Subgroups — #36525119

Order 22784.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030822	C₂	1	1	yes	maximal, characteristic
2	#26029403	C₂	1	2	yes	maximal, complemented
3	#26019156	C₂	1	2	yes	maximal, complemented
4	#26019150	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019138	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019066	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018718	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152132	C₂²	1	2	yes	—
13	#19152000	C₄	1	1	yes	characteristic
14	#19151982	C₄	1	1	yes	characteristic
15	#19151953	C₄	1	2	yes	—
16	#19151839	C₄	1	1	yes	characteristic
17	#19151834	C₄	1	1	yes	characteristic
18	#19151827	C₄	1	2	yes	—
19	#19151728	C₂²	1	2	yes	—
20	C₈×Dic₁₇₈	C₂²	1	2	yes	—
21	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	C₄×#13083563	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	#19151677	C₂²	1	2	yes	—
25	#19151673	—	2	2	no	—
26	#19151672	—	2	2	no	—
27	C₈×D₃₅₆	C₂²	1	2	yes	—
28	#19151658	—	2	2	no	—
29	#19151623	—	2	2	no	—
30	#19151622	C₂²	1	2	yes	—
31	#19151351	C₂²	1	1	yes	characteristic
32	#19151351	C₂²	1	1	yes	characteristic
33	#19151351	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	#19151330	C₂²	1	2	yes	—
36	#19151327	C₂²	1	2	yes	—
37	#19151327	C₂²	1	2	yes	—
38	#431×C₈₉	C₂²	1	1	yes	characteristic
39	#431×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
41	#322×C₈₉	C₂²	1	1	yes	characteristic
42	#322×C₈₉	C₂²	1	1	yes	characteristic
43	C₈²×C₈₉	C₂²	1	1	yes	characteristic
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083563	C₂×C₄	1	2	yes	—
47	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
48	C₂×#13083558	—	2	2	no	—
49	C₂×#13083557	—	2	2	no	—
50	C₂×C₈×D₈₉	—	2	2	no	—
51	#15668006	—	2	2	no	—
52	#15668004	C₂×C₄	1	2	yes	—
53	#15668000	C₂×C₄	1	2	yes	—
54	#15667995	—	2	2	no	—
55	#15667991	C₂×C₄	1	2	yes	—
56	#15667986	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₇₈	C₂³	1	2	yes	—
58	C₄×D₃₅₆	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₈₉	—	2	2	no	—
60	#15667969	—	2	2	no	—
61	#15667946	C₂×C₄	1	1	yes	characteristic
62	#15667946	C₂×C₄	1	1	yes	characteristic
63	#15667944	C₂×C₄	1	2	yes	—
64	C₄⋊#10623494	—	2	2	no	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	C₄⋊#10623494	C₂³	1	2	yes	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	2	no	—
69	#15667937	C₂×C₄	1	2	yes	—
70	#15667937	C₂×C₄	1	2	yes	—
71	#15667934	C₂×C₄	1	2	yes	—
72	#15667934	C₂×C₄	1	2	yes	—
73	#15667924	—	2	2	no	—
74	#15667924	—	2	2	no	—
75	#15667922	—	2	2	no	—
76	C₈×Dic₈₉	—	2	2	no	—
77	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₈₉	—	2	1	no	—
81	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
82	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
83	C₂²⋊C₈×C₈₉	—	2	1	no	—
84	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
85	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
95	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₈₉	—	2	2	no	—
97	C₂×C₄×D₈₉	—	2	2	no	—
98	Dic₃₅₆	—	2	2	no	—
99	#13083563	—	2	4	no	—
100	D₇₁₂	—	2	2	no	—
101	#13083558	—	8	2	no	—
102	#13083557	—	8	2	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₈×D₈₉	—	8	2	no	—
106	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
107	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
108	#13083553	—	2	2	no	—
109	#13083553	—	2	2	no	—
110	#13083553	—	2	2	no	—
111	#13083553	—	2	2	no	—
112	C₂²⋊Dic₈₉	—	2	2	no	—
113	#13083551	—	2	2	no	—
114	#13083551	—	2	2	no	—
115	#13083551	—	2	2	no	—
116	#13083551	—	2	2	no	—
117	C₄×Dic₈₉	—	2	2	no	—
