Subgroups — #36447267

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020232	C₂	1	2	yes	maximal
2	C₂×#19151351	C₂	1	2	yes	maximal, complemented
3	#26018087	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
6	C₂×C₂²⋊#10623494	—	2	2	no	—
7	#19151353	—	2	2	no	—
8	#19151351	—	2	4	no	—
9	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	—	2	4	no	—
13	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	4	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄⋊#10623494	—	2	4	no	—
20	C₂²⋊#10623494	—	4	4	no	—
21	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	—	2	2	no	—
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	—	2	2	no	—
29	C₂²×C₈×C₈₉	C₈	1	4	yes	complemented
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	—	2	2	no	—
35	C₂×C₄²×C₈₉	—	2	2	no	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	—	2	2	no	—
42	C₄×C₈×C₈₉	—	2	2	no	—
43	C₄×C₈×C₈₉	D₄	1	4	yes	—
44	C₂×#10623494	—	4	4	no	—
45	C₂×#10623494	—	4	4	no	—
46	C₂×#10623494	—	4	4	no	—
47	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	—	2	2	no	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	1	no	—
55	C₂²×C₄×C₈₉	—	2	1	no	—
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
62	C₂²×C₄×C₈₉	D₈	1	2	yes	—
63	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	8	no	—
68	C₂×C₈×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	—	2	1	no	—
70	C₂×C₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	—	2	1	no	—
72	C₂×C₈×C₈₉	—	2	1	no	—
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	—	2	8	no	—
75	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
80	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
81	C₄²×C₈₉	—	2	1	no	—
82	C₄²×C₈₉	—	2	1	no	—
83	C₄²×C₈₉	—	2	2	no	—
84	C₄²×C₈₉	—	2	2	no	—
85	C₄²×C₈₉	C₂×D₄	1	2	yes	—
86	C₄²×C₈₉	#41	1	2	yes	—
87	C₂³×C₈₉	—	2	1	no	—
88	C₂³×C₈₉	—	2	1	no	—
89	C₂³×C₈₉	—	2	1	no	—
90	C₂³×C₈₉	—	2	1	no	—
91	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
92	C₂³×C₈₉	#107	1	2	yes	—
93	C₂³×C₈₉	#108	1	2	yes	—
94	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	—	2	8	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	4	no	—
109	C₂×C₄×C₈₉	—	2	4	no	—
110	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₈₉	#107	1	2	yes	—
112	C₂×C₄×C₈₉	#108	1	2	yes	—
113	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#121	1	2	yes	—
116	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#135	1	2	yes	—
124	C₈×C₈₉	—	2	2	no	—
125	C₈×C₈₉	—	2	2	no	—
126	C₈×C₈₉	—	2	2	no	—
127	C₈×C₈₉	C₂²⋊C₈	1	4	yes	complemented
128	#10623494	—	8	4	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	#335	1	4	yes	complemented
136	C₂²×C₈₉	#380	1	1	yes	characteristic
137	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₈₉	#423	1	1	yes	characteristic
139	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
140	C₂²×C₈₉	#427	1	1	yes	characteristic
141	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
142	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₈₉	#407	1	1	yes	characteristic
149	C₄×C₈₉	#431	1	1	yes	characteristic
150	C₄×C₈₉	#432	1	1	yes	characteristic
151	C₄×C₈₉	#466	1	2	yes	—
152	C₄×C₈₉	#467	1	2	yes	—
153	#8596	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	2	4	no	—
155	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
156	C₂×C₈₉	#1564	1	1	yes	characteristic
157	C₂×C₈₉	#1567	1	1	yes	characteristic
158	C₂×C₈₉	#1568	1	1	yes	characteristic
159	C₂×C₈₉	#1797	1	1	yes	characteristic
160	C₂×C₈₉	#1808	1	1	yes	characteristic
161	C₂×C₈₉	#1810	1	1	yes	characteristic
162	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
163	#1797	—	89	2	no	—
164	C₂×#335	—	89	2	no	—
165	#1277	—	89	2	no	—
166	C₈₉	#8596	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
168	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
169	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
