Subgroups — #36453345

Order 22784.

478 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083563	C₂	1	1	yes	maximal, characteristic
2	#26018648	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018644	C₂	1	1	yes	maximal, characteristic
4	#26018575	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018572	C₂	1	1	yes	maximal, characteristic
6	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083563	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
11	C₄×#13083563	—	2	1	no	—
12	C₄×#13083563	—	2	1	no	—
13	C₄×#13083563	—	2	1	no	—
14	C₄×#13083563	—	2	1	no	—
15	#19151651	C₂²	1	1	yes	characteristic
16	#19151651	C₂²	1	1	yes	characteristic
17	C₂×#15667944	C₄	1	1	yes	characteristic
18	C₂×#15667937	C₄	1	1	yes	characteristic
19	C₂×#15667934	C₄	1	1	yes	characteristic
20	#19151330	—	2	2	no	—
21	#19151327	—	2	2	no	—
22	#19151323	—	2	2	no	—
23	#19151321	—	2	2	no	—
24	#19151319	—	2	1	no	—
25	C₈×#10623494	—	2	1	no	—
26	#431×C₈₉	C₂²	1	1	yes	characteristic
27	#431×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#335×C₈₉	—	2	1	no	—
30	#334×C₈₉	—	2	1	no	—
31	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₈₉	—	2	1	no	—
34	C₂×#13083563	—	2	1	no	—
35	C₂×#13083563	—	2	1	no	—
36	C₂×#13083563	—	2	2	no	—
37	C₂×#13083563	C₈	1	4	yes	complemented
38	C₄×Dic₁₇₈	—	2	1	no	—
39	C₄×Dic₁₇₈	—	2	1	no	—
40	C₄×Dic₁₇₈	—	2	1	no	—
41	C₄×Dic₁₇₈	—	2	1	no	—
42	C₄×D₃₅₆	—	2	1	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	C₄×D₃₅₆	—	2	1	no	—
45	C₄×D₃₅₆	—	2	1	no	—
46	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×#13083553	—	2	1	no	—
49	C₂×#13083551	—	2	1	no	—
50	C₂×C₄×Dic₈₉	—	2	1	no	—
51	#15667944	—	2	1	no	—
52	#15667944	—	2	1	no	—
53	C₄⋊#10623494	—	2	1	no	—
54	C₄⋊#10623494	D₄	1	2	yes	—
55	#15667937	—	2	1	no	—
56	#15667937	—	2	1	no	—
57	#15667934	—	2	1	no	—
58	#15667934	—	2	1	no	—
59	C₂²⋊#10623494	—	2	1	no	—
60	C₂²⋊#10623494	D₄	1	2	yes	—
61	C₄×#10623494	—	2	1	no	—
62	C₄×#10623494	—	2	1	no	—
63	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₈₉	—	2	1	no	—
67	C₄⋊C₈×C₈₉	—	2	1	no	—
68	C₂²⋊C₈×C₈₉	—	2	1	no	—
69	C₂²⋊C₈×C₈₉	—	2	1	no	—
70	C₄×C₈×C₈₉	—	2	1	no	—
71	C₄×C₈×C₈₉	—	2	1	no	—
72	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₂³×D₈₉	—	2	1	no	—
77	C₂×Dic₁₇₈	—	2	1	no	—
78	C₂×Dic₁₇₈	—	2	1	no	—
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	2	1	no	—
81	C₂×Dic₁₇₈	—	2	1	no	—
82	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂×D₃₅₆	—	2	1	no	—
86	C₂×D₃₅₆	—	2	1	no	—
87	C₂×D₃₅₆	—	2	1	no	—
88	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
89	C₂²×Dic₈₉	—	2	1	no	—
90	C₂²×Dic₈₉	—	2	1	no	—
91	C₂²×Dic₈₉	—	2	1	no	—
92	C₂×C₄×D₈₉	—	4	1	no	—
93	C₂×C₄×D₈₉	—	4	1	no	—
94	C₂×C₄×D₈₉	—	8	2	no	—
95	#13083563	—	2	4	no	—
