Subgroups — #35504368

Order 21248.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127824	C₂	1	1	yes	maximal, characteristic
2	#24908632	C₂	1	4	yes	maximal, complemented
3	#24908359	C₂	1	2	yes	maximal
4	#24907586	C₂	1	2	yes	maximal, complemented
5	#24907514	C₂	1	2	yes	maximal, complemented
6	#24907506	C₂	1	2	yes	maximal, complemented
7	#24907347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581814	C₂²	1	1	yes	characteristic
10	C₂×#14581798	C₂²	1	2	yes	—
11	#19127824	—	2	4	no	—
12	C₂×#14581728	C₂²	1	1	yes	characteristic
13	C₂×#14581728	C₂²	1	1	yes	characteristic
14	#19127739	C₂²	1	1	yes	characteristic
15	#19127719	C₂²	1	4	yes	—
16	C₂×#14581720	C₂²	1	2	yes	—
17	#19127674	C₂²	1	4	yes	—
18	#19127656	C₂²	1	2	yes	—
19	#19127471	C₂²	1	4	yes	—
20	#19127461	C₂²	1	2	yes	—
21	#19127457	C₂²	1	4	yes	—
22	#533×C₈₃	C₂²	1	2	yes	—
23	#501×C₈₃	—	2	4	no	—
24	#427×C₈₃	C₂²	1	2	yes	—
25	#427×C₈₃	C₂²	1	2	yes	—
26	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#13050590	C₂³	1	2	yes	—
28	#14581814	—	2	2	no	—
29	#14581814	—	2	2	no	—
30	C₂×Dic₃₃₂	Q₈	1	1	yes	characteristic, complemented
31	C₂×#13050579	Q₈	1	2	yes	complemented
32	C₂×D₆₆₄	Q₈	1	1	yes	characteristic, complemented
33	#14581798	—	2	2	no	—
34	#14581798	—	2	2	no	—
35	#14581798	—	2	2	no	—
36	#14581798	—	2	2	no	—
37	C₂×#13050574	C₂³	1	2	yes	—
38	C₂×#13050574	C₂³	1	2	yes	—
39	C₂×#13050573	—	2	2	no	—
40	C₂×C₈×D₈₃	—	2	2	no	—
41	#14581751	C₂³	1	2	yes	—
42	#14581728	—	2	2	no	—
43	#14581728	—	2	2	no	—
44	#14581726	—	2	1	no	—
45	#14581724	—	2	1	no	—
46	#14581720	—	2	4	no	—
47	#14581720	—	4	4	no	—
48	#14581706	—	4	4	no	—
49	#14581702	—	2	2	no	—
50	C₈×Dic₈₃	—	2	2	no	—
51	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₃	—	2	2	no	—
54	#125×C₈₃	—	2	2	no	—
55	C₂²⋊Q₈×C₈₃	—	4	2	no	—
56	C₄×Q₈×C₈₃	—	4	2	no	—
57	#117×C₈₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
59	#107×C₈₃	—	2	2	no	—
60	#107×C₈₃	—	2	2	no	—
61	#106×C₈₃	—	2	2	no	—
62	#106×C₈₃	—	2	2	no	—
63	#97×C₈₃	D₄	1	4	yes	—
64	#13050590	—	2	2	no	—
65	#13050590	—	2	2	no	—
66	#13050590	—	2	2	no	—
67	#13050590	—	2	2	no	—
68	C₂×Dic₁₆₆	C₂×Q₈	1	2	yes	—
69	C₂×D₃₃₂	C₂×Q₈	1	2	yes	—
70	C₂×C₂²⋊D₈₃	—	2	2	no	—
71	C₂×C₄×D₈₃	—	2	2	no	—
72	Dic₃₃₂	—	2	2	no	—
73	#13050579	—	2	4	no	—
74	D₆₆₄	—	2	2	no	—
75	#13050574	—	2	2	no	—
76	#13050574	—	2	2	no	—
77	#13050574	—	2	2	no	—
78	#13050574	—	2	2	no	—
79	#13050573	—	4	2	no	—
80	#13050573	—	4	2	no	—
81	C₂×C₈₃⋊C₈	—	2	2	no	—
82	C₂×C₈₃⋊C₈	—	2	2	no	—
83	C₈×D₈₃	—	4	2	no	—
84	C₈×D₈₃	—	4	2	no	—
85	C₄⋊Dic₈₃	—	2	2	no	—
86	C₂²⋊Dic₈₃	—	2	2	no	—
87	C₄×Dic₈₃	—	2	2	no	—
88	C₂×Q₈×C₈₃	—	4	2	no	—
89	C₂²×C₄×C₈₃	—	2	1	no	—
90	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₈₃	#41	1	2	yes	—
96	C₂×C₈×C₈₃	#41	1	2	yes	—
97	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	—	2	4	no	—
100	C₄⋊C₄×C₈₃	—	4	4	no	—
101	C₄⋊C₄×C₈₃	—	4	2	no	—
102	C₂²⋊C₄×C₈₃	—	2	4	no	—
103	C₄²×C₈₃	—	2	1	no	—
104	C₄²×C₈₃	—	2	4	no	—
105	C₄²×C₈₃	C₂×D₄	1	2	yes	—
106	C₂³×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×D₈₃	—	2	2	no	—
