Subgroups — C₂×#24909510 #35558651

Order 21248.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14581738	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#24909510	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#19127505	C₂	1	2	yes	maximal
6	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₆	—	2	1	no	—
9	C₂×#14581738	—	2	8	no	—
10	C₂×#14581738	C₂²	1	8	yes	—
11	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
12	C₄×Dic₃₃₂	—	2	2	no	—
13	C₄×Dic₃₃₂	C₂²	1	4	yes	—
14	C₂×#14581726	C₂²	1	1	yes	characteristic
15	C₂×#14581717	—	2	1	no	—
16	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	4	yes	—
17	#19127505	C₂²	1	8	yes	—
18	C₂²×Q₁₆×C₈₃	—	2	1	no	—
19	C₂²×Q₁₆×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₈₃	—	2	1	no	—
21	#494×C₈₃	—	2	2	no	—
22	#494×C₈₃	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₆₆	—	2	1	no	—
25	C₂×Dic₃₃₂	—	2	1	no	—
26	C₂×Dic₃₃₂	—	2	2	no	—
27	C₂×Dic₃₃₂	D₄	1	4	yes	—
28	C₂×Dic₃₃₂	C₂³	1	2	yes	—
29	C₄×Dic₁₆₆	—	2	4	no	—
30	C₄×Dic₁₆₆	—	4	2	no	—
31	C₂×C₄⋊Dic₈₃	—	2	1	no	—
32	C₂×Dic₈₃⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₈₃	—	4	1	no	—
34	#14581738	—	2	16	no	—
35	C₂×C₈₃⋊C₁₆	—	2	4	no	—
36	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
37	#14581726	—	2	1	no	—
38	#14581726	C₂³	1	2	yes	—
39	#14581717	—	2	4	no	—
40	C₂²×Q₈×C₈₃	—	2	2	no	—
41	C₂×Q₁₆×C₈₃	—	2	4	no	—
42	C₂×Q₁₆×C₈₃	—	2	2	no	—
43	C₂×Q₁₆×C₈₃	—	2	4	no	—
44	C₂×Q₁₆×C₈₃	—	2	4	no	—
45	C₂×Q₁₆×C₈₃	—	4	4	no	—
46	C₂×Q₁₆×C₈₃	C₂³	1	4	yes	—
47	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₈₃	—	2	4	no	—
50	C₄⋊Q₈×C₈₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
52	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
53	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₈₃	D₄	1	2	yes	—
55	C₄×C₈×C₈₃	C₂³	1	2	yes	—
56	C₂×Dic₁₆₆	—	2	4	no	—
57	C₂×Dic₁₆₆	—	2	2	no	—
58	C₂×Dic₁₆₆	—	4	1	no	—
59	C₂×Dic₁₆₆	—	4	2	no	—
60	C₂²×Dic₈₃	—	4	1	no	—
61	C₂²×Dic₈₃	—	4	1	no	—
62	Dic₃₃₂	—	2	4	no	—
63	Dic₃₃₂	—	2	2	no	—
64	Dic₃₃₂	C₂×D₄	1	4	yes	—
65	C₄⋊Dic₈₃	—	2	2	no	—
66	C₄⋊Dic₈₃	—	4	1	no	—
67	Dic₈₃⋊C₄	—	4	4	no	—
68	C₄×Dic₈₃	—	4	4	no	—
69	C₈₃⋊C₁₆	—	2	4	no	—
70	C₂×Q₈×C₈₃	—	2	4	no	—
71	C₂×Q₈×C₈₃	—	4	2	no	—
72	C₂×Q₈×C₈₃	—	4	4	no	—
73	C₂×Q₈×C₈₃	—	4	4	no	—
74	C₂²×C₄×C₈₃	—	4	2	no	—
75	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₈₃	—	2	4	no	—
79	Q₁₆×C₈₃	—	2	4	no	—
80	Q₁₆×C₈₃	—	4	8	no	—
81	Q₁₆×C₈₃	—	4	4	no	—
82	C₂×C₈×C₈₃	—	2	2	no	—
83	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₃	#41	1	2	yes	—
87	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₈₃	—	4	4	no	—
89	C₄⋊C₄×C₈₃	—	4	4	no	—
90	C₄²×C₈₃	D₈	1	2	yes	—
91	C₄²×C₈₃	C₂×D₄	1	2	yes	—