96	#13083563	—	2	4	no	—
97	C₂×#10623494	—	2	2	no	—
98	C₂×#10623494	—	2	2	no	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	2	1	no	—
102	C₄⋊Dic₈₉	—	2	1	no	—
103	#13083553	—	4	1	no	—
104	#13083553	—	4	1	no	—
105	#13083551	—	4	1	no	—
106	#13083551	—	4	1	no	—
107	C₄×Dic₈₉	—	4	1	no	—
108	C₄×Dic₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	—	2	1	no	—
113	C₂×C₈×C₈₉	—	2	1	no	—
114	C₂×C₈×C₈₉	—	4	1	no	—
115	C₂×C₈×C₈₉	—	4	1	no	—
116	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
121	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₉	#41	1	1	yes	characteristic
126	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
127	C₂²×D₈₉	—	2	2	no	—
128	C₂²×D₈₉	—	4	1	no	—
129	C₂²×D₈₉	—	4	1	no	—
130	C₂²×D₈₉	—	8	1	no	—
131	C₂²×D₈₉	—	8	1	no	—
132	C₂×C₄×C₈₉	—	2	2	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	#121	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	4	1	no	—
146	C₂×Dic₈₉	—	4	1	no	—
147	C₂×Dic₈₉	—	4	1	no	—
148	C₂×Dic₈₉	—	4	1	no	—
149	C₂×Dic₈₉	—	8	1	no	—
150	C₂×Dic₈₉	—	8	1	no	—
151	D₃₅₆	—	2	2	no	—
152	D₃₅₆	—	2	2	no	—
153	D₃₅₆	—	4	1	no	—
154	D₃₅₆	—	4	1	no	—
155	C₄×D₈₉	—	8	2	no	—
156	C₄×D₈₉	—	8	2	no	—
157	Dic₁₇₈	—	2	2	no	—
158	Dic₁₇₈	—	2	2	no	—
159	Dic₁₇₈	—	4	1	no	—
160	Dic₁₇₈	—	4	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	C₈×C₈₉	—	8	1	no	—
164	C₈×C₈₉	—	8	1	no	—
165	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
166	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
167	#10623494	—	4	2	no	—
168	#10623494	—	4	2	no	—
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	#390	1	1	yes	characteristic
172	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
173	C₂²×C₈₉	#436	1	1	yes	characteristic
174	C₂×D₈₉	—	4	2	no	—
175	C₂×D₈₉	—	8	2	no	—
176	C₂×D₈₉	—	8	1	no	—
177	C₂×D₈₉	—	8	1	no	—
178	C₂×D₈₉	—	8	2	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₄×C₈₉	#408	1	1	yes	characteristic
184	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
185	C₄×C₈₉	#436	1	1	yes	characteristic
186	C₄×C₈₉	#471	1	2	yes	—
187	Dic₈₉	—	4	2	no	—
188	Dic₈₉	—	8	1	no	—
189	Dic₈₉	—	8	1	no	—
190	#9667	—	89	1	no	maximal, Sylow
191	C₂×C₈₉	—	2	1	no	—
192	C₂×C₈₉	—	2	1	no	—
193	C₂×C₈₉	#1954	1	1	yes	characteristic
194	C₂×C₈₉	#1976	1	1	yes	characteristic
195	C₂×C₈₉	#1977	1	1	yes	characteristic
196	D₈₉	—	8	2	no	—
197	C₂×C₄×SD₁₆	—	89	1	no	—
198	#1564	D₈₉	1	1	yes	characteristic, complemented
199	#1479	—	89	1	no	—
200	#1478	—	89	1	no	—
201	#1461	—	89	1	no	—
202	#1460	—	89	1	no	—
203	#1450	—	89	1	no	—
204	C₈₉	#9667	1	1	yes	characteristic, complemented, Sylow
205	C₂²×SD₁₆	—	89	1	no	—
206	C₂×C₄×Q₈	—	89	1	no	—
207	C₂×C₄×D₄	—	89	1	no	—
208	C₄×SD₁₆	—	178	1	no	—
209	C₄×SD₁₆	—	178	1	no	—