108	Q₈×C₈₃	—	8	4	no	—
109	C₂×C₄×C₈₃	—	2	2	no	—
110	C₂×C₄×C₈₃	—	4	4	no	—
111	C₂×C₄×C₈₃	—	8	1	no	—
112	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²⋊D₈₃	—	4	2	no	—
119	C₂²⋊D₈₃	—	4	2	no	—
120	C₂×Dic₈₃	—	2	2	no	—
121	C₂×Dic₈₃	—	4	2	no	—
122	D₃₃₂	—	2	2	no	—
123	D₃₃₂	—	2	2	no	—
124	C₄×D₈₃	—	4	2	no	—
125	C₄×D₈₃	—	4	2	no	—
126	Dic₁₆₆	—	2	2	no	—
127	Dic₁₆₆	—	2	2	no	—
128	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
129	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₈₃	#124	1	2	yes	—
131	C₈×C₈₃	#124	1	2	yes	—
132	C₈₃⋊C₈	—	4	2	no	—
133	C₈₃⋊C₈	—	4	2	no	—
134	C₂²×C₈₃	—	4	1	no	—
135	C₂²×C₈₃	#535	1	1	yes	characteristic
136	C₂²×C₈₃	#554	1	1	yes	characteristic
137	C₂²×C₈₃	#555	1	1	yes	characteristic
138	C₂×D₈₃	—	4	2	no	—
139	C₂×D₈₃	—	8	2	no	—
140	C₄×C₈₃	—	4	2	no	—
141	C₄×C₈₃	—	8	4	no	—
142	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
144	C₄×C₈₃	#554	1	1	yes	characteristic
145	C₄×C₈₃	#555	1	1	yes	characteristic
146	Dic₈₃	—	4	2	no	—
147	Dic₈₃	—	8	2	no	—
148	#29155	—	83	1	no	maximal, Sylow
149	C₂×C₈₃	—	2	1	no	—
150	C₂×C₈₃	—	2	1	no	—
151	C₂×C₈₃	—	2	1	no	—
152	C₂×C₈₃	#2563	1	1	yes	characteristic
153	D₈₃	—	8	2	no	—
154	#3162	D₈₃	1	1	yes	characteristic, complemented
155	C₂×#444	—	83	1	no	—
156	#2031	—	83	4	no	—
157	#1976	—	83	2	no	—
158	#1833	—	83	2	no	—
159	#1814	—	83	2	no	—
160	#1812	—	83	2	no	—
161	#1773	—	83	1	no	—
162	C₈₃	#29155	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	—	83	1	no	—
164	C₂×#131	—	83	2	no	—
165	#533	C₂×D₈₃	1	2	yes	—
166	#501	—	2	4	no	—
167	#444	—	166	4	no	—
168	C₂×#108	—	83	1	no	—
169	C₂×#108	—	83	1	no	—
170	#427	—	83	1	no	—
171	#427	C₂×D₈₃	1	2	yes	—
172	#427	C₂×D₈₃	1	2	yes	—
173	#421	—	83	4	no	—
174	C₂×#104	—	83	2	no	—
175	#410	—	83	4	no	—
176	#405	—	83	2	no	—
177	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
178	#344	—	83	4	no	—
179	#339	—	83	2	no	—
180	H(3,ℤ/4)	—	83	4	no	—
181	C₂×#41	—	83	2	no	—
182	#135	—	166	2	no	—
183	#135	—	166	2	no	—
184	C₂×Q₁₆	—	83	1	no	—
185	C₂×SD₁₆	—	83	2	no	—
186	C₂×D₈	—	83	1	no	—
187	#131	—	166	2	no	—
188	#131	—	166	2	no	—
189	#131	—	166	2	no	—
190	#131	—	166	2	no	—
191	C₂×M₁₆	—	83	2	no	—
192	C₂×M₁₆	—	83	2	no	—
193	C₂×M₁₆	—	166	2	no	—
194	C₂²×C₈	—	166	2	no	—
195	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	#125	—	2	2	no	—
199	C₂²⋊Q₈	—	4	2	no	—
200	C₄×Q₈	—	4	2	no	—
201	#117	—	83	2	no	—
202	#117	C₂²×D₈₃	1	4	yes	—
203	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
204	#108	—	166	2	no	—
205	#108	—	166	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	2	no	—
208	#107	—	166	1	no	—
209	#106	—	2	2	no	—
210	#106	—	2	2	no	—
211	#106	—	166	1	no	—
212	#104	—	332	4	no	—
213	#104	—	166	4	no	—
214	#99	—	332	4	no	—
215	#97	—	166	2	no	—
216	#97	C₂²⋊D₈₃	1	4	yes	—
217	C₄×C₈	—	166	2	no	—
218	#41	—	166	2	no	—
219	#41	—	166	2	no	—
220	#41	—	166	2	no	—
221	#41	—	166	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	83	2	no	—
224	C₂×D₄	—	83	2	no	—
225	C₂×D₄	—	166	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	166	2	no	—
