Subgroups — #36447127

Order 22784.

568 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	C₄²×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151330	C₂	1	2	yes	maximal
4	#26018092	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
10	C₄×#13083553	—	2	1	no	—
11	C₄×#13083553	—	2	1	no	—
12	C₄×#13083553	—	2	1	no	—
13	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
15	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄²×Dic₈₉	—	2	1	no	—
17	#19151353	—	2	1	no	—
18	#19151353	—	2	1	no	—
19	#19151330	—	2	4	no	—
20	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
22	C₄³×C₈₉	C₂²	1	1	yes	characteristic
23	#336×C₈₉	—	2	1	no	—
24	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10623494	—	2	2	no	—
26	C₄×Dic₁₇₈	—	2	2	no	—
27	C₄×Dic₁₇₈	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	C₄×Dic₁₇₈	—	2	1	no	—
30	C₄×Dic₁₇₈	—	2	1	no	—
31	C₄×Dic₁₇₈	C₈	1	4	yes	complemented
32	C₄×Dic₁₇₈	D₄	1	2	yes	—
33	C₄×Dic₁₇₈	Q₈	1	2	yes	—
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083553	—	2	1	no	—
38	C₂×#13083553	—	2	1	no	—
39	C₂×#13083553	—	2	1	no	—
40	C₂×C₄×Dic₈₉	—	2	1	no	—
41	C₂×C₄×Dic₈₉	—	2	1	no	—
42	C₂×C₄×Dic₈₉	—	2	1	no	—
43	C₂×C₄×Dic₈₉	—	2	1	no	—
44	C₂×C₄×Dic₈₉	—	2	1	no	—
45	C₂×C₄×Dic₈₉	—	2	1	no	—
46	C₄⋊#10623494	—	2	2	no	—
47	C₄⋊#10623494	—	2	2	no	—
48	#15667919	—	2	1	no	—
49	#15667919	—	2	1	no	—
50	#15667919	—	2	1	no	—
51	C₂²×C₈×C₈₉	—	2	2	no	—
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₈₉	—	2	4	no	—
60	C₄×C₈×C₈₉	—	2	2	no	—
61	C₄×C₈×C₈₉	—	2	2	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
67	C₂×Dic₁₇₈	M₁₆	1	2	yes	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂²×Dic₈₉	—	2	1	no	—
71	C₂²×Dic₈₉	—	2	1	no	—
72	C₂×#10623494	—	4	2	no	—
73	C₂×#10623494	—	4	2	no	—
74	C₂×#10623494	—	4	2	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
79	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
80	C₄⋊Dic₈₉	M₁₆	1	2	yes	—
81	#13083553	—	2	4	no	—
82	#13083553	—	2	4	no	—
83	#13083553	—	2	2	no	—
84	#13083553	—	2	2	no	—
85	C₄×Dic₈₉	—	2	4	no	—
86	C₄×Dic₈₉	—	2	4	no	—
87	C₄×Dic₈₉	—	2	2	no	—
88	C₄×Dic₈₉	—	2	2	no	—
89	C₄×Dic₈₉	—	4	2	no	—
90	C₄×Dic₈₉	—	4	2	no	—
91	C₄×Dic₈₉	—	4	2	no	—
92	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	4	no	—
100	C₂×C₈×C₈₉	—	4	2	no	—
101	C₂×C₈×C₈₉	—	4	2	no	—
102	C₄²×C₈₉	—	2	2	no	—
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	—	2	2	no	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×C₈₉	—	4	1	no	—
108	C₄²×C₈₉	C₂×C₈	1	2	yes	—
109	C₄²×C₈₉	M₁₆	1	2	yes	—
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₄²×C₈₉	#41	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
128	C₂×C₄×C₈₉	#104	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#135	1	2	yes	—
140	C₂×C₄×C₈₉	#137	1	2	yes	—
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	4	1	no	—
148	C₂×Dic₈₉	—	4	1	no	—
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	1	no	—
151	C₂×Dic₈₉	—	4	1	no	—
152	C₂×Dic₈₉	—	4	1	no	—
153	Dic₁₇₈	—	2	2	no	—
154	Dic₁₇₈	—	2	2	no	—
155	Dic₁₇₈	—	2	2	no	—
156	Dic₁₇₈	C₄⋊C₈	1	4	yes	complemented
157	C₈×C₈₉	—	4	4	no	—
