Subgroups — C₂×#24906868 #35386767

Order 21248.

638 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#19127997	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
4	#24906868	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19127449	C₂	1	2	yes	maximal
6	C₂×#335×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14581792	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
9	#19128106	—	2	2	no	—
10	#19128106	C₂²	1	4	yes	—
11	#19127997	—	2	4	no	—
12	#19127997	C₂²	1	8	yes	—
13	C₈×Dic₁₆₆	—	2	2	no	—
14	C₈×Dic₁₆₆	C₂²	1	4	yes	—
15	C₂×#14581726	—	2	2	no	—
16	C₂×#14581726	—	2	1	no	—
17	C₂×#14581726	—	2	1	no	—
18	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₈₃⋊C₈	—	2	1	no	—
20	C₂×#14581717	—	2	2	no	—
21	C₂×C₈×Dic₈₃	—	2	1	no	—
22	#19127449	C₂²	1	8	yes	—
23	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	#335×C₈₃	C₂²	1	4	yes	—
26	C₂²×Dic₁₆₆	—	2	2	no	—
27	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
28	C₂²×C₈₃⋊C₈	—	2	1	no	—
29	#14581792	—	2	2	no	—
30	#14581792	—	2	2	no	—
31	#14581792	—	2	2	no	—
32	#14581792	C₂³	1	4	yes	—
33	C₄×Dic₁₆₆	—	2	1	no	—
34	C₄×Dic₁₆₆	—	2	1	no	—
35	C₄×Dic₁₆₆	D₄	1	2	yes	—
36	C₄×Dic₁₆₆	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₈₃	—	2	2	no	—
38	C₂×C₄⋊Dic₈₃	—	2	2	no	—
39	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
40	C₂×Dic₈₃⋊C₄	—	2	1	no	—
41	C₂×C₄×Dic₈₃	—	2	1	no	—
42	#14581726	—	2	8	no	—
43	#14581726	—	2	4	no	—
44	#14581726	—	2	4	no	—
45	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
47	Dic₈₃⋊C₈	—	2	4	no	—
48	#14581717	—	2	8	no	—
49	C₈×Dic₈₃	—	2	4	no	—
50	C₂²×C₈×C₈₃	—	2	2	no	—
51	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₃	—	2	2	no	—
55	C₄⋊C₈×C₈₃	C₂³	1	4	yes	—
56	C₄×C₈×C₈₃	D₄	1	2	yes	—
57	C₄×C₈×C₈₃	C₂³	1	2	yes	—
58	C₂×Dic₁₆₆	—	2	1	no	—
59	C₂×Dic₁₆₆	—	2	2	no	—
60	C₂×Dic₁₆₆	—	2	4	no	—
61	C₂×Dic₁₆₆	—	4	2	no	—
62	C₂×Dic₁₆₆	—	8	4	no	—
63	C₂×Dic₁₆₆	Q₁₆	1	4	yes	—
64	C₂×Dic₁₆₆	C₂×D₄	1	2	yes	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂²×Dic₈₃	—	4	2	no	—
68	C₂×C₈₃⋊C₈	—	2	2	no	—
69	C₂×C₈₃⋊C₈	—	4	2	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	4	no	—
72	C₄⋊Dic₈₃	—	2	4	no	—
73	C₄⋊Dic₈₃	—	4	2	no	—
74	C₄⋊Dic₈₃	—	4	2	no	—
75	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
76	Dic₈₃⋊C₄	—	2	2	no	—
77	Dic₈₃⋊C₄	—	2	2	no	—
78	C₄×Dic₈₃	—	2	2	no	—
79	C₄×Dic₈₃	—	2	2	no	—
80	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₈₃	—	2	2	no	—
84	C₂×C₈×C₈₃	—	2	4	no	—
85	C₂×C₈×C₈₃	—	4	4	no	—
86	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
87	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
89	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₈₃	#41	1	1	yes	characteristic
92	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
93	C₂³×C₈₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₈₃	#135	1	2	yes	—
101	C₂×C₄×C₈₃	#137	1	2	yes	—
