Subgroups — #35298146

Order 21248.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907579	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127453	C₂	1	2	yes	maximal
4	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
5	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
12	C₄²×Dic₈₃	C₂²	1	1	yes	characteristic
13	#19127455	—	2	4	no	—
14	#19127453	—	2	4	no	—
15	#19127452	—	2	2	no	—
16	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	#334×C₈₃	—	2	2	no	—
19	C₂²×C₈₃⋊C₈	—	2	2	no	—
20	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₈₃	D₄	1	2	yes	—
25	C₂×C₄×Dic₈₃	Q₈	1	2	yes	—
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
28	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
29	C₄×C₈₃⋊C₈	Q₈	1	2	yes	—
30	C₈×Dic₈₃	—	2	2	no	—
31	C₈×Dic₈₃	—	2	2	no	—
32	C₈×Dic₈₃	—	2	2	no	—
33	C₈×Dic₈₃	C₈	1	4	yes	complemented
34	C₂²×C₈×C₈₃	—	2	2	no	—
35	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₈₃	—	2	2	no	—
39	C₄⋊C₈×C₈₃	—	2	2	no	—
40	C₄×C₈×C₈₃	D₄	1	2	yes	—
41	C₄×C₈×C₈₃	Q₈	1	2	yes	—
42	C₂²×Dic₈₃	C₄⋊C₄	1	2	yes	—
43	C₂²×Dic₈₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈₃⋊C₈	—	2	1	no	—
45	C₂×C₈₃⋊C₈	—	2	1	no	—
46	C₂×C₈₃⋊C₈	—	4	2	no	—
47	C₂×C₈₃⋊C₈	—	4	4	no	—
48	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
51	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
52	C₄×Dic₈₃	—	2	1	no	—
53	C₄×Dic₈₃	—	2	1	no	—
54	C₄×Dic₈₃	—	2	4	no	—
55	C₄×Dic₈₃	—	2	4	no	—
56	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
57	C₄×Dic₈₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₃	—	2	1	no	—
62	C₂×C₈×C₈₃	—	2	1	no	—
63	C₂×C₈×C₈₃	—	4	2	no	—
64	C₂×C₈×C₈₃	—	4	4	no	—
65	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
69	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₃	—	2	2	no	—
75	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₈₃	#135	1	4	yes	—
86	C₂×Dic₈₃	—	2	2	no	—
87	C₂×Dic₈₃	—	2	2	no	—
88	C₂×Dic₈₃	—	2	2	no	—
89	C₂×Dic₈₃	—	2	2	no	—
90	C₂×Dic₈₃	C₄⋊C₈	1	4	yes	—
91	C₂×Dic₈₃	#108	1	4	yes	—
92	C₈×C₈₃	—	2	2	no	—
93	C₈×C₈₃	—	8	2	no	—
94	C₈×C₈₃	C₄⋊C₈	1	4	yes	—
95	C₈₃⋊C₈	—	2	2	no	—
96	C₈₃⋊C₈	—	8	2	no	—
97	C₈₃⋊C₈	C₄⋊C₈	1	4	yes	—
98	C₂²×C₈₃	#384	1	1	yes	characteristic
99	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₈₃	#423	1	1	yes	characteristic
101	C₂²×C₈₃	#427	1	2	yes	—
102	C₂²×C₈₃	C₂×#108	1	2	yes	—
103	C₄×C₈₃	—	2	1	no	—
104	C₄×C₈₃	—	2	1	no	—
105	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₈₃	#423	1	1	yes	characteristic
108	C₄×C₈₃	#423	1	1	yes	characteristic
109	C₄×C₈₃	#495	1	2	yes	—
110	C₄×C₈₃	#499	1	2	yes	—
111	Dic₈₃	—	2	4	no	—
112	Dic₈₃	—	2	4	no	—
113	#8807	—	83	1	no	maximal, Sylow
114	C₂×C₈₃	#1564	1	2	yes	—
115	C₂×C₈₃	#1568	1	2	yes	—
116	C₂×C₈₃	#1842	1	1	yes	characteristic
117	C₂×C₈₃	#1846	1	1	yes	characteristic
118	C₂×C₈₃	#1848	1	1	yes	characteristic
119	#1831	—	83	2	no	—
120	C₄²×C₈	—	83	1	no	—
121	C₂×#335	—	83	2	no	—
122	C₂×#334	—	83	1	no	—
123	C₂×#334	D₈₃	1	1	yes	characteristic
124	C₈₃	#8807	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊C₈	—	83	2	no	—
126	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
127	C₂×C₄×C₈	—	83	1	no	—
128	C₂×C₄×C₈	—	83	1	no	—
129	C₂×C₄×C₈	—	83	1	no	—
130	C₂×C₄×C₈	—	83	1	no	—
131	C₂×C₄×C₈	—	83	1	no	—
132	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
133	C₄³	—	83	1	no	—
134	#336	—	166	4	no	—