170	C₂×C₄⋊C₈	—	89	4	no	—
171	C₂×C₂²⋊C₈	—	178	2	no	—
172	C₂×C₄×C₈	—	2	4	no	—
173	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
174	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
177	#336	—	178	2	no	—
178	#335	—	178	4	no	—
179	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
180	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
181	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	178	4	no	—
188	C₂²×C₈	#10623494	1	4	yes	complemented
189	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
196	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
199	C₄⋊C₈	—	178	4	no	—
200	C₄⋊C₈	—	178	4	no	—
201	C₂²⋊C₈	—	356	4	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	C₂²⋊D₈₉	1	4	yes	—
206	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×#10623494	1	2	yes	—
220	C₂²×C₄	#13083558	1	2	yes	—
221	C₂²×C₄	D₇₁₂	1	2	yes	—
222	C₂²×C₄	Dic₃₅₆	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
226	C₂×C₈	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	356	4	no	—
235	C₂×C₈	—	356	4	no	—
236	C₂×C₈	—	356	4	no	—
237	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
238	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
239	C₂×C₈	C₂×#10623494	1	2	yes	—
240	C₂×C₈	C₂×#10623494	1	2	yes	—
241	C₂×C₈	#13083558	1	2	yes	—
242	C₂×C₈	#13083558	1	2	yes	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
248	C₄²	#13083574	1	2	yes	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	C₄⋊#10623494	1	2	yes	—
254	C₂³	#15667946	1	2	yes	—
255	C₂³	#15667948	1	2	yes	—
256	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	C₄⋊#10623494	1	2	yes	—
273	C₂×C₄	#15667946	1	2	yes	—
274	C₂×C₄	#15667948	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
276	C₂×C₄	#15667971	1	1	yes	characteristic
277	C₂×C₄	#15667986	1	2	yes	—
278	C₂×C₄	#15667991	1	2	yes	—
279	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
280	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
281	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
282	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
283	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
284	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
285	C₂×C₄	#15668034	1	2	yes	—
286	C₈	—	2	2	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	712	4	no	—
290	C₈	C₂²⋊#10623494	1	4	yes	complemented
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	#19151351	1	4	yes	complemented
298	C₂²	#19151458	1	1	yes	characteristic
299	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
300	C₂²	#19151622	1	1	yes	characteristic
301	C₂²	C₂×#15667946	1	1	yes	characteristic
302	C₂²	#19151636	1	1	yes	characteristic
303	C₂²	C₂×#15667948	1	1	yes	characteristic
304	C₂²	C₂×#15667948	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	C₂×C₂²⋊#10623494	1	1	yes	characteristic
310	C₄	#19151562	1	1	yes	characteristic
311	C₄	#19151651	1	1	yes	characteristic
312	C₄	#19151655	1	1	yes	characteristic
313	C₄	#19151834	1	2	yes	—
314	C₄	#19151839	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	C₂×#19151351	1	1	yes	characteristic
317	C₂	#26019000	1	1	yes	characteristic
318	C₂	#26019011	1	1	yes	characteristic
319	C₂	#26019015	1	1	yes	characteristic
320	C₂	#26020232	1	1	yes	characteristic
321	C₂	#26020281	1	1	yes	characteristic
322	C₂	#26020290	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#8596
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19151351	C₂	semidirect
3. #26018087	C₂	semidirect
29. C₂²×C₈×C₈₉	C₈	semidirect
127. C₈×C₈₉	C₂²⋊C₈	semidirect
135. C₂²×C₈₉	#335	semidirect
166. C₈₉	#8596	semidirect
188. C₂²×C₈	#10623494	semidirect
290. C₈	C₂²⋊#10623494	semidirect
297. C₂²	#19151351	semidirect

Maximal subgroups

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

subgroup	index
#26020232	2
C₂×#19151351	2
#26018087	2
C₂²×C₄×C₈×C₈₉	2
#8596	89