118	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	1	no	—
125	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₈₉	#41	1	1	yes	characteristic
136	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂²×D₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	—	2	1	no	—
142	C₂×C₄×C₈₉	—	2	1	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₈₉	—	8	2	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	2	2	no	—
154	D₃₅₆	—	2	2	no	—
155	D₃₅₆	—	2	2	no	—
156	C₄×D₈₉	—	4	2	no	—
157	C₄×D₈₉	—	4	2	no	—
158	Dic₁₇₈	—	2	2	no	—
159	Dic₁₇₈	—	2	2	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	—	2	1	no	—
165	C₈×C₈₉	—	2	1	no	—
166	C₈×C₈₉	—	4	1	no	—
167	C₈×C₈₉	—	4	1	no	—
168	C₈×C₈₉	#131	1	2	yes	—
169	C₈×C₈₉	#131	1	2	yes	—
170	#10623494	—	4	2	no	—
171	#10623494	—	4	2	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	—	2	1	no	—
174	C₂²×C₈₉	—	2	1	no	—
175	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₈₉	—	4	2	no	—
177	C₂×D₈₉	—	4	2	no	—
178	C₂×D₈₉	—	4	2	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
184	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
185	C₄×C₈₉	#576	1	1	yes	characteristic
186	C₄×C₈₉	#578	1	1	yes	characteristic
187	Dic₈₉	—	4	2	no	—
188	Dic₈₉	—	4	2	no	—
189	Dic₈₉	—	4	2	no	—
190	#17973	—	89	1	no	maximal, Sylow
191	C₂×C₈₉	—	4	1	no	—
192	C₂×C₈₉	#2931	1	1	yes	characteristic
193	C₂×C₈₉	#2951	1	1	yes	characteristic
194	C₂×C₈₉	#2957	1	1	yes	characteristic
195	D₈₉	—	8	2	no	—
196	#3168	—	89	1	no	—
197	#2993	—	89	2	no	—
198	#1592	—	89	2	no	—
199	#1591	—	89	1	no	—
200	#1589	—	89	1	no	—
201	SD₁₆⋊C₈	—	89	2	no	—
202	C₈×Q₁₆	—	89	1	no	—
203	C₈×D₈	—	89	1	no	—
204	#1567	—	89	1	no	—
205	#1495	—	89	2	no	—
206	#1450	D₈₉	1	1	yes	characteristic, complemented
207	C₈₉	#17973	1	1	yes	characteristic, complemented, Sylow
208	#529	—	89	2	no	—
209	#499	—	89	1	no	—
210	#495	—	89	1	no	—
211	#488	—	89	2	no	—
212	#467	—	89	1	no	—
213	#466	—	89	1	no	—
214	#465	—	89	2	no	—
215	#446	—	89	2	no	—
216	C₈×Q₈	—	89	2	no	—
217	C₄×Q₁₆	—	89	1	no	—
218	C₄×SD₁₆	—	89	2	no	—
219	C₄×D₈	—	89	1	no	—
220	#436	—	89	2	no	—
221	#436	—	178	2	no	—
222	#435	—	178	2	no	—
223	C₈×D₄	—	89	2	no	—
224	#433	—	178	2	no	—
225	#431	C₂×D₈₉	1	1	yes	characteristic
226	#431	C₂×D₈₉	1	1	yes	characteristic
227	#424	—	178	2	no	—
228	#423	—	89	2	no	—
229	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
230	#335	—	89	1	no	—
231	#335	—	89	1	no	—
232	#335	—	89	2	no	—
233	Q₈⋊C₈	—	89	2	no	—
234	Q₈⋊C₈	—	89	2	no	—
235	D₄⋊C₈	—	89	2	no	—
236	D₄⋊C₈	—	89	2	no	—
237	#322	C₂×D₈₉	1	1	yes	characteristic
238	#322	C₂×D₈₉	1	1	yes	characteristic
239	C₈²	C₂×D₈₉	1	1	yes	characteristic
240	C₈²	C₂×D₈₉	1	1	yes	characteristic
241	C₂×Q₁₆	—	89	1	no	—
242	C₂×SD₁₆	—	89	2	no	—
243	C₂×D₈	—	89	1	no	—
244	C₂×M₁₆	—	178	2	no	—
245	C₂×M₁₆	—	178	2	no	—
246	C₂²×C₈	—	178	2	no	—
247	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
248	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
249	#126	—	178	2	no	—
250	#125	—	89	2	no	—
251	#124	—	89	2	no	—
252	#123	—	178	2	no	—
253	C₂²⋊Q₈	—	89	2	no	—
254	#121	—	89	2	no	—
255	C₄×Q₈	—	89	2	no	—
256	C₄×D₄	—	89	2	no	—
257	C₄×D₄	—	178	2	no	—
258	#117	—	178	2	no	—
259	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
260	#107	—	89	1	no	—
261	#107	—	89	1	no	—
262	#106	—	89	2	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	89	2	no	—
265	C₄⋊C₈	—	89	2	no	—
266	C₄⋊C₈	—	178	2	no	—
267	C₄⋊C₈	—	178	2	no	—
268	C₄⋊C₈	—	178	2	no	—
269	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
270	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
271	#103	—	89	2	no	—