102	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₈₃	—	2	2	no	—
106	C₂×Dic₈₃	—	2	4	no	—
107	C₂×Dic₈₃	—	2	2	no	—
108	C₂×Dic₈₃	—	4	2	no	—
109	C₂×Dic₈₃	—	4	4	no	—
110	C₂×Dic₈₃	—	8	4	no	—
111	Dic₁₆₆	—	2	2	no	—
112	Dic₁₆₆	—	2	4	no	—
113	Dic₁₆₆	—	8	4	no	—
114	Dic₁₆₆	—	8	4	no	—
115	Dic₁₆₆	C₂×Q₁₆	1	4	yes	—
116	C₈×C₈₃	—	2	2	no	—
117	C₈×C₈₃	—	4	4	no	—
118	C₈×C₈₃	C₂×Q₁₆	1	4	yes	—
119	C₈₃⋊C₈	—	4	2	no	—
120	C₂²×C₈₃	#462	1	2	yes	—
121	C₂²×C₈₃	#467	1	2	yes	—
122	C₂²×C₈₃	#472	1	2	yes	—
123	C₂²×C₈₃	C₂×#121	1	1	yes	characteristic
124	C₄×C₈₃	—	2	1	no	—
125	C₄×C₈₃	—	2	1	no	—
126	C₄×C₈₃	#462	1	2	yes	—
127	C₄×C₈₃	#467	1	2	yes	—
128	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
129	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
130	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
131	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
132	Dic₈₃	—	2	4	no	—
133	Dic₈₃	—	4	2	no	—
134	Dic₈₃	—	8	4	no	—
135	C₂×#1676	—	83	1	no	maximal, Sylow
136	C₂×C₈₃	#1676	1	4	yes	complemented
137	C₂×C₈₃	C₂×#462	1	1	yes	characteristic
138	C₂×C₈₃	C₂×#467	1	1	yes	characteristic
139	C₂×C₈₃	C₂×#472	1	1	yes	characteristic
140	C₂×#500	—	83	1	no	—
141	C₂×#477	—	83	2	no	—
142	C₂×C₈×Q₈	—	83	1	no	—
143	#1676	—	83	8	no	—
144	C₂×#335	D₈₃	1	1	yes	characteristic
145	C₂×#333	—	83	2	no	—
146	C₈₃	C₂×#1676	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄⋊Q₈	—	83	2	no	—
148	C₂×C₄×Q₈	—	83	1	no	—
149	#500	—	83	4	no	—
150	#500	—	166	2	no	—
151	#477	—	83	8	no	—
152	#477	—	166	4	no	—
153	C₈×Q₈	—	83	4	no	—
154	C₈×Q₈	—	166	2	no	—
155	C₂×#107	—	166	2	no	—
156	C₂×#107	—	166	1	no	—
157	C₂×#107	—	166	1	no	—
158	C₂×C₄⋊C₈	—	83	1	no	—
159	C₂×C₄⋊C₈	—	166	1	no	—
160	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
161	C₂×#103	—	166	2	no	—
162	C₂×C₄×C₈	—	166	1	no	—
163	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
164	#335	C₂×D₈₃	1	4	yes	—
165	#333	—	83	8	no	—
166	C₂²×Q₈	—	83	1	no	—
167	C₂²×Q₈	—	166	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	166	1	no	—
170	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
171	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
172	C₄⋊Q₈	—	83	4	no	—
173	C₄⋊Q₈	—	166	2	no	—
174	C₄⋊Q₈	—	166	2	no	—
175	C₄⋊Q₈	—	166	2	no	—
176	C₄×Q₈	—	83	2	no	—
177	C₄×Q₈	—	83	2	no	—
178	C₄×Q₈	—	166	1	no	—
179	C₄×Q₈	—	166	1	no	—
180	C₂×C₄⋊C₄	—	83	1	no	—
181	C₂×C₄⋊C₄	—	166	2	no	—
182	C₂×C₄⋊C₄	—	166	2	no	—
183	C₂×C₄⋊C₄	—	166	1	no	—
184	C₂×C₄²	—	166	1	no	—
185	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
186	#107	—	166	8	no	—
187	#107	—	166	4	no	—
188	#107	—	166	4	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	83	2	no	—
191	C₄⋊C₈	—	166	4	no	—
192	C₄⋊C₈	—	166	1	no	—
193	C₄⋊C₈	C₂²×D₈₃	1	4	yes	—
194	#103	—	166	8	no	—
195	C₄×C₈	—	166	4	no	—
196	C₄×C₈	D₃₃₂	1	2	yes	—
197	C₄×C₈	C₂²×D₈₃	1	2	yes	—
198	C₂×Q₈	—	332	2	no	—
199	C₂×Q₈	—	83	4	no	—
200	C₂×Q₈	—	83	2	no	—
201	C₂×Q₈	—	664	4	no	—
