Subgroups — #35557548

Order 21248.

528 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581798	C₂	1	1	yes	maximal, characteristic
2	#24909278	C₂	1	4	yes	maximal, complemented
3	#24909274	C₂	1	4	yes	maximal, complemented
4	C₂×#19127522	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909083	C₂	1	2	yes	maximal
6	#24909067	C₂	1	2	yes	maximal, complemented
7	C₂×#364×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581798	C₄	1	1	yes	characteristic
9	C₂×#14581798	C₂²	1	1	yes	characteristic
10	C₄×#13050590	C₄	1	1	yes	characteristic
11	C₂×#14581732	C₂²	1	2	yes	—
12	#19127656	C₄	1	1	yes	characteristic
13	#19127656	C₂²	1	1	yes	characteristic
14	#19127655	C₄	1	2	yes	—
15	#19127655	C₂²	1	2	yes	—
16	C₄×#13050574	C₄	1	1	yes	characteristic
17	C₄×#13050573	C₂²	1	2	yes	—
18	C₄×C₈×D₈₃	C₄	1	2	yes	—
19	#19127522	C₂²	1	2	yes	—
20	#19127522	C₂²	1	2	yes	—
21	#19127482	C₂²	1	8	yes	—
22	#19127478	C₂²	1	8	yes	—
23	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	#364×C₈₃	C₂²	1	2	yes	—
26	#364×C₈₃	C₂²	1	2	yes	—
27	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
28	#14581798	—	2	2	no	—
29	#14581798	—	2	2	no	—
30	#14581798	—	2	1	no	—
31	#14581798	—	2	1	no	—
32	#14581798	—	2	1	no	—
33	#14581798	—	2	1	no	—
34	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
35	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
36	C₂×#13050573	C₂×C₄	1	2	yes	—
37	C₂×#13050573	C₂³	1	2	yes	—
38	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
40	C₄×Dic₁₆₆	—	2	1	no	—
41	C₄×D₃₃₂	—	2	1	no	—
42	C₄×C₂²⋊D₈₃	—	2	2	no	—
43	#14581751	C₂×C₄	1	1	yes	characteristic
44	#14581749	C₂×C₄	1	2	yes	—
45	C₄²×D₈₃	C₂×C₄	1	2	yes	—
46	#14581732	—	2	2	no	—
47	#14581732	C₂³	1	4	yes	—
48	C₂×C₈₃⋊C₁₆	—	2	2	no	—
49	#14581703	C₂×C₄	1	2	yes	—
50	#14581702	C₂×C₄	1	2	yes	—
51	#14581702	C₂³	1	2	yes	—
52	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₈₃	—	2	2	no	—
59	M₃₂×C₈₃	C₂³	1	4	yes	—
60	C₂×C₁₆×C₈₃	—	2	2	no	—
61	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
65	#13050590	—	2	1	no	—
66	#13050590	—	2	1	no	—
67	#13050590	—	2	1	no	—
68	#13050590	—	2	1	no	—
69	C₂×Dic₁₆₆	M₁₆	1	1	yes	characteristic
70	C₂×D₃₃₂	M₁₆	1	1	yes	characteristic
71	C₂×C₂²⋊D₈₃	M₁₆	1	2	yes	—
72	C₂×C₄×D₈₃	—	2	2	no	—
73	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
74	#13050574	—	2	1	no	—
75	#13050574	—	2	1	no	—
76	#13050574	—	2	1	no	—
77	#13050574	—	2	1	no	—
78	#13050573	—	2	4	no	—
79	#13050573	—	2	2	no	—
80	#13050573	—	2	2	no	—
81	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
83	C₈×D₈₃	—	2	2	no	—
84	C₈×D₈₃	—	2	2	no	—
85	C₈×D₈₃	—	2	4	no	—
86	C₄⋊Dic₈₃	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	2	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	2	no	—
90	C₄×Dic₈₃	—	2	2	no	—
91	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
92	C₈₃⋊C₁₆	—	2	4	no	—
93	C₂²×C₄×C₈₃	—	2	1	no	—
94	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
96	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
97	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
105	C₄²×C₈₃	—	2	1	no	—
106	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
108	C₁₆×C₈₃	—	2	4	no	—
