Subgroups — #35374964

Order 21248.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918049	C₂	1	1	yes	maximal, characteristic
2	#24916397	C₂	1	2	yes	maximal
3	#24906373	C₂	1	4	yes	maximal, complemented
4	#24906283	C₂	1	4	yes	maximal, complemented
5	#24906219	C₂	1	1	yes	maximal, characteristic
6	#24905919	C₂	1	2	yes	maximal
7	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic
8	#19128258	C₂²	1	2	yes	—
9	#19128106	C₄	1	1	yes	characteristic
10	#19128081	C₄	1	1	yes	characteristic
11	#19128052	C₄	1	2	yes	—
12	#19127942	C₄	1	4	yes	—
13	#19127831	—	2	2	no	—
14	#19127831	C₂²	1	4	yes	—
15	C₈×Dic₁₆₆	—	2	2	no	—
16	C₈×Dic₁₆₆	C₂²	1	4	yes	—
17	C₄×Dic₃₃₂	C₂²	1	4	yes	—
18	#19127727	—	2	2	no	—
19	#19127725	C₂²	1	2	yes	—
20	#19127656	—	2	2	no	—
21	#19127452	C₂²	1	4	yes	—
22	#19127433	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
26	#322×C₈₃	C₂²	1	2	yes	—
27	#322×C₈₃	C₂²	1	2	yes	—
28	C₂×Dic₃₃₂	C₂×C₄	1	4	yes	—
29	C₂×#13050574	—	2	2	no	—
30	#14581792	C₂×C₄	1	2	yes	—
31	#14581792	C₂×C₄	1	2	yes	—
32	#14581784	—	2	2	no	—
33	#14581771	—	2	2	no	—
34	#14581771	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₆₆	—	2	2	no	—
36	C₄×Dic₁₆₆	C₂³	1	4	yes	—
37	#14581751	—	2	2	no	—
38	#14581726	C₂×C₄	1	4	yes	—
39	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
41	Dic₈₃⋊C₈	—	2	4	no	—
42	Dic₈₃⋊C₈	—	2	4	no	—
43	#14581717	C₂×C₄	1	8	yes	—
44	#14581702	—	2	4	no	—
45	C₈×Dic₈₃	—	2	4	no	—
46	C₂²×C₈×C₈₃	—	2	1	no	—
47	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₈₃	D₄	1	2	yes	—
49	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₈₃	—	2	1	no	—
51	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₈₃	D₄	1	2	yes	—
54	C₄×C₈×C₈₃	C₂³	1	2	yes	—
55	C₄×C₈×C₈₃	C₂³	1	2	yes	—
56	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₆₆	—	2	2	no	—
59	C₂×Dic₁₆₆	C₂²×C₄	1	4	yes	—
60	Dic₃₃₂	—	2	8	no	—
61	#13050574	—	4	4	no	—
62	C₂×C₈₃⋊C₈	—	2	4	no	—
63	C₄⋊Dic₈₃	—	2	2	no	—
64	C₄⋊Dic₈₃	—	2	2	no	—
65	C₄⋊Dic₈₃	—	2	2	no	—
66	C₄⋊Dic₈₃	C₂²×C₄	1	4	yes	—
67	Dic₈₃⋊C₄	—	2	4	no	—
68	Dic₈₃⋊C₄	—	2	4	no	—
69	C₂²⋊Dic₈₃	—	2	4	no	—
70	C₄×Dic₈₃	—	2	4	no	—
71	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₈₃	—	2	4	no	—
75	C₂×C₈×C₈₃	—	2	4	no	—
76	C₂×C₈×C₈₃	—	2	2	no	—
77	C₂×C₈×C₈₃	—	4	2	no	—
78	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
82	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₃	#41	1	1	yes	characteristic
85	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
86	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₃	—	2	2	no	—
88	C₂×C₄×C₈₃	—	2	2	no	—
89	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₃	#137	1	4	yes	—
93	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₈₃	—	2	4	no	—
97	C₂×Dic₈₃	—	2	4	no	—
98	Dic₁₆₆	—	2	8	no	—
99	Dic₁₆₆	—	4	4	no	—
100	C₈×C₈₃	—	2	2	no	—
101	C₈×C₈₃	—	2	2	no	—
102	C₈×C₈₃	—	2	2	no	—
103	C₈×C₈₃	—	4	2	no	—
104	C₈×C₈₃	#131	1	4	yes	—
105	C₈₃⋊C₈	—	4	4	no	—
106	C₂²×C₈₃	—	2	1	no	—
107	C₂²×C₈₃	—	2	1	no	—
108	C₂²×C₈₃	#441	1	2	yes	—
109	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₈₃	—	2	1	no	—
111	C₄×C₈₃	—	2	1	no	—
112	C₄×C₈₃	#441	1	2	yes	—
113	C₄×C₈₃	#444	1	2	yes	—
114	C₄×C₈₃	C₂×#131	1	2	yes	—
115	C₄×C₈₃	C₂×#137	1	2	yes	—
116	Dic₈₃	—	4	4	no	—
117	Dic₈₃	—	4	4	no	—
118	#17961	—	83	1	no	maximal, Sylow
119	C₂×C₈₃	—	2	2	no	—
120	C₂×C₈₃	#2931	1	1	yes	characteristic
121	C₂×C₈₃	C₂×#441	1	1	yes	characteristic
122	C₂×C₈₃	C₂×#444	1	1	yes	characteristic
123	#3170	—	83	1	no	—
124	#2960	—	83	2	no	—
125	#1593	—	83	4	no	—
126	#1579	—	83	4	no	—
127	#1566	—	83	1	no	—
128	#1493	—	83	2	no	—
129	C₂×#322	D₈₃	1	1	yes	characteristic