228	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
229	Q₁₆	—	166	2	no	—
230	SD₁₆	—	166	4	no	—
231	D₈	—	166	2	no	—
232	M₁₆	—	332	2	no	—
233	M₁₆	—	332	2	no	—
234	M₁₆	—	166	2	no	—
235	M₁₆	—	166	2	no	—
236	M₁₆	—	166	2	no	—
237	M₁₆	—	166	2	no	—
238	C₂×C₈	—	332	2	no	—
239	C₂×C₈	—	332	2	no	—
240	C₂×C₈	—	166	2	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
244	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
245	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
246	C₂×C₈	#13050591	1	2	yes	—
247	C₂×C₈	#13050591	1	1	yes	characteristic
248	C₂×C₈	#13050591	1	1	yes	characteristic
249	C₂×C₈	#13050591	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	166	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	166	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	166	2	no	—
259	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
260	C₂³	—	166	2	no	—
261	C₂³	#14581837	1	1	yes	characteristic
262	Q₈	—	8	4	no	—
263	Q₈	—	166	2	no	—
264	Q₈	—	166	2	no	—
265	D₄	—	332	2	no	—
266	D₄	—	332	2	no	—
267	D₄	—	166	2	no	—
268	D₄	—	166	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	332	2	no	—
273	C₂×C₄	—	332	2	no	—
274	C₂×C₄	—	332	2	no	—
275	C₂×C₄	—	166	2	no	—
276	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
277	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
279	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
280	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
281	C₂×C₄	#14581837	1	1	yes	characteristic
282	C₈	—	332	2	no	—
283	C₈	—	332	2	no	—
284	C₈	#14581768	1	2	yes	—
285	C₈	#14581768	1	2	yes	—
286	C₈	#14581781	1	2	yes	—
287	C₈	#14581781	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	332	2	no	—
290	C₂²	—	664	2	no	—
291	C₂²	#19128268	1	1	yes	characteristic
292	C₂²	#19128412	1	1	yes	characteristic
293	C₂²	#19128417	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	8	4	no	—
296	C₄	—	332	2	no	—
297	C₄	—	664	2	no	—
298	C₄	C₂×#14581768	1	1	yes	characteristic
299	C₄	C₂×#14581781	1	1	yes	characteristic
300	C₄	#19128412	1	1	yes	characteristic
301	C₄	#19128417	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	664	2	no	—
306	C₂	#24912637	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19128417
commutator, cosocle	C₂²×C₄×C₈₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈₃×#3162	C₂
socle	C₂×C₈₃	#2563
prime radical	C₂×#14581814	C₂²
hypercenter	#3162	D₈₃
hypocenter	C₈₃	#29155
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3162

Complements

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

N	H ≅ G/N	type
2. #24908632	C₂	semidirect
4. #24907586	C₂	semidirect
5. #24907514	C₂	semidirect
6. #24907506	C₂	semidirect
7. #24907347	C₂	semidirect
8. C₈₃×#3162	C₂	semidirect
30. C₂×Dic₃₃₂	Q₈	semidirect
31. C₂×#13050579	Q₈	semidirect
32. C₂×D₆₆₄	Q₈	semidirect
154. #3162	D₈₃	semidirect
162. C₈₃	#29155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127824	2
#24908632	2
#24908359	2
#24907586	2
#24907514	2
#24907506	2
#24907347	2
C₈₃×#3162	2
#29155	83