109	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂²×D₈₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	4	2	no	—
114	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₈₃	—	2	2	no	—
121	C₂²⋊D₈₃	—	2	2	no	—
122	C₂×Dic₈₃	—	2	2	no	—
123	C₂×Dic₈₃	C₂×M₁₆	1	2	yes	—
124	D₃₃₂	—	2	1	no	—
125	D₃₃₂	—	2	1	no	—
126	C₄×D₈₃	—	2	2	no	—
127	C₄×D₈₃	—	2	2	no	—
128	C₄×D₈₃	—	4	4	no	—
129	Dic₁₆₆	—	2	1	no	—
130	Dic₁₆₆	—	2	1	no	—
131	C₈×C₈₃	#117	1	1	yes	characteristic
132	C₈×C₈₃	#117	1	1	yes	characteristic
133	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
134	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
135	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
137	C₈₃⋊C₈	—	2	2	no	—
138	C₈₃⋊C₈	—	2	2	no	—
139	C₂²×C₈₃	—	4	1	no	—
140	C₂²×C₈₃	#432	1	1	yes	characteristic
141	C₂²×C₈₃	#432	1	1	yes	characteristic
142	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₂×D₈₃	—	2	2	no	—
144	C₂×D₈₃	—	4	2	no	—
145	C₄×C₈₃	—	2	2	no	—
146	C₄×C₈₃	—	4	1	no	—
147	C₄×C₈₃	#432	1	1	yes	characteristic
148	C₄×C₈₃	#432	1	1	yes	characteristic
149	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
150	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
151	Dic₈₃	—	2	2	no	—
152	Dic₈₃	—	4	2	no	—
153	#33428	—	83	1	no	maximal, Sylow
154	C₂×C₈₃	—	2	1	no	—
155	C₂×C₈₃	—	2	1	no	—
156	C₂×C₈₃	—	2	1	no	—
157	C₂×C₈₃	C₂×#432	1	1	yes	characteristic
158	D₈₃	—	4	2	no	—
159	C₄×#131	—	83	1	no	—
160	#2166	—	83	4	no	—
161	#2165	—	83	4	no	—
162	C₂×#364	—	83	1	no	—
163	C₂×#364	D₈₃	1	1	yes	characteristic, complemented
164	#2115	—	83	2	no	—
165	#2110	—	83	2	no	—
166	C₈₃	#33428	1	1	yes	characteristic, complemented, Sylow
167	C₂×#131	—	83	1	no	—
168	C₂×#131	—	83	1	no	—
169	C₄×#41	—	83	1	no	—
170	C₂×M₃₂	—	83	2	no	—
171	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
172	#405	—	83	1	no	—
173	#405	—	83	1	no	—
174	#405	—	83	2	no	—
175	#405	—	83	2	no	—
176	C₄×M₁₆	—	83	1	no	—
177	C₄×M₁₆	—	83	2	no	—
178	C₂×C₄×C₈	—	83	2	no	—
179	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
180	#364	—	83	2	no	—
181	#364	—	83	2	no	—
182	#364	C₂×D₈₃	1	2	yes	—
183	#364	C₂×D₈₃	1	2	yes	—
184	#349	—	83	8	no	—
185	#347	—	83	8	no	—
186	C₂×#41	—	83	1	no	—
187	#131	—	166	2	no	—
188	#131	—	166	2	no	—
189	#131	—	166	1	no	—
190	#131	—	166	1	no	—
191	#131	—	166	1	no	—
192	#131	—	166	1	no	—
193	C₂×M₁₆	—	83	1	no	—
194	C₂×M₁₆	—	83	1	no	—
195	C₂×M₁₆	—	83	2	no	—
196	C₂×M₁₆	—	83	2	no	—
197	C₂²×C₈	—	83	2	no	—
198	C₂²×C₈	—	83	2	no	—
199	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
201	C₄×Q₈	—	166	1	no	—
202	C₄×D₄	—	166	1	no	—
203	C₄×D₄	—	166	2	no	—
204	#117	—	83	1	no	—
205	#117	—	83	2	no	—
206	C₂×C₄²	—	83	2	no	—
207	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
208	M₃₂	—	2	2	no	—
209	M₃₂	—	83	4	no	—
210	M₃₂	—	166	2	no	—
211	M₃₂	C₂²×D₈₃	1	4	yes	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	166	2	no	—
214	#97	—	83	2	no	—
215	#97	—	83	2	no	—
216	#97	—	83	2	no	—
217	C₄×C₈	—	83	2	no	—
218	C₄×C₈	—	83	2	no	—
219	C₄×C₈	—	83	2	no	—
220	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
221	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
224	#41	—	166	1	no	—
