Subgroups — #35295083

Order 21248.

514 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581799	C₂	1	1	yes	maximal, characteristic
2	#24905845	C₂	1	1	yes	maximal, characteristic
3	#24905747	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19127423	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905300	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1476	C₂	1	1	yes	maximal, characteristic
8	C₂×#14581799	C₄	1	2	yes	—
9	#19127754	C₂²	1	1	yes	characteristic
10	#19127657	—	2	2	no	—
11	#19127657	C₄	1	2	yes	—
12	C₄×#13050574	—	2	1	no	—
13	C₄×#13050574	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈₃⋊C₈	—	2	1	no	—
15	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
18	D₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
19	#19127423	—	2	2	no	—
20	#19127423	—	2	1	no	—
21	C₈×C₈₃⋊C₈	—	2	1	no	—
22	C₄×#41×C₈₃	C₂²	1	1	yes	characteristic
23	#431×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	Q₈⋊C₈×C₈₃	C₄	1	2	yes	—
26	D₄⋊C₈×C₈₃	C₄	1	2	yes	—
27	#14581799	—	2	4	no	—
28	#14581799	C₈	1	8	yes	complemented
29	C₂×#13050574	—	2	2	no	—
30	C₂×#13050574	C₂×C₄	1	2	yes	—
31	C₂²×C₈₃⋊C₈	—	2	2	no	—
32	C₂²×C₈₃⋊C₈	—	2	1	no	—
33	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
34	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
35	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
36	#14581703	—	2	2	no	—
37	#14581703	—	2	1	no	—
38	C₄×C₈₃⋊C₈	—	2	2	no	—
39	C₄×C₈₃⋊C₈	—	2	1	no	—
40	C₄×C₈₃⋊C₈	—	2	1	no	—
41	C₄×C₈₃⋊C₈	—	2	1	no	—
42	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
44	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
45	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
46	C₂×#41×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	—	2	1	no	—
48	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
50	C₄×D₄×C₈₃	—	2	1	no	—
51	C₄×D₄×C₈₃	—	2	1	no	—
52	C₄×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₄×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
54	#117×C₈₃	—	2	1	no	—
55	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₃	—	2	1	no	—
57	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₈₃	—	2	1	no	—
60	C₄×C₈×C₈₃	—	2	1	no	—
61	C₄×C₈×C₈₃	—	2	1	no	—
62	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
63	#13050574	—	2	2	no	—
64	#13050574	—	2	8	no	—
65	#13050574	C₂×C₈	1	4	yes	—
66	C₂×C₈₃⋊C₈	—	2	2	no	—
67	C₂×C₈₃⋊C₈	—	2	8	no	—
68	C₂×C₈₃⋊C₈	—	2	2	no	—
69	C₂×C₈₃⋊C₈	—	2	2	no	—
70	C₂×C₈₃⋊C₈	—	4	2	no	—
71	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
74	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
75	#41×C₈₃	—	2	2	no	—
76	#41×C₈₃	C₂×C₈	1	4	yes	—
77	C₂×Q₈×C₈₃	C₄²	1	1	yes	characteristic
78	C₂×D₄×C₈₃	—	2	1	no	—
79	C₂×D₄×C₈₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₈₃	—	2	1	no	—
81	C₂²×C₄×C₈₃	—	2	1	no	—
82	C₂²×C₄×C₈₃	—	2	1	no	—
83	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₈₃	—	2	2	no	—
87	C₂×C₈×C₈₃	—	2	2	no	—
88	C₂×C₈×C₈₃	—	4	2	no	—
89	C₄⋊C₄×C₈₃	—	2	1	no	—
90	C₄⋊C₄×C₈₃	—	2	1	no	—
91	C₄⋊C₄×C₈₃	C₄²	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₈₃	—	2	1	no	—
94	C₂²⋊C₄×C₈₃	—	2	1	no	—
95	C₂²⋊C₄×C₈₃	—	2	1	no	—
96	C₄²×C₈₃	—	2	1	no	—
97	C₄²×C₈₃	—	2	1	no	—
98	C₄²×C₈₃	—	2	1	no	—
99	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₈₃	—	2	1	no	—
104	C₂³×C₈₃	#117	1	1	yes	characteristic
105	Q₈×C₈₃	—	2	1	no	—
106	Q₈×C₈₃	C₄×C₈	1	2	yes	—
107	D₄×C₈₃	—	2	4	no	—
108	D₄×C₈₃	—	2	1	no	—
109	D₄×C₈₃	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	4	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	2	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	4	2	no	—
117	C₂×C₄×C₈₃	—	4	2	no	—
118	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
126	C₈×C₈₃	—	4	2	no	—
127	C₈×C₈₃	—	4	2	no	—
128	C₈₃⋊C₈	—	2	4	no	—
129	C₈₃⋊C₈	—	4	2	no	—
130	C₈₃⋊C₈	—	4	2	no	—
131	C₈₃⋊C₈	C₂²⋊C₈	1	4	yes	—
132	C₈₃⋊C₈	C₂²⋊C₈	1	4	yes	—
133	C₂²×C₈₃	—	2	1	no	—
134	C₂²×C₈₃	—	2	1	no	—
135	C₂²×C₈₃	—	2	2	no	—
136	C₂²×C₈₃	—	4	1	no	—
137	C₂²×C₈₃	—	4	1	no	—
138	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₈₃	#431	1	2	yes	—
140	C₄×C₈₃	—	2	2	no	—
141	C₄×C₈₃	—	2	1	no	—
142	C₄×C₈₃	—	2	1	no	—
143	C₄×C₈₃	—	4	1	no	—
144	C₄×C₈₃	—	4	1	no	—
145	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₈₃	C₂×C₂²⋊C₈	1	2	yes	—
148	C₄×C₈₃	#444	1	2	yes	—
149	C₄×C₈₃	#444	1	2	yes	—
150	#8185	—	83	1	no	maximal, Sylow
151	C₂×C₈₃	—	2	2	no	—
152	C₂×C₈₃	—	4	2	no	—
