Subgroups — C₂×#24906854 #35386757

Order 21248.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127908	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#24906854	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
5	C₂×#19127447	C₂	1	2	yes	maximal
6	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13050580	—	2	2	no	—
8	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
9	#19127908	—	2	4	no	—
10	#19127908	C₂²	1	8	yes	—
11	C₈×Dic₁₆₆	—	2	2	no	—
12	C₈×Dic₁₆₆	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₈₃⋊C₈	—	2	1	no	—
16	C₂×#14581716	—	2	2	no	—
17	C₂×C₈×Dic₈₃	—	2	1	no	—
18	#19127454	C₂²	1	4	yes	—
19	#19127447	C₂²	1	8	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	1	no	—
24	C₂×#107×C₈₃	—	2	1	no	—
25	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
27	C₂×#13050580	—	2	8	no	—
28	C₂×#13050580	—	4	8	no	—
29	C₂²×C₈₃⋊C₈	—	2	1	no	—
30	C₂²×C₈₃⋊C₈	—	2	2	no	—
31	C₄×Dic₁₆₆	—	2	1	no	—
32	C₄×Dic₁₆₆	—	2	1	no	—
33	C₄×Dic₁₆₆	D₄	1	2	yes	—
34	C₄×Dic₁₆₆	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
36	C₂×Dic₈₃⋊C₄	—	2	1	no	—
37	C₂×C₄×Dic₈₃	—	2	1	no	—
38	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
41	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
42	Dic₈₃⋊C₈	—	2	4	no	—
43	#14581716	—	2	8	no	—
44	C₈×Dic₈₃	—	2	4	no	—
45	C₂²×Q₈×C₈₃	—	2	2	no	—
46	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₃	—	2	2	no	—
49	C₄⋊Q₈×C₈₃	—	2	2	no	—
50	C₄⋊Q₈×C₈₃	—	2	2	no	—
51	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
54	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
55	#107×C₈₃	—	2	4	no	—
56	#107×C₈₃	—	2	4	no	—
57	C₄×C₈×C₈₃	D₄	1	2	yes	—
58	C₄×C₈×C₈₃	C₂³	1	2	yes	—
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×Dic₁₆₆	D₈	1	4	yes	—
62	C₂×Dic₁₆₆	C₂×D₄	1	2	yes	—
63	C₂²×Dic₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	#13050580	—	4	16	no	—
66	C₂×C₈₃⋊C₈	—	2	4	no	—
67	C₂×C₈₃⋊C₈	—	2	2	no	—
68	C₂×C₈₃⋊C₈	—	4	4	no	—
69	C₂×C₈₃⋊C₈	—	4	2	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
72	Dic₈₃⋊C₄	—	2	2	no	—
73	Dic₈₃⋊C₄	—	2	2	no	—
74	C₄×Dic₈₃	—	2	2	no	—
75	C₄×Dic₈₃	—	2	2	no	—
76	C₂×Q₈×C₈₃	—	2	4	no	—
77	C₂×Q₈×C₈₃	—	4	2	no	—
78	C₂×Q₈×C₈₃	—	4	4	no	—
79	C₂×Q₈×C₈₃	—	4	4	no	—
80	C₂²×C₄×C₈₃	—	4	2	no	—
81	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₃	—	2	2	no	—
85	C₂×C₈×C₈₃	Q₁₆	1	4	yes	complemented
86	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₈₃	—	2	4	no	—
89	C₄⋊C₄×C₈₃	—	2	4	no	—
90	C₄⋊C₄×C₈₃	—	4	2	no	—
91	C₄⋊C₄×C₈₃	—	4	2	no	—
92	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₃	#41	1	1	yes	characteristic
95	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
96	C₂³×C₈₃	#121	1	1	yes	characteristic
97	Q₈×C₈₃	—	4	4	no	—
98	Q₈×C₈₃	—	4	4	no	—
99	Q₈×C₈₃	—	8	4	no	—