158	#10623494	—	8	2	no	—
159	C₂²×C₈₉	#380	1	1	yes	characteristic
160	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂²×C₈₉	#407	1	1	yes	characteristic
162	C₂²×C₈₉	#416	1	1	yes	characteristic
163	C₂²×C₈₉	#419	1	1	yes	characteristic
164	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
165	C₂²×C₈₉	#421	1	1	yes	characteristic
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	2	2	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
173	C₄×C₈₉	#423	1	1	yes	characteristic
174	C₄×C₈₉	#431	1	1	yes	characteristic
175	C₄×C₈₉	#432	1	1	yes	characteristic
176	C₄×C₈₉	#464	1	2	yes	—
177	C₄×C₈₉	#479	1	2	yes	—
178	Dic₈₉	—	2	4	no	—
179	Dic₈₉	—	4	2	no	—
180	Dic₈₉	—	4	2	no	—
181	Dic₈₉	—	4	2	no	—
182	#8566	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	#1476	1	1	yes	characteristic
184	C₂×C₈₉	#1479	1	1	yes	characteristic
185	C₂×C₈₉	#1488	1	1	yes	characteristic
186	C₂×C₈₉	#1491	1	1	yes	characteristic
187	C₂×C₈₉	#1797	1	1	yes	characteristic
188	C₂×C₈₉	#1805	1	1	yes	characteristic
189	C₂×C₈₉	#1806	1	1	yes	characteristic
190	C₄²×Q₈	—	89	1	no	—
191	#1831	—	89	1	no	—
192	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
193	C₂×Q₈⋊C₈	—	89	2	no	—
194	#1279	—	89	2	no	—
195	C₈₉	#8566	1	1	yes	characteristic, complemented, Sylow
196	C₂×C₄×Q₈	—	89	1	no	—
197	C₂×C₄×Q₈	—	89	1	no	—
198	C₂×C₄×Q₈	—	89	1	no	—
199	C₂×C₄⋊C₈	—	2	1	no	—
200	C₂×C₄⋊C₈	—	89	2	no	—
201	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
202	C₄×C₄⋊C₄	—	89	1	no	—
203	C₄×C₄⋊C₄	—	89	1	no	—
204	C₄×C₄⋊C₄	—	89	1	no	—
205	C₄×C₄⋊C₄	—	178	1	no	—
206	C₄×C₄⋊C₄	—	178	1	no	—
207	C₄×C₄⋊C₄	—	178	1	no	—
208	C₄³	—	178	1	no	—
209	C₄³	C₂×D₈₉	1	1	yes	characteristic
210	#336	—	2	1	no	—
211	#336	—	178	1	no	—
212	#336	—	178	1	no	—
213	Q₈⋊C₈	—	178	4	no	—
214	C₂²×Q₈	—	89	1	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	178	2	no	—
217	C₄×Q₈	—	89	4	no	—
218	C₄×Q₈	—	89	2	no	—
219	C₄×Q₈	—	89	2	no	—
220	C₄×Q₈	—	178	2	no	—
221	C₄×Q₈	—	178	2	no	—
222	C₄×Q₈	—	178	2	no	—
223	C₄×Q₈	—	178	1	no	—
224	C₄×Q₈	—	178	1	no	—
225	C₂×C₄⋊C₄	—	89	1	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	C₂×C₄⋊C₄	—	89	1	no	—
228	C₂×C₄⋊C₄	—	178	1	no	—
229	C₂×C₄⋊C₄	—	178	1	no	—
230	C₂×C₄⋊C₄	—	178	1	no	—
231	C₂×C₄²	—	178	1	no	—
232	C₂×C₄²	—	178	1	no	—
233	C₂×C₄²	—	178	1	no	—
234	C₂×C₄²	—	178	1	no	—
235	C₂×C₄²	—	178	1	no	—
236	C₂×C₄²	—	178	1	no	—
237	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
238	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
239	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
240	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
241	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
244	C₄⋊C₈	—	2	4	no	—
245	C₄⋊C₈	—	178	2	no	—
246	C₄⋊C₈	—	178	2	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	#95	—	178	1	no	—
250	#95	—	178	1	no	—
251	#95	—	178	1	no	—
252	C₂×Q₈	—	89	2	no	—
253	C₂×Q₈	—	89	2	no	—
254	C₂×Q₈	—	89	2	no	—
255	C₂×Q₈	—	178	1	no	—
256	C₂×Q₈	—	178	1	no	—
257	C₂×Q₈	—	178	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	#13083551	1	1	yes	characteristic
263	C₂²×C₄	#13083551	1	1	yes	characteristic
