Subgroups — C₂×#24905565 #35319313

Order 21248.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127739	C₂	1	2	yes	maximal
2	#24905565	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581751	—	2	2	no	—
5	#19127739	—	2	2	no	—
6	#19127739	—	2	2	no	—
7	#19127739	C₄	1	4	yes	complemented
8	#19127739	C₂²	1	4	yes	—
9