1	no	—
95	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₈₉	—	2	2	no	—
97	C₂²×Dic₈₉	—	2	1	no	—
98	C₂²×Dic₈₉	—	2	1	no	—
99	C₂×C₄×D₈₉	—	2	2	no	—
100	#13083563	—	2	8	no	—
101	#13083558	—	4	2	no	—
102	#13083557	—	4	4	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₂×#10623494	—	4	2	no	—
106	C₈×D₈₉	—	4	4	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₄⋊Dic₈₉	—	2	1	no	—
109	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
111	#13083553	—	2	2	no	—
112	#13083553	—	2	2	no	—
113	#13083551	—	2	2	no	—
114	#13083551	—	2	2	no	—
115	C₄×Dic₈₉	—	2	2	no	—
116	C₄×Dic₈₉	—	2	2	no	—
117	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₈₉	#41	1	1	yes	characteristic
131	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₈₉	#137	1	2	yes	—
141	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₈₉	—	4	4	no	—
145	C₂×Dic₈₉	—	2	4	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	D₃₅₆	—	2	4	no	—
151	D₃₅₆	—	4	2	no	—
152	D₃₅₆	—	4	2	no	—
153	C₄×D₈₉	—	4	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	Dic₁₇₈	—	2	4	no	—
156	Dic₁₇₈	—	2	2	no	—
157	Dic₁₇₈	—	4	2	no	—
158	Dic₁₇₈	#131	1	4	yes	complemented
159	C₈×C₈₉	—	2	4	no	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	4	2	no	—
165	#10623494	—	4	2	no	—
166	#10623494	—	4	2	no	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	#440	1	2	yes	—
170	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₈₉	—	4	2	no	—
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	#440	1	2	yes	—
178	C₄×C₈₉	C₂×#131	1	2	yes	—
179	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
181	Dic₈₉	—	2	4	no	—
182	Dic₈₉	—	4	2	no	—
183	Dic₈₉	—	4	2	no	—
184	#17908	—	89	1	no	maximal, Sylow
185	C₂×C₈₉	—	2	2	no	—
186	C₂×C₈₉	#2929	1	1	yes	characteristic
187	C₂×C₈₉	C₂×#440	1	1	yes	characteristic
188	C₂×C₈₉	#2955	1	1	yes	characteristic
189	D₈₉	—	8	2	no	—
190	#3103	—	89	1	no	—
191	C₂×C₈×Q₈	—	89	1	no	—
192	#2931	—	89	1	no	—
193	#1588	—	89	4	no	—
194	SD₁₆⋊C₈	—	89	4	no	—
195	#1571	—	89	1	no	—
196	#1487	—	89	1	no	—
197	#1480	—	89	1	no	—
198	#1452	D₈₉	1	1	yes	characteristic, complemented
199	C₈₉	#17908	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	178	1	no	—
201	#532	—	89	1	no	—
202	C₂×C₄×Q₈	—	89	1	no	—
203	#481	—	89	2	no	—
204	#475	—	89	2	no	—
205	#460	—	89	2	no	—
206	#448	—	89	2	no	—
207	C₈×Q₈	—	89	2	no	—
208	C₈×Q₈	—	89	2	no	—
209	C₈×Q₈	—	178	1	no	—
210	C₈×Q₈	—	178	1	no	—
211	C₄×SD₁₆	—	89	4	no	—
212	#436	—	89	2	no	—
213	#436	—	178	1	no	—
214	C₈×D₄	—	89	2	no	—
215	C₈×D₄	—	178	1	no	—
216	#431	C₂×D₈₉	1	1	yes	characteristic
217	#431	C₂×D₈₉	1	1	yes	characteristic
218	#423	—	89	1	no	—
219	C₂×C₄⋊C₈	—	89	1	no	—
220	C₂×C₄⋊C₈	—	178	1	no	—
221	#408	—	178	1	no	—
222	C₂×C₄×C₈	—	178	1	no	—
223	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
224	#335	—	89	4	no	—
225	Q₈⋊C₈	—	89	4	no	—
226	D₄⋊C₈	—	89	4	no	—
227	#322	C₂×D₈₉	1	2	yes	—
228	#322	C₂×D₈₉	1	2	yes	—
229	C₂×#41	—	178	1	no	—
230	C₂²×Q₈	—	89	1	no	—
231	C₂×SD₁₆	—	89	4	no	—
232	#131	—	356	4	no	—
233	C₂×M₁₆	—	178	1	no	—
234	C₂×M₁₆	—	178	2	no	—
235	C₂²×C₈	—	178	1	no	—
236	C₂²×C₈	—	178	2	no	—
237	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
238	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
239	C₄⋊Q₈	—	89	1	no	—
240	#127	—	89	1	no	—
241	#124	—	178	1	no	—
242	#121	—	89	2	no	—
243	#121	—	178	1	no	—
244	C₄×Q₈	—	89	2	no	—
245	C₄×Q₈	—	89	2	no	—
246	C₄×Q₈	—	178	1	no	—
247	C₄×Q₈	—	178	1	no	—
248	C₄×D₄	—	89	2	no	—
249	C₄×D₄	—	178	1	no	—
250	C₂×C₄⋊C₄	—	89	1	no	—
251	C₂×C₄⋊C₄	—	178	1	no	—
252	C₂×C₄²	—	178	1	no	—
253	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
254	#106	—	89	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	89	2	no	—
257	C₄⋊C₈	—	89	2	no	—
258	C₄⋊C₈	—	178	2	no	—
259	C₄⋊C₈	—	178	2	no	—
260	C₄⋊C₈	—	178	1	no	—
261	C₄⋊C₈	—	178	1	no	—