272	#103	—	89	2	no	—
273	#102	—	89	2	no	—
274	#102	—	89	2	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	—	178	2	no	—
277	C₂²⋊C₈	—	178	2	no	—
278	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
279	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
280	#97	—	178	2	no	—
281	C₄×C₈	—	178	2	no	—
282	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
283	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	C₂×Q₈	—	89	2	no	—
291	C₂×D₄	—	89	2	no	—
292	C₂×D₄	—	178	2	no	—
293	C₂²×C₄	—	178	2	no	—
294	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
296	C₂²×C₄	#13083574	1	1	yes	characteristic
297	Q₁₆	—	178	2	no	—
298	SD₁₆	—	178	4	no	—
299	D₈	—	178	2	no	—
300	M₁₆	—	712	2	no	—
301	M₁₆	—	712	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	—	712	2	no	—
309	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
311	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
313	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
314	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
315	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
316	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
317	C₄⋊C₄	—	89	2	no	—
318	C₄⋊C₄	—	89	2	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	—	178	2	no	—
321	C₄⋊C₄	—	178	2	no	—
322	C₄⋊C₄	—	178	2	no	—
323	C₂²⋊C₄	—	178	2	no	—
324	C₂²⋊C₄	—	178	2	no	—
325	C₂²⋊C₄	—	178	2	no	—
326	C₂²⋊C₄	—	178	2	no	—
327	C₂²⋊C₄	—	178	2	no	—
328	C₄²	—	178	2	no	—
329	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
330	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
331	C₄²	#13083574	1	1	yes	characteristic
332	C₄²	C₂³×D₈₉	1	1	yes	characteristic
333	C₂³	—	178	2	no	—
334	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
335	Q₈	—	178	2	no	—
336	Q₈	—	178	2	no	—
337	D₄	—	178	2	no	—
338	D₄	—	178	2	no	—
339	D₄	—	712	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	178	2	no	—
348	C₂×C₄	—	178	2	no	—
349	C₂×C₄	—	178	2	no	—
350	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
351	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
352	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
355	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	356	2	no	—
365	C₈	—	356	2	no	—
366	C₈	#15668018	1	2	yes	—
367	C₈	#15668018	1	2	yes	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	356	2	no	—
372	C₂²	—	356	2	no	—
373	C₂²	—	356	2	no	—
374	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	356	2	no	—
380	C₄	—	356	2	no	—
381	C₄	—	356	2	no	—
382	C₄	C₂×#15668018	1	1	yes	characteristic
383	C₄	C₂×#15668018	1	1	yes	characteristic
384	C₄	#19152403	1	1	yes	characteristic
385	C₄	#19152415	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	712	2	no	—
388	C₂	#26028965	1	1	yes	characteristic
389	C₂	#26029129	1	1	yes	characteristic
390	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19151834	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1450	D₈₉
hypocenter	C₈₉	#17973
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #26029403	C₂	semidirect
3. #26019156	C₂	semidirect
4. #26019150	C₂	semidirect
5. #26019138	C₂	semidirect
6. #26019066	C₂	semidirect
7. C₈×Dic₃₅₆	C₂	semidirect
8. C₈×D₇₁₂	C₂	semidirect
9. #26019011	C₂	semidirect
10. #26018718	C₂	semidirect
11. C₈₉×#1450	C₂	semidirect
206. #1450	D₈₉	semidirect
207. C₈₉	#17973	semidirect

Maximal subgroups

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

subgroup	index
#26030822	2
#26029403	2
#26019156	2
#26019150	2
#26019138	2
#26019066	2
C₈×Dic₃₅₆	2
C₈×D₇₁₂	2
#26019011	2
#26018718	2
C₈₉×#1450	2
#17973	89