130	C₈₃	#17961	1	1	yes	characteristic, complemented, Sylow
131	#533	—	83	2	no	—
132	#500	—	83	1	no	—
133	#494	—	83	1	no	—
134	#487	—	83	2	no	—
135	#467	—	83	4	no	—
136	#446	—	83	4	no	—
137	#446	—	166	2	no	—
138	C₈×Q₈	—	83	4	no	—
139	C₈×Q₈	—	166	2	no	—
140	C₄×Q₁₆	—	83	4	no	—
141	#424	—	166	2	no	—
142	#423	—	83	2	no	—
143	#405	—	166	2	no	—
144	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
147	#334	—	83	4	no	—
148	Q₈⋊C₈	—	83	8	no	—
149	#322	C₂×D₈₃	1	2	yes	—
150	#322	C₂×D₈₃	1	2	yes	—
151	C₂×Q₁₆	—	83	4	no	—
152	C₂×M₁₆	—	166	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₄×D₈₃	1	2	yes	—
155	C₂²×C₈	D₃₃₂	1	2	yes	—
156	C₄⋊Q₈	—	83	2	no	—
157	C₄⋊Q₈	—	83	2	no	—
158	#125	—	166	2	no	—
159	C₂²⋊Q₈	—	83	4	no	—
160	C₂²⋊Q₈	—	166	2	no	—
161	C₄×Q₈	—	83	4	no	—
162	C₄×Q₈	—	166	2	no	—
163	#117	—	166	2	no	—
164	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
165	#107	—	83	4	no	—
166	C₄⋊C₈	—	83	4	no	—
167	C₄⋊C₈	—	166	4	no	—
168	C₄⋊C₈	—	166	4	no	—
169	C₄⋊C₈	—	166	2	no	—
170	#103	—	83	8	no	—
171	#97	—	166	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	166	4	no	—
174	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
175	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
176	C₄×C₈	D₃₃₂	1	2	yes	—
177	C₄×C₈	C₂²×D₈₃	1	2	yes	—
178	C₄×C₈	C₂²×D₈₃	1	2	yes	—
179	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
181	C₂×Q₈	—	83	4	no	—
182	C₂×Q₈	—	166	2	no	—
183	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
185	C₂²×C₄	#13050590	1	1	yes	characteristic
186	Q₁₆	—	166	8	no	—
187	M₁₆	—	332	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	166	4	no	—
193	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
195	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
196	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
197	C₄⋊C₄	—	83	4	no	—
198	C₄⋊C₄	—	166	4	no	—
199	C₄⋊C₄	—	166	2	no	—
200	C₄⋊C₄	—	166	2	no	—
201	C₄⋊C₄	—	166	4	no	—
202	C₄⋊C₄	—	166	2	no	—
203	C₂²⋊C₄	—	166	4	no	—
204	C₄²	—	166	4	no	—
205	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
206	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
207	C₄²	#13050590	1	1	yes	characteristic
208	C₄²	C₂³×D₈₃	1	1	yes	characteristic
209	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
210	Q₈	—	332	4	no	—
211	Q₈	—	166	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	166	4	no	—
215	C₂×C₄	—	166	4	no	—
216	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
217	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
218	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
219	C₂×C₄	#14581825	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
222	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	332	4	no	—
228	C₈	#14581798	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#19127809	1	2	yes	—
232	C₂²	C₂×C₄×D₃₃₂	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	332	4	no	—
236	C₄	—	332	4	no	—
237	C₄	#19127809	1	2	yes	—
238	C₄	#19127824	1	2	yes	—
239	C₄	C₂×#14581798	1	2	yes	—
240	C₄	C₂×#14581825	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#24916176	1	1	yes	characteristic
243	C₂	C₂×#19127809	1	1	yes	characteristic
244	C₂	C₂×#19127824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#322×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₂×#322	D₈₃
hypocenter	C₈₃	#17961
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #24906373	C₂	semidirect
4. #24906283	C₂	semidirect
130. C₈₃	#17961	semidirect

Maximal subgroups

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

subgroup	index
#24918049	2
#24916397	2
#24906373	2
#24906283	2
#24906219	2
#24905919	2
C₂×#322×C₈₃	2
#17961	83