153	C₂×C₈₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₈₃	#1765	1	1	yes	characteristic
155	C₂×C₈₃	#1765	1	1	yes	characteristic
156	C₄×#131	—	83	1	no	—
157	#1476	—	83	1	no	—
158	#1476	D₈₃	1	1	yes	characteristic
159	#1450	—	83	1	no	—
160	C₂×#322	—	83	1	no	—
161	#1278	—	83	1	no	—
162	#1278	—	83	1	no	—
163	C₈₃	#8185	1	1	yes	characteristic, complemented, Sylow
164	C₂×#131	—	83	2	no	—
165	C₄×#41	C₂×D₈₃	1	1	yes	characteristic
166	#431	—	83	1	no	—
167	#431	C₂×D₈₃	1	1	yes	characteristic
168	#405	—	83	2	no	—
169	#405	—	166	2	no	—
170	C₄×M₁₆	—	83	1	no	—
171	C₄×M₁₆	—	166	1	no	—
172	C₂×C₄×C₈	—	83	1	no	—
173	C₂×C₄×C₈	—	83	1	no	—
174	C₂×C₄×C₈	—	166	1	no	—
175	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
176	Q₈⋊C₈	—	83	2	no	—
177	Q₈⋊C₈	Dic₈₃	1	2	yes	—
178	D₄⋊C₈	—	83	2	no	—
179	D₄⋊C₈	Dic₈₃	1	2	yes	—
180	#322	—	166	2	no	—
181	#322	—	166	1	no	—
182	C₈²	—	166	1	no	—
183	C₂×#41	C₄×D₈₃	1	1	yes	characteristic
184	#131	—	83	8	no	—
185	#131	—	166	4	no	—
186	C₂×M₁₆	—	83	2	no	—
187	C₂×M₁₆	—	166	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	83	2	no	—
190	C₂²×C₈	—	166	1	no	—
191	C₂²×C₈	—	166	2	no	—
192	C₄×Q₈	C₄×D₈₃	1	1	yes	characteristic
193	C₄×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	C₄×D₈₃	1	1	yes	characteristic
197	C₄×D₄	C₂×Dic₈₃	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#117	C₄×D₈₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
202	C₄⋊C₈	—	83	2	no	—
203	C₄⋊C₈	C₂×Dic₈₃	1	2	yes	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	166	1	no	—
206	#97	—	166	2	no	—
207	#97	—	166	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	83	2	no	—
211	C₄×C₈	—	83	2	no	—
212	C₄×C₈	—	83	1	no	—
213	C₄×C₈	—	83	1	no	—
214	C₄×C₈	—	166	2	no	—
215	C₄×C₈	—	166	1	no	—
216	C₄×C₈	—	166	1	no	—
217	C₄×C₈	—	166	1	no	—
218	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
219	#41	—	2	2	no	—
220	#41	C₈×D₈₃	1	4	yes	—
221	C₂×Q₈	C₄×Dic₈₃	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄×Dic₈₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
228	C₂²×C₄	#13050589	1	1	yes	characteristic
229	C₂²×C₄	#13050589	1	1	yes	characteristic
230	M₁₆	—	83	4	no	—
231	M₁₆	—	166	8	no	—
232	M₁₆	—	166	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	332	2	no	—
237	C₂×C₈	—	83	4	no	—
238	C₂×C₈	—	83	4	no	—
239	C₂×C₈	—	83	2	no	—
240	C₂×C₈	—	83	2	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	—	166	2	no	—
243	C₂×C₈	—	166	8	no	—
244	C₂×C₈	—	166	2	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	C₄×Dic₈₃	1	1	yes	characteristic
248	C₄⋊C₄	C₄×Dic₈₃	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
256	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
257	C₄²	D₄×D₈₃	1	1	yes	characteristic
258	C₄²	D₄×D₈₃	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	C₂³	#14581748	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	C₈×Dic₈₃	1	2	yes	—
263	D₄	—	2	4	no	—
264	D₄	—	2	1	no	—
265	D₄	C₈×Dic₈₃	1	2	yes	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
275	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
276	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
279	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
281	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	332	2	no	—
285	C₈	—	332	2	no	—
286	C₈	—	83	4	no	—
287	C₈	—	83	4	no	—
288	C₈	—	166	4	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₂²⋊C₄×Dic₈₃	1	1	yes	characteristic
295	C₂²	#19127751	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×C₈×Dic₈₃	1	1	yes	characteristic
302	C₄	M₁₆×Dic₈₃	1	1	yes	characteristic
303	C₄	C₂²⋊C₈×D₈₃	1	2	yes	—
304	C₄	#19127820	1	2	yes	—
305	C₄	#19127823	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	C₂²⋊C₈×Dic₈₃	1	1	yes	characteristic
309	C₂	#24907308	1	1	yes	characteristic
310	C₂	#24907310	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1476	C₂
socle	C₂²×C₈₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₈₃	C₂×C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1476	D₈₃
hypocenter	C₈₃	#8185
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1476

Complements

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

N	H ≅ G/N	type
3. #24905747	C₂	semidirect
4. C₂×#19127423	C₂	semidirect
5. #24905300	C₂	semidirect
6. #24905299	C₂	semidirect
28. #14581799	C₈	semidirect
163. C₈₃	#8185	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581799	2
#24905845	2
#24905747	2
C₂×#19127423	2
#24905300	2
#24905299	2
C₈₃×#1476	2
#8185	83