92	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₈₃	—	4	4	no	—
94	Q₈×C₈₃	—	4	4	no	—
95	Q₈×C₈₃	—	8	4	no	—
96	C₂×C₄×C₈₃	—	2	2	no	—
97	C₂×C₄×C₈₃	—	4	4	no	—
98	C₂×C₄×C₈₃	—	8	4	no	—
99	C₂×C₄×C₈₃	Q₃₂	1	4	yes	complemented
100	C₂×C₄×C₈₃	#121	1	2	yes	—
101	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₈₃	—	4	2	no	—
107	C₂×Dic₈₃	—	4	4	no	—
108	C₂×Dic₈₃	—	4	2	no	—
109	C₂×Dic₈₃	—	8	2	no	—
110	Dic₁₆₆	—	2	4	no	—
111	Dic₁₆₆	—	4	2	no	—
112	Dic₁₆₆	—	4	4	no	—
113	C₈×C₈₃	—	2	2	no	—
114	C₈×C₈₃	#121	1	2	yes	—
115	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₈₃	#459	1	2	yes	—
118	C₂²×C₈₃	C₂×Q₃₂	1	2	yes	—
119	C₂²×C₈₃	#510	1	2	yes	—
120	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	8	4	no	—
123	C₄×C₈₃	#459	1	2	yes	—
124	C₄×C₈₃	C₂×Q₃₂	1	4	yes	complemented
125	C₄×C₈₃	C₂×#121	1	1	yes	characteristic
126	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	Dic₈₃	—	4	4	no	—
128	Dic₈₃	—	8	2	no	—
129	C₂×#2209	—	83	1	no	maximal, Sylow
130	C₂×C₈₃	#2209	1	4	yes	complemented
131	C₂×C₈₃	C₂×#459	1	1	yes	characteristic
132	C₂×C₈₃	C₂²×Q₃₂	1	1	yes	characteristic
133	C₂×C₈₃	C₂×#510	1	1	yes	characteristic
134	C₂²×Q₃₂	—	83	2	no	—
135	C₂×#494	D₈₃	1	1	yes	characteristic
136	C₂×C₄×Q₁₆	—	83	1	no	—
137	#2209	—	83	8	no	—
138	C₂×C₄⋊C₁₆	—	83	1	no	—
139	C₂×#358	—	83	2	no	—
140	C₈₃	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₁₆	—	2	1	no	—
142	C₂²×Q₁₆	—	83	1	no	—
143	C₂²×Q₁₆	C₂×D₈₃	1	2	yes	—
144	C₂×C₄⋊Q₈	—	2	1	no	—
145	C₂×C₄×Q₈	—	166	1	no	—
146	C₂×Q₃₂	—	83	8	no	—
147	C₂×Q₃₂	—	166	8	no	—
148	C₂²×C₁₆	—	83	2	no	—
149	#494	—	2	2	no	—
150	#494	C₂×D₈₃	1	4	yes	—
151	C₄×Q₁₆	—	83	4	no	—
152	C₄×Q₁₆	—	166	2	no	—
153	C₂×#107	—	83	1	no	—
154	C₂×#103	—	166	1	no	—
155	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	83	4	no	—
157	#358	—	83	8	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	—	166	1	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×Q₁₆	—	83	4	no	—
166	C₂×Q₁₆	—	83	2	no	—
167	C₂×Q₁₆	—	166	1	no	—
168	C₂×Q₁₆	—	166	2	no	—
169	C₂×Q₁₆	C₂²×D₈₃	1	4	yes	—
170	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	—	4	2	no	—
174	C₄×Q₈	—	332	2	no	—
175	C₄×Q₈	—	166	4	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	332	1	no	—
179	C₂×C₄⋊C₄	—	166	1	no	—
180	C₂×C₄²	—	332	1	no	—
181	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
182	Q₃₂	—	166	16	no	—
183	C₂×C₁₆	—	83	4	no	—
184	C₂×C₁₆	—	166	4	no	—
185	#107	—	83	2	no	—
186	#107	—	166	1	no	—
187	#103	—	166	4	no	—
188	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
189	C₄×C₈	C₂²×D₈₃	1	2	yes	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	332	1	no	—
195	C₂×Q₈	—	332	2	no	—