210	C₄×SD₁₆	—	178	1	no	—
211	C₄×SD₁₆	—	178	1	no	—
212	#431	—	89	1	no	—
213	#431	—	89	1	no	—
214	#431	C₂×D₈₉	1	1	yes	characteristic
215	#431	C₂×D₈₉	1	1	yes	characteristic
216	C₂×#106	—	89	1	no	—
217	C₂×#103	—	89	1	no	—
218	C₂×#102	—	89	1	no	—
219	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
220	#335	—	2	1	no	—
221	#334	—	2	1	no	—
222	Q₈⋊C₈	—	178	2	no	—
223	D₄⋊C₈	—	178	2	no	—
224	#324	—	178	2	no	—
225	#323	—	178	2	no	—
226	#322	—	178	1	no	—
227	C₈²	—	178	1	no	—
228	C₂²×Q₈	—	89	1	no	—
229	C₂²×D₄	—	89	1	no	—
230	C₂³×C₄	—	178	1	no	—
231	C₂×SD₁₆	—	89	4	no	—
232	C₂×SD₁₆	—	178	1	no	—
233	C₂×SD₁₆	—	178	1	no	—
234	C₂×SD₁₆	—	178	2	no	—
235	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
237	C₄×Q₈	—	178	1	no	—
238	C₄×Q₈	—	178	1	no	—
239	C₄×Q₈	—	178	1	no	—
240	C₄×Q₈	—	178	1	no	—
241	C₄×D₄	—	178	1	no	—
242	C₄×D₄	—	178	1	no	—
243	C₄×D₄	—	178	1	no	—
244	C₄×D₄	—	178	1	no	—
245	C₂×C₄⋊C₄	—	89	1	no	—
246	C₂×C₄⋊C₄	—	89	1	no	—
247	C₂×C₄⋊C₄	—	178	1	no	—
248	C₂×C₂²⋊C₄	—	178	1	no	—
249	C₂×C₄²	—	178	1	no	—
250	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
251	#106	—	178	1	no	—
252	#106	—	178	1	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	178	1	no	—
257	#103	—	178	1	no	—
258	#103	—	178	1	no	—
259	#102	—	178	1	no	—
260	#102	—	178	1	no	—
261	C₂²⋊C₈	—	2	1	no	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	89	2	no	—
264	C₂²⋊C₈	—	178	1	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	178	1	no	—
268	C₄×C₈	—	178	1	no	—
269	C₄×C₈	D₃₅₆	1	1	yes	characteristic
270	C₄×C₈	D₃₅₆	1	1	yes	characteristic
271	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
272	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
273	C₂⁴	—	178	1	no	—
274	C₂×Q₈	—	89	2	no	—
275	C₂×Q₈	—	178	1	no	—
276	C₂×Q₈	—	178	1	no	—
277	C₂×Q₈	—	178	1	no	—
278	C₂×Q₈	—	178	1	no	—
279	C₂×Q₈	—	178	1	no	—
280	C₂×D₄	—	89	2	no	—
281	C₂×D₄	—	178	1	no	—
282	C₂×D₄	—	178	1	no	—
283	C₂×D₄	—	178	1	no	—
284	C₂×D₄	—	178	1	no	—
285	C₂×D₄	—	178	1	no	—
286	C₂²×C₄	—	356	1	no	—
287	C₂²×C₄	—	356	1	no	—
288	C₂²×C₄	—	178	1	no	—
289	C₂²×C₄	—	178	1	no	—
290	C₂²×C₄	—	178	1	no	—
291	C₂²×C₄	—	712	2	no	—
292	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
293	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
294	C₂²×C₄	#13083575	1	1	yes	characteristic
295	SD₁₆	—	178	4	no	—
296	SD₁₆	—	178	4	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	4	1	no	—
300	C₂×C₈	—	4	1	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	#13083551	1	1	yes	characteristic
304	C₂×C₈	#13083551	1	1	yes	characteristic
305	C₂×C₈	#13083551	1	1	yes	characteristic
306	C₂×C₈	#13083551	1	1	yes	characteristic
307	C₂×C₈	C₈×D₈₉	1	2	yes	—
308	C₂×C₈	#13083557	1	2	yes	—
309	C₄⋊C₄	—	356	1	no	—