100	C₂×C₄×C₈₃	—	2	2	no	—
101	C₂×C₄×C₈₃	—	4	4	no	—
102	C₂×C₄×C₈₃	—	8	4	no	—
103	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₈₃	#135	1	2	yes	—
109	C₂×C₄×C₈₃	#137	1	2	yes	—
110	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₈₃	—	2	2	no	—
114	C₂×Dic₈₃	—	2	4	no	—
115	C₂×Dic₈₃	—	2	2	no	—
116	C₂×Dic₈₃	—	4	2	no	—
117	Dic₁₆₆	—	2	2	no	—
118	Dic₁₆₆	—	2	4	no	—
119	Dic₁₆₆	C₂×D₈	1	4	yes	—
120	C₈×C₈₃	—	2	2	no	—
121	C₈×C₈₃	C₂×Q₁₆	1	4	yes	complemented
122	C₈₃⋊C₈	—	4	4	no	—
123	C₈₃⋊C₈	—	4	2	no	—
124	C₂²×C₈₃	#462	1	2	yes	—
125	C₂²×C₈₃	#466	1	2	yes	—
126	C₂²×C₈₃	#473	1	2	yes	—
127	C₂²×C₈₃	C₂×#121	1	1	yes	characteristic
128	C₄×C₈₃	—	2	1	no	—
129	C₄×C₈₃	—	2	1	no	—
130	C₄×C₈₃	—	8	4	no	—
131	C₄×C₈₃	#462	1	2	yes	—
132	C₄×C₈₃	#466	1	2	yes	—
133	C₄×C₈₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
135	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
136	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
137	Dic₈₃	—	2	4	no	—
138	Dic₈₃	—	4	2	no	—
139	C₂×#1674	—	83	1	no	maximal, Sylow
140	C₂×C₈₃	#1674	1	4	yes	complemented
141	C₂×C₈₃	C₂×#462	1	1	yes	characteristic
142	C₂×C₈₃	C₂×#466	1	1	yes	characteristic
143	C₂×C₈₃	C₂×#473	1	1	yes	characteristic
144	C₂×#500	D₈₃	1	1	yes	characteristic
145	C₂×#462	—	83	2	no	—
146	C₂×C₈×Q₈	—	83	1	no	—
147	#1674	—	83	8	no	—
148	C₂×#335	—	83	1	no	—
149	C₂×#332	—	83	2	no	—
150	C₈₃	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	166	2	no	—
152	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
153	C₂×C₄×Q₈	—	83	1	no	—
154	#500	—	2	2	no	—
155	#500	C₂×D₈₃	1	4	yes	—
156	#462	—	83	8	no	—
157	#462	—	166	4	no	—
158	C₈×Q₈	—	83	4	no	—
159	C₈×Q₈	—	166	2	no	—
160	C₂×#107	—	2	1	no	—
161	C₂×#107	—	2	1	no	—
162	C₂×C₄⋊C₈	—	83	1	no	—
163	C₂×C₄⋊C₈	—	83	2	no	—
164	C₂×C₄⋊C₈	—	166	1	no	—
165	C₂×#103	—	166	2	no	—
166	C₂×C₄×C₈	—	166	1	no	—
167	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
168	#335	—	83	4	no	—
169	#332	—	83	8	no	—
170	C₂²×Q₈	—	2	2	no	—
171	C₂²×Q₈	—	83	1	no	—
172	C₂×Q₁₆	—	332	8	no	—
173	C₂×Q₁₆	—	166	8	no	—
174	C₂²×C₈	—	166	2	no	—
175	C₂²×C₈	—	166	1	no	—
176	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂²×D₈₃	1	4	yes	—
182	C₄×Q₈	—	83	2	no	—
183	C₄×Q₈	—	83	2	no	—
184	C₄×Q₈	—	166	1	no	—
185	C₄×Q₈	—	166	1	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	83	1	no	—
189	C₂×C₄⋊C₄	—	166	1	no	—
190	C₂×C₄²	—	166	1	no	—
191	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	C₄⋊C₈	—	83	2	no	—
195	C₄⋊C₈	—	83	4	no	—
196	C₄⋊C₈	—	166	4	no	—
197	C₄⋊C₈	—	166	2	no	—
198	C₄⋊C₈	—	166	1	no	—
199	#103	—	166	8	no	—
200	C₄×C₈	—	166	4	no	—
201	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
202	C₄×C₈	C₂²×D₈₃	1	2	yes	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	83	4	no	—
208	C₂×Q₈	—	83	2	no	—
