Subgroups — C₂×#24907072 #35392076

Order 21248.

750 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128106	C₂	1	1	yes	maximal, characteristic
2	C₂×#19128053	C₂	1	2	yes	maximal
3	C₂×#19128051	C₂	1	2	yes	maximal
4	#24907072	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14581792	C₂²	1	2	yes	—
8	#19128106	—	2	2	no	—
9	#19128106	C₂²	1	4	yes	—
10	#19128053	—	2	4	no	—
11	#19128053	C₂²	1	8	yes	—
12	#19128051	—	2	4	no	—
13	#19128051	C₂²	1	8	yes	—
14	C₂×#14581726	—	2	2	no	—
15	C₂×#14581716	—	2	4	no	—
16	C₂×#14581715	—	2	4	no	—
17	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×C₈₃⋊C₈	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
21	#500×C₈₃	—	2	2	no	—
22	#500×C₈₃	C₂²	1	4	yes	—
23	C₂×#107×C₈₃	—	2	2	no	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₆₆	—	2	2	no	—
26	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
27	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
28	#14581792	—	2	2	no	—
29	#14581792	—	2	2	no	—
30	#14581792	—	2	2	no	—
31	#14581792	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₈₃	—	2	2	no	—
33	C₂×C₄⋊Dic₈₃	—	2	2	no	—
34	#14581726	—	2	8	no	—
35	#14581716	—	2	16	no	—
36	#14581715	—	2	16	no	—
37	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
38	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
39	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
40	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
41	C₂²×Q₈×C₈₃	—	2	2	no	—
42	C₂²×C₈×C₈₃	D₄	1	2	yes	—
43	C₄⋊Q₈×C₈₃	—	2	2	no	—
44	C₄⋊Q₈×C₈₃	—	2	2	no	—
45	C₄⋊Q₈×C₈₃	—	2	2	no	—
46	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
49	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
50	#107×C₈₃	—	2	8	no	—
51	C₄×C₈×C₈₃	D₄	1	2	yes	—
52	C₄×C₈×C₈₃	C₂³	1	2	yes	—
53	C₂×Dic₁₆₆	—	2	4	no	—
54	C₂×Dic₁₆₆	—	4	8	no	—
55	C₂×Dic₁₆₆	—	4	2	no	—
56	C₂²×Dic₈₃	—	4	2	no	—
57	C₂×C₈₃⋊C₈	—	2	4	no	—
58	C₂×C₈₃⋊C₈	—	2	4	no	—
59	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
60	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
61	C₄⋊Dic₈₃	—	2	4	no	—
62	C₄⋊Dic₈₃	—	2	4	no	—
63	C₄⋊Dic₈₃	—	4	2	no	—
64	C₄⋊Dic₈₃	—	4	2	no	—
65	C₂×Q₈×C₈₃	—	2	4	no	—
66	C₂×Q₈×C₈₃	—	4	2	no	—
67	C₂×Q₈×C₈₃	—	4	8	no	—
68	C₂²×C₄×C₈₃	—	4	2	no	—
69	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₃	Q₁₆	1	8	yes	—
73	C₂×C₈×C₈₃	C₂×D₄	1	4	yes	—
74	C₄⋊C₄×C₈₃	—	2	4	no	—
75	C₄⋊C₄×C₈₃	—	2	4	no	—
76	C₄⋊C₄×C₈₃	—	4	2	no	—
77	C₄⋊C₄×C₈₃	—	4	2	no	—
78	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₈₃	#127	1	1	yes	characteristic
83	Q₈×C₈₃	—	4	8	no	—
84	Q₈×C₈₃	—	8	4	no	—
85	C₂×C₄×C₈₃	—	4	4	no	—
86	C₂×C₄×C₈₃	—	8	4	no	—
87	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×Q₁₆	1	4	yes	—
91	C₂×C₄×C₈₃	#135	1	4	yes	—
92	C₂×C₄×C₈₃	#135	1	4	yes	—
93	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₈₃	—	4	4	no	—
97	C₂×Dic₈₃	—	8	4	no	—
98	Dic₁₆₆	—	4	8	no	—
99	Dic₁₆₆	—	8	4	no	—
100	C₈×C₈₃	C₂×Q₁₆	1	8	yes	—
101	C₈₃⋊C₈	—	2	4	no	—
102	C₈₃⋊C₈	—	2	4	no	—
103	C₂²×C₈₃	#494	1	2	yes	—
104	C₂²×C₈₃	#495	1	2	yes	—
105	C₂²×C₈₃	#495	1	2	yes	—
106	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
107	C₄×C₈₃	—	8	4	no	—
108	C₄×C₈₃	#494	1	2	yes	—
109	C₄×C₈₃	#495	1	2	yes	—
110	C₄×C₈₃	#495	1	2	yes	—
111	C₄×C₈₃	C₂²×Q₁₆	1	2	yes	—
112	C₄×C₈₃	C₂×#135	1	2	yes	—
113	C₄×C₈₃	C₂×#135	1	2	yes	—
114	Dic₈₃	—	8	4	no	—
115	C₂×#1710	—	83	1	no	maximal, Sylow
116	C₂×C₈₃	#1710	1	4	yes	complemented
117	C₂×C₈₃	C₂×#494	1	1	yes	characteristic
118	C₂×C₈₃	C₂×#495	1	1	yes	characteristic
119	C₂×C₈₃	C₂×#495	1	1	yes	characteristic
120	C₂×#500	—	83	1	no	—
121	C₂×#500	D₈₃	1	1	yes	characteristic
122	C₂×#487	—	83	2	no	—
123	C₂×#487	—	83	2	no	—