135	#335	—	166	4	no	—
136	#334	—	2	2	no	—
137	#334	—	166	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	83	1	no	—
140	C₂²×C₈	—	83	1	no	—
141	C₂²×C₈	—	166	2	no	—
142	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
143	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
144	C₂×C₄²	—	83	1	no	—
145	C₂×C₄²	—	83	1	no	—
146	C₂×C₄²	—	83	2	no	—
147	C₂×C₄²	—	83	2	no	—
148	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	166	2	no	—
152	C₄⋊C₈	—	166	2	no	—
153	C₄×C₈	—	83	4	no	—
154	C₄×C₈	—	83	2	no	—
155	C₄×C₈	—	83	2	no	—
156	C₄×C₈	—	166	2	no	—
157	C₄×C₈	—	166	2	no	—
158	C₄×C₈	—	166	2	no	—
159	C₄×C₈	Dic₁₆₆	1	2	yes	—
160	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
161	C₂²×C₄	—	83	2	no	—
162	C₂²×C₄	—	83	2	no	—
163	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
164	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
165	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	—	332	2	no	—
171	C₂×C₈	—	332	4	no	—
172	C₂×C₈	—	83	2	no	—
173	C₂×C₈	—	83	2	no	—
174	C₂×C₈	—	83	2	no	—
175	C₂×C₈	—	83	2	no	—
176	C₂×C₈	—	166	1	no	—
177	C₂×C₈	—	166	1	no	—
178	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
179	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
180	C₂×C₈	C₈×D₈₃	1	2	yes	—
181	C₂×C₈	#13050573	1	2	yes	—
182	C₄²	—	83	2	no	—
183	C₄²	—	83	2	no	—
184	C₄²	—	166	4	no	—
185	C₄²	—	166	1	no	—
186	C₄²	—	166	1	no	—
187	C₄²	—	166	4	no	—
188	C₄²	D₄×D₈₃	1	1	yes	characteristic
189	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
190	C₄²	Q₈×D₈₃	1	1	yes	characteristic
191	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
192	C₂³	C₄⋊C₄×D₈₃	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	83	4	no	—
195	C₂×C₄	—	83	4	no	—
196	C₂×C₄	—	166	2	no	—
197	C₂×C₄	—	166	2	no	—
198	C₂×C₄	—	166	2	no	—
199	C₂×C₄	—	166	2	no	—
200	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
201	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
202	C₂×C₄	#14581788	1	1	yes	characteristic
203	C₂×C₄	#14581790	1	1	yes	characteristic
204	C₂×C₄	#14581791	1	1	yes	characteristic
205	C₂×C₄	#14581791	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
209	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
210	C₂×C₄	#14581813	1	4	yes	—
211	C₈	—	2	2	no	—
212	C₈	—	8	2	no	—
213	C₈	—	83	4	no	—
214	C₈	—	664	2	no	—
215	C₈	—	166	2	no	—
216	C₈	Dic₈₃⋊C₈	1	4	yes	—
217	C₂²	#19127570	1	1	yes	characteristic
218	C₂²	C₄⋊C₈×D₈₃	1	1	yes	characteristic
219	C₂²	#19127723	1	1	yes	characteristic
220	C₂²	#19127736	1	2	yes	—
221	C₂²	#108×D₈₃	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	166	4	no	—
225	C₄	—	166	4	no	—
226	C₄	C₄⋊C₈×D₈₃	1	1	yes	characteristic
227	C₄	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
228	C₄	#19127723	1	1	yes	characteristic
229	C₄	#19127724	1	1	yes	characteristic
230	C₄	#19128082	1	2	yes	—
231	C₄	#19128101	1	2	yes	—
232	C₂	#24906208	1	2	yes	—
233	C₂	#24906223	1	2	yes	—
234	C₂	#24907614	1	1	yes	characteristic
235	C₂	#24907624	1	1	yes	characteristic
236	C₂	#24907633	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	#8807
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
33. C₈×Dic₈₃	C₈	semidirect
124. C₈₃	#8807	semidirect

Maximal subgroups

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

subgroup	index
#24907579	2
C₄×C₈×Dic₈₃	2
C₂×#19127453	2
C₂×#19127452	2
C₂×#334×C₈₃	2
#8807	83