264	C₂²×C₄	#13083551	1	1	yes	characteristic
265	C₂²×C₄	#13083551	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	356	2	no	—
273	C₂×C₈	—	356	2	no	—
274	C₂×C₈	—	356	2	no	—
275	C₄⋊C₄	—	89	2	no	—
276	C₄⋊C₄	—	89	2	no	—
277	C₄⋊C₄	—	89	2	no	—
278	C₄⋊C₄	—	178	4	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₄⋊C₄	—	178	2	no	—
282	C₄⋊C₄	—	178	1	no	—
283	C₄⋊C₄	—	178	1	no	—
284	C₄⋊C₄	—	178	4	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	356	2	no	—
292	C₄²	—	356	2	no	—
293	C₄²	—	356	2	no	—
294	C₄²	—	178	4	no	—
295	C₄²	—	178	2	no	—
296	C₄²	—	178	2	no	—
297	C₄²	—	178	4	no	—
298	C₄²	C₈×D₈₉	1	2	yes	—
299	C₄²	#13083557	1	2	yes	—
300	C₄²	D₄×D₈₉	1	1	yes	characteristic
301	C₄²	Q₈×D₈₉	1	1	yes	characteristic
302	C₄²	#13083573	1	1	yes	characteristic
303	C₄²	#13083575	1	1	yes	characteristic
304	C₂³	C₂×#13083551	1	1	yes	characteristic
305	Q₈	—	89	4	no	—
306	Q₈	—	178	2	no	—
307	Q₈	—	178	2	no	—
308	Q₈	—	178	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	356	1	no	—
321	C₂×C₄	—	356	1	no	—
322	C₂×C₄	—	356	1	no	—
323	C₂×C₄	—	356	1	no	—
324	C₂×C₄	—	356	1	no	—
325	C₂×C₄	—	356	1	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	—	178	2	no	—
332	C₂×C₄	#15667924	1	2	yes	—
333	C₂×C₄	#15667924	1	2	yes	—
334	C₂×C₄	#15667940	1	2	yes	—
335	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
336	C₂×C₄	#15667970	1	1	yes	characteristic
337	C₂×C₄	#15667983	1	1	yes	characteristic
338	C₂×C₄	#15667987	1	1	yes	characteristic
339	C₂×C₄	#15667988	1	1	yes	characteristic
340	C₂×C₄	#15667992	1	1	yes	characteristic
341	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
342	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
343	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
344	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
345	C₂×C₄	#15668034	1	2	yes	—
346	C₂×C₄	#15668045	1	2	yes	—
347	C₈	—	4	4	no	—
348	C₈	—	712	2	no	—
349	C₂²	#19151455	1	1	yes	characteristic
350	C₂²	C₂×#15667924	1	1	yes	characteristic
351	C₂²	#19151561	1	1	yes	characteristic
352	C₂²	#19151595	1	1	yes	characteristic
353	C₂²	#19151608	1	1	yes	characteristic
354	C₂²	C₂×#15667940	1	1	yes	characteristic
355	C₂²	#19151616	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	356	2	no	—
363	C₄	—	356	2	no	—
364	C₄	—	356	2	no	—
365	C₄	—	178	4	no	—
366	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
367	C₄	#19151620	1	1	yes	characteristic
368	C₄	#19151650	1	1	yes	characteristic
369	C₄	#19151654	1	1	yes	characteristic
370	C₄	#19151821	1	2	yes	—
371	C₄	#19151906	1	2	yes	—
372	C₂	#26018635	1	1	yes	characteristic
373	C₂	#26018648	1	1	yes	characteristic
374	C₂	#26018687	1	1	yes	characteristic
375	C₂	#26018700	1	1	yes	characteristic
376	C₂	#26020229	1	1	yes	characteristic
377	C₂	#26020268	1	1	yes	characteristic
378	C₂	#26020273	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#8566
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
31. C₄×Dic₁₇₈	C₈	semidirect
156. Dic₁₇₈	C₄⋊C₈	semidirect
195. C₈₉	#8566	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₁₇₈	2
#26020369	2
C₂×#19151330	2
#26018092	2
C₄×C₄⋊C₈×C₈₉	2
#8566	89