196	C₂×Q₈	—	166	4	no	—
197	C₂×Q₈	—	166	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	332	1	no	—
200	C₂²×C₄	—	332	1	no	—
201	C₂²×C₄	#13050575	1	1	yes	characteristic
202	C₂²×C₄	#13050575	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	8	no	—
207	Q₁₆	—	4	4	no	—
208	Q₁₆	—	83	4	no	—
209	Q₁₆	—	166	4	no	—
210	Q₁₆	—	166	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	D₄×D₈₃	1	2	yes	—
213	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
214	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
215	C₂×C₈	#13050589	1	2	yes	—
216	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	332	4	no	—
220	C₄⋊C₄	—	332	1	no	—
221	C₄⋊C₄	—	166	2	no	—
222	C₄²	—	332	4	no	—
223	C₄²	#13050575	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
225	C₁₆	—	166	4	no	—
226	C₂³	C₂×#13050575	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	8	4	no	—
230	Q₈	—	332	4	no	—
231	Q₈	—	332	2	no	—
232	Q₈	—	166	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	332	4	no	—
237	C₂×C₄	—	332	2	no	—
238	C₂×C₄	—	332	2	no	—
239	C₂×C₄	—	664	2	no	—
240	C₂×C₄	#14581738	1	4	yes	complemented
241	C₂×C₄	#14581763	1	2	yes	—
242	C₂×C₄	C₂×#13050575	1	2	yes	—
243	C₂×C₄	C₂×#13050575	1	2	yes	—
244	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
245	C₂×C₄	#14581818	1	2	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#14581763	1	2	yes	—
249	C₈	C₂×D₄×D₈₃	1	1	yes	characteristic
250	C₈	C₂×#13050589	1	1	yes	characteristic
251	C₂²	#19127893	1	2	yes	—
252	C₂²	C₂×#14581738	1	2	yes	—
253	C₂²	#19128142	1	2	yes	—
254	C₂²	C₂²×#13050575	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	8	4	no	—
257	C₄	—	332	4	no	—
258	C₄	—	664	2	no	—
259	C₄	#19127893	1	2	yes	—
260	C₄	C₂×#14581738	1	4	yes	complemented
261	C₄	C₂×#14581763	1	1	yes	characteristic
262	C₄	C₂×#14581818	1	1	yes	characteristic
263	C₂	#24909510	1	4	yes	direct product factor
264	C₂	C₂×#19127893	1	1	yes	characteristic
265	C₂	C₂²×#14581738	1	1	yes	characteristic
266	C₂	C₂×#19128142	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	C₂×#2209
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #24909510	C₂	direct (N × H)
99. C₂×C₄×C₈₃	Q₃₂	semidirect
124. C₄×C₈₃	C₂×Q₃₂	semidirect
130. C₂×C₈₃	#2209	semidirect
140. C₈₃	C₂×#2209	semidirect
240. C₂×C₄	#14581738	semidirect
260. C₄	C₂×#14581738	semidirect
263. C₂	#24909510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14581738	2
C₂×C₄×Dic₃₃₂	2
#24909510	2
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₂×#19127505	2
C₂×#494×C₈₃	2
C₂×#2209	83

Subgroups — C2×#24909510 #35558651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909510 #35558651