262	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
263	#103	—	89	4	no	—
264	#102	—	89	4	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	178	2	no	—
267	C₂²⋊C₈	—	178	2	no	—
268	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
269	C₄×C₈	—	178	2	no	—
270	C₄×C₈	—	178	2	no	—
271	C₄×C₈	D₃₅₆	1	1	yes	characteristic
272	C₄×C₈	D₃₅₆	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₈₉	1	2	yes	—
274	C₄×C₈	C₂²×D₈₉	1	2	yes	—
275	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
277	#41	—	356	2	no	—
278	#41	—	356	2	no	—
279	C₂×Q₈	—	89	2	no	—
280	C₂×Q₈	—	178	4	no	—
281	C₂×Q₈	—	178	1	no	—
282	C₂×Q₈	—	178	1	no	—
283	C₂×D₄	—	89	2	no	—
284	C₂×D₄	—	178	2	no	—
285	C₂×D₄	—	178	1	no	—
286	C₂×D₄	—	178	1	no	—
287	C₂²×C₄	—	178	1	no	—
288	C₂²×C₄	—	178	2	no	—
289	C₂²×C₄	—	178	1	no	—
290	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
292	C₂²×C₄	#13083574	1	1	yes	characteristic
293	SD₁₆	—	178	8	no	—
294	M₁₆	—	356	4	no	—
295	M₁₆	—	356	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	356	4	no	—
300	C₂×C₈	—	356	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
304	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
305	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
306	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
307	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
308	C₄⋊C₄	—	89	2	no	—
309	C₄⋊C₄	—	89	2	no	—
310	C₄⋊C₄	—	178	1	no	—
311	C₄⋊C₄	—	178	2	no	—
312	C₄⋊C₄	—	178	1	no	—
313	C₄⋊C₄	—	178	2	no	—
314	C₂²⋊C₄	—	178	2	no	—
315	C₂²⋊C₄	—	178	2	no	—
316	C₄²	—	178	2	no	—
317	C₄²	—	178	2	no	—
318	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
319	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
320	C₄²	#13083574	1	1	yes	characteristic
321	C₄²	C₂³×D₈₉	1	1	yes	characteristic
322	C₂³	—	178	2	no	—
323	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
324	Q₈	—	89	4	no	—
325	Q₈	—	356	2	no	—
326	Q₈	—	178	4	no	—
327	Q₈	—	178	2	no	—
328	D₄	—	356	2	no	—
329	D₄	—	356	4	no	—
330	D₄	—	356	2	no	—
331	D₄	—	178	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	—	178	2	no	—
339	C₂×C₄	—	178	4	no	—
340	C₂×C₄	—	178	2	no	—
341	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
342	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
343	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
344	C₂×C₄	#15668040	1	2	yes	—
345	C₂×C₄	#15668045	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
348	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	4	no	—
354	C₈	—	4	2	no	—
355	C₈	—	356	2	no	—
356	C₈	—	356	2	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	356	2	no	—
360	C₂²	—	356	2	no	—
361	C₂²	—	356	2	no	—
362	C₂²	#19151700	1	2	yes	—
363	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	356	2	no	—
368	C₄	—	356	2	no	—
369	C₄	—	178	4	no	—
370	C₄	#19151700	1	2	yes	—
371	C₄	#131×D₈₉	1	2	yes	—
372	C₄	C₂×#15668040	1	1	yes	characteristic
373	C₄	C₂×#15668045	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	712	2	no	—
376	C₂	#26028948	1	1	yes	characteristic
377	C₂	C₂×#19151700	1	1	yes	characteristic
378	C₂	#26029156	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1452	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19152148	C₂²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1452	D₈₉
hypocenter	C₈₉	#17908
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Dic₁₇₈	C₂	semidirect
4. #26019132	C₂	semidirect
5. #26019066	C₂	semidirect
6. #26019027	C₂	semidirect
7. #26018682	C₂	semidirect
9. C₈₉×#1452	C₂	semidirect
158. Dic₁₇₈	#131	semidirect
198. #1452	D₈₉	semidirect
199. C₈₉	#17908	semidirect

Maximal subgroups

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

subgroup	index
#26030284	2
C₂×C₈×Dic₁₇₈	2
#26028965	2
#26019132	2
#26019066	2
#26019027	2
#26018682	2
#26018652	2
C₈₉×#1452	2
#17908	89