310	C₄⋊C₄	—	356	1	no	—
311	C₄⋊C₄	—	178	1	no	—
312	C₄⋊C₄	—	178	1	no	—
313	C₄⋊C₄	—	178	1	no	—
314	C₄⋊C₄	—	178	1	no	—
315	C₂²⋊C₄	—	356	1	no	—
316	C₂²⋊C₄	—	356	1	no	—
317	C₄²	—	356	1	no	—
318	C₄²	—	356	1	no	—
319	C₄²	D₄×D₈₉	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
322	C₄²	#13083575	1	1	yes	characteristic
323	C₂³	—	356	1	no	—
324	C₂³	—	356	1	no	—
325	C₂³	—	178	2	no	—
326	C₂³	—	712	1	no	—
327	C₂³	—	712	1	no	—
328	C₂³	#15667974	1	1	yes	characteristic
329	Q₈	—	356	1	no	—
330	Q₈	—	356	1	no	—
331	Q₈	—	178	2	no	—
332	Q₈	—	178	2	no	—
333	D₄	—	356	1	no	—
334	D₄	—	356	1	no	—
335	D₄	—	178	2	no	—
336	D₄	—	178	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	356	1	no	—
340	C₂×C₄	—	356	1	no	—
341	C₂×C₄	—	356	1	no	—
342	C₂×C₄	—	356	1	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	—	712	2	no	—
345	C₂×C₄	—	712	2	no	—
346	C₂×C₄	—	712	1	no	—
347	C₂×C₄	—	712	1	no	—
348	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
349	C₂×C₄	#15667974	1	1	yes	characteristic
350	C₂×C₄	#15667987	1	1	yes	characteristic
351	C₂×C₄	#15667987	1	1	yes	characteristic
352	C₂×C₄	#15668001	1	1	yes	characteristic
353	C₂×C₄	#15668008	1	1	yes	characteristic
354	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
355	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
356	C₂×C₄	#15668020	1	1	yes	characteristic
357	C₂×C₄	#15668020	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	8	1	no	—
361	C₈	—	8	1	no	—
362	C₈	—	356	2	no	—
363	C₈	—	356	2	no	—
364	C₈	#15667924	1	2	yes	—
365	C₈	#15667924	1	2	yes	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	356	2	no	—
369	C₂²	—	712	2	no	—
370	C₂²	—	712	2	no	—
371	C₂²	—	712	1	no	—
372	C₂²	—	712	1	no	—
373	C₂²	#19151499	1	1	yes	characteristic
374	C₂²	#19151663	1	1	yes	characteristic
375	C₂²	#19151675	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	356	2	no	—
380	C₄	—	712	1	no	—
381	C₄	—	712	1	no	—
382	C₄	C₂×#15667924	1	1	yes	characteristic
383	C₄	#19151566	1	1	yes	characteristic
384	C₄	#19151663	1	1	yes	characteristic
385	C₄	#19151675	1	1	yes	characteristic
386	C₄	#19151862	1	2	yes	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	712	2	no	—
390	C₂	#26021045	1	1	yes	characteristic
391	C₂	#26021148	1	1	yes	characteristic
392	C₂	#26021153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#9667
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #26018648	C₂	semidirect
4. #26018575	C₂	semidirect
6. #26018535	C₂	semidirect
7. C₈₉×#1564	C₂	semidirect
37. C₂×#13083563	C₈	semidirect
198. #1564	D₈₉	semidirect
204. C₈₉	#9667	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083563	2
#26018648	2
#26018644	2
#26018575	2
#26018572	2
#26018535	2
C₈₉×#1564	2
#9667	89