209	C₂×Q₈	—	166	1	no	—
210	C₂×Q₈	—	166	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	166	1	no	—
213	C₂²×C₄	—	166	1	no	—
214	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
216	C₂²×C₄	#13050589	1	1	yes	characteristic
217	Q₁₆	—	332	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	332	4	no	—
220	C₂×C₈	—	332	2	no	—
221	C₂×C₈	—	166	4	no	—
222	C₂×C₈	—	166	2	no	—
223	C₂×C₈	#13050580	1	4	yes	complemented
224	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	83	2	no	—
231	C₄⋊C₄	—	166	1	no	—
232	C₄⋊C₄	—	166	2	no	—
233	C₄⋊C₄	—	166	2	no	—
234	C₄²	—	166	2	no	—
235	C₄²	—	166	2	no	—
236	C₄²	D₄×D₈₃	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
238	C₄²	#13050589	1	1	yes	characteristic
239	C₄²	C₂³×D₈₃	1	1	yes	characteristic
240	C₂³	#14581763	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	8	4	no	—
244	Q₈	—	83	4	no	—
245	Q₈	—	166	4	no	—
246	Q₈	—	166	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	332	2	no	—
251	C₂×C₄	—	166	2	no	—
252	C₂×C₄	—	166	4	no	—
253	C₂×C₄	—	166	2	no	—
254	C₂×C₄	#14581763	1	1	yes	characteristic
255	C₂×C₄	#14581763	1	1	yes	characteristic
256	C₂×C₄	#14581763	1	1	yes	characteristic
257	C₂×C₄	D₈×D₈₃	1	2	yes	—
258	C₂×C₄	C₂×#13050580	1	2	yes	—
259	C₂×C₄	#14581811	1	2	yes	—
260	C₂×C₄	#14581823	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#13050589	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	332	4	no	—
266	C₈	—	332	2	no	—
267	C₈	C₂×#13050580	1	4	yes	complemented
268	C₂²	#19127908	1	2	yes	—
269	C₂²	#19127934	1	2	yes	—
270	C₂²	#19127973	1	2	yes	—
271	C₂²	C₂×#14581763	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	4	no	—
275	C₄	—	332	2	no	—
276	C₄	—	166	4	no	—
277	C₄	#19127908	1	2	yes	—
278	C₄	#19127934	1	2	yes	—
279	C₄	C₂×D₈×D₈₃	1	1	yes	characteristic
280	C₄	C₂²×#13050580	1	1	yes	characteristic
281	C₄	C₂×#14581811	1	1	yes	characteristic
282	C₄	C₂×#14581823	1	1	yes	characteristic
283	C₂	#24906854	1	4	yes	direct product factor
284	C₂	C₂×#19127908	1	1	yes	characteristic
285	C₂	C₂×#19127934	1	1	yes	characteristic
286	C₂	C₂×#19127973	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581763
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	C₂×#1674
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
3. #24906854	C₂	direct (N × H)
85. C₂×C₈×C₈₃	Q₁₆	semidirect
121. C₈×C₈₃	C₂×Q₁₆	semidirect
140. C₂×C₈₃	#1674	semidirect
150. C₈₃	C₂×#1674	semidirect
223. C₂×C₈	#13050580	semidirect
267. C₈	C₂×#13050580	semidirect
283. C₂	#24906854	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×#19127908	2
C₂×C₈×Dic₁₆₆	2
#24906854	2
C₂×#19127454	2
C₂×#19127447	2
C₂×#500×C₈₃	2
C₂×#1674	83

Subgroups — C2×#24906854 #35386757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24906854 #35386757