124	#1710	—	83	8	no	—
125	C₂×C₈²	—	83	1	no	—
126	C₈₃	C₂×#1710	1	1	yes	characteristic, complemented, Sylow
127	C₂×C₄⋊Q₈	—	83	2	no	—
128	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
129	#500	—	2	2	no	—
130	#500	—	83	4	no	—
131	#500	—	166	2	no	—
132	#500	C₂×D₈₃	1	4	yes	—
133	#487	—	83	8	no	—
134	#487	—	83	8	no	—
135	#487	—	166	4	no	—
136	#487	—	166	4	no	—
137	C₂×#107	—	2	2	no	—
138	C₂×#107	—	166	2	no	—
139	C₂×#103	—	166	4	no	—
140	C₂×#103	—	166	4	no	—
141	C₂×C₄×C₈	—	83	1	no	—
142	C₂×C₄×C₈	—	83	1	no	—
143	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
144	C₈²	—	83	4	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×Q₈	—	166	2	no	—
147	C₂²×C₈	—	83	2	no	—
148	C₂²×C₈	—	83	2	no	—
149	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	83	4	no	—
154	C₄⋊Q₈	—	166	2	no	—
155	C₄⋊Q₈	—	166	2	no	—
156	C₄⋊Q₈	—	166	2	no	—
157	C₄⋊Q₈	C₂²×D₈₃	1	4	yes	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	166	2	no	—
161	C₂×C₄⋊C₄	—	166	2	no	—
162	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
163	#107	—	2	8	no	—
164	#107	—	166	8	no	—
165	#103	—	166	16	no	—
166	#103	—	166	16	no	—
167	C₄×C₈	—	83	2	no	—
168	C₄×C₈	—	83	2	no	—
169	C₄×C₈	—	83	2	no	—
170	C₄×C₈	—	83	2	no	—
171	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
172	C₄×C₈	C₂²×D₈₃	1	2	yes	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	8	no	—
176	C₂×Q₈	—	332	2	no	—
177	C₂×Q₈	—	332	8	no	—
178	C₂×Q₈	—	166	4	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	332	2	no	—
181	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
184	C₂×C₈	—	83	4	no	—
185	C₂×C₈	—	83	4	no	—
186	C₂×C₈	—	166	4	no	—
187	C₂×C₈	—	166	4	no	—
188	C₂×C₈	#13050580	1	8	yes	—
189	C₂×C₈	C₂×C₂²⋊D₈₃	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	332	2	no	—
195	C₄⋊C₄	—	332	2	no	—
196	C₄⋊C₄	—	166	4	no	—
197	C₄⋊C₄	—	166	4	no	—
198	C₄²	D₄×D₈₃	1	1	yes	characteristic
199	C₄²	D₄×D₈₃	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
201	C₄²	C₂³×D₈₃	1	1	yes	characteristic
202	C₂³	#14581788	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	332	8	no	—
206	Q₈	—	664	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	332	4	no	—
210	C₂×C₄	—	664	4	no	—
211	C₂×C₄	#14581788	1	1	yes	characteristic
212	C₂×C₄	#14581788	1	1	yes	characteristic
213	C₂×C₄	#14581788	1	1	yes	characteristic
214	C₂×C₄	C₂×#13050580	1	4	yes	—
215	C₂×C₄	#14581811	1	4	yes	—
216	C₂×C₄	#14581815	1	4	yes	—
217	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
220	C₈	—	166	4	no	—
221	C₈	—	166	4	no	—
222	C₈	C₂×#13050580	1	8	yes	—
223	C₂²	#19128080	1	2	yes	—
224	C₂²	#19128084	1	2	yes	—
225	C₂²	#19128086	1	2	yes	—
226	C₂²	C₂×#14581788	1	1	yes	characteristic
227	C₄	—	8	4	no	—
228	C₄	—	664	4	no	—
229	C₄	#19128080	1	2	yes	—
230	C₄	#19128084	1	2	yes	—
231	C₄	#19128086	1	2	yes	—
232	C₄	C₂²×#13050580	1	2	yes	—
233	C₄	C₂×#14581811	1	2	yes	—
234	C₄	C₂×#14581815	1	2	yes	—
235	C₂	#24907072	1	4	yes	direct product factor
236	C₂	C₂×#19128080	1	1	yes	characteristic
237	C₂	C₂×#19128084	1	1	yes	characteristic
238	C₂	C₂×#19128086	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581788
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	C₂×#1710
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #24907072	C₂	direct (N × H)
116. C₂×C₈₃	#1710	semidirect
126. C₈₃	C₂×#1710	semidirect
235. C₂	#24907072	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19128106	2
C₂×#19128053	2
C₂×#19128051	2
#24907072	2
C₂×C₈×C₈₃⋊C₈	2
C₂×#500×C₈₃	2
C₂×#1710	83

Subgroups — C2×#24907072 #35392076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24907072 #35392076