202	C₂×Q₈	—	166	4	no	—
203	C₂×Q₈	—	166	1	no	—
204	C₂×Q₈	—	166	2	no	—
205	C₂²×C₄	—	332	2	no	—
206	C₂²×C₄	—	166	1	no	—
207	C₂²×C₄	—	166	1	no	—
208	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
210	C₂²×C₄	#13050591	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	332	2	no	—
215	C₂×C₈	—	166	2	no	—
216	C₂×C₈	Dic₃₃₂	1	4	yes	—
217	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
218	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
219	C₄⋊C₄	—	332	2	no	—
220	C₄⋊C₄	—	332	2	no	—
221	C₄⋊C₄	—	83	2	no	—
222	C₄⋊C₄	—	166	4	no	—
223	C₄⋊C₄	—	166	4	no	—
224	C₄⋊C₄	—	166	1	no	—
225	C₄⋊C₄	—	166	2	no	—
226	C₄⋊C₄	—	166	2	no	—
227	C₄²	—	166	2	no	—
228	C₄²	—	166	2	no	—
229	C₄²	D₄×D₈₃	1	1	yes	characteristic
230	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
231	C₄²	#13050591	1	1	yes	characteristic
232	C₄²	C₂³×D₈₃	1	1	yes	characteristic
233	C₂³	#14581767	1	1	yes	characteristic
234	Q₈	—	83	4	no	—
235	Q₈	—	664	4	no	—
236	Q₈	—	664	4	no	—
237	Q₈	—	166	4	no	—
238	Q₈	—	166	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	332	2	no	—
241	C₂×C₄	—	332	4	no	—
242	C₂×C₄	—	664	4	no	—
243	C₂×C₄	—	166	2	no	—
244	C₂×C₄	—	166	2	no	—
245	C₂×C₄	—	166	4	no	—
246	C₂×C₄	#14581767	1	1	yes	characteristic
247	C₂×C₄	#14581767	1	1	yes	characteristic
248	C₂×C₄	#14581767	1	1	yes	characteristic
249	C₂×C₄	Q₁₆×D₈₃	1	2	yes	—
250	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
251	C₂×C₄	#14581811	1	2	yes	—
252	C₂×C₄	#14581825	1	2	yes	—
253	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
255	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	332	2	no	—
259	C₈	C₂×Dic₃₃₂	1	4	yes	—
260	C₂²	#19127912	1	2	yes	—
261	C₂²	#19127943	1	2	yes	—
262	C₂²	#19127972	1	2	yes	—
263	C₂²	C₂×#14581767	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	332	2	no	—
267	C₄	—	664	4	no	—
268	C₄	—	166	4	no	—
269	C₄	#19127912	1	2	yes	—
270	C₄	#19127943	1	2	yes	—
271	C₄	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
272	C₄	C₂²×Dic₃₃₂	1	1	yes	characteristic
273	C₄	C₂×#14581811	1	1	yes	characteristic
274	C₄	C₂×#14581825	1	1	yes	characteristic
275	C₂	#24906868	1	4	yes	direct product factor
276	C₂	C₂×#19127912	1	1	yes	characteristic
277	C₂	C₂×#19127943	1	1	yes	characteristic
278	C₂	C₂×#19127972	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581767
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#335×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
hypercenter	C₂×#335	D₈₃
hypocenter	C₈₃	C₂×#1676
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #24906868	C₂	direct (N × H)
136. C₂×C₈₃	#1676	semidirect
146. C₈₃	C₂×#1676	semidirect
275. C₂	#24906868	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×#19127997	2
C₂×C₈×Dic₁₆₆	2
#24906868	2
C₂×#19127449	2
C₂×#335×C₈₃	2
C₂×#1676	83

Subgroups — C2×#24906868 #35386767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24906868 #35386767