225	#41	—	166	1	no	—
226	#41	—	166	1	no	—
227	#41	—	166	1	no	—
228	C₂×Q₈	—	83	1	no	—
229	C₂×D₄	—	83	2	no	—
230	C₂×D₄	—	83	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	83	2	no	—
233	C₂²×C₄	—	166	2	no	—
234	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
235	M₁₆	—	166	1	no	—
236	M₁₆	—	166	1	no	—
237	M₁₆	—	166	1	no	—
238	M₁₆	—	166	1	no	—
239	M₁₆	—	166	4	no	—
240	M₁₆	—	166	2	no	—
241	M₁₆	—	166	2	no	—
242	C₂×C₈	—	83	2	no	—
243	C₂×C₈	—	83	2	no	—
244	C₂×C₈	—	166	4	no	—
245	C₂×C₈	—	166	2	no	—
246	C₂×C₈	—	166	2	no	—
247	C₂×C₈	#13050573	1	2	yes	—
248	C₂×C₈	#13050573	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
252	C₂×C₈	#13050589	1	1	yes	characteristic
253	C₂×C₈	#13050589	1	1	yes	characteristic
254	C₂×C₈	#13050591	1	1	yes	characteristic
255	C₂×C₈	#13050591	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	—	166	2	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₂²⋊C₄	—	166	2	no	—
260	C₂²⋊C₄	—	166	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	83	2	no	—
263	C₄²	—	166	2	no	—
264	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
266	C₁₆	—	2	4	no	—
267	C₁₆	—	166	4	no	—
268	C₂³	—	83	2	no	—
269	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
270	Q₈	—	166	1	no	—
271	Q₈	—	166	1	no	—
272	D₄	—	166	2	no	—
273	D₄	—	166	2	no	—
274	D₄	—	166	1	no	—
275	D₄	—	166	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	332	4	no	—
280	C₂×C₄	—	83	2	no	—
281	C₂×C₄	—	166	2	no	—
282	C₂×C₄	—	166	2	no	—
283	C₂×C₄	—	166	2	no	—
284	C₂×C₄	#14581750	1	1	yes	characteristic
285	C₂×C₄	#14581750	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
287	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
288	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
290	C₈	—	166	2	no	—
291	C₈	—	166	2	no	—
292	C₈	#14581750	1	1	yes	characteristic
293	C₈	#14581750	1	1	yes	characteristic
294	C₈	C₂×#13050573	1	2	yes	—
295	C₈	C₂×#13050573	1	2	yes	—
296	C₈	C₂×#13050589	1	1	yes	characteristic
297	C₈	C₂×#13050591	1	1	yes	characteristic
298	C₂²	—	4	1	no	—
299	C₂²	—	332	2	no	—
300	C₂²	—	166	2	no	—
301	C₂²	#19127757	1	1	yes	characteristic
302	C₂²	#19127757	1	1	yes	characteristic
303	C₂²	C₂×M₁₆×D₈₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	332	2	no	—
307	C₄	—	166	2	no	—
308	C₄	#19127757	1	1	yes	characteristic
309	C₄	#19127757	1	1	yes	characteristic
310	C₄	C₂×#14581750	1	1	yes	characteristic
311	C₄	C₂²×#13050573	1	1	yes	characteristic
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	332	2	no	—
316	C₂	C₂×#19127757	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14581750
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#364×C₈₃	C₂
socle	C₂×C₈₃	C₂×#432
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#364	D₈₃
hypocenter	C₈₃	#33428
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #24909278	C₂	semidirect
3. #24909274	C₂	semidirect
4. C₂×#19127522	C₂	semidirect
6. #24909067	C₂	semidirect
7. C₂×#364×C₈₃	C₂	semidirect
163. C₂×#364	D₈₃	semidirect
166. C₈₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581798	2
#24909278	2
#24909274	2
C₂×#19127522	2
#24909083	2
#24909067	2
C₂×#364×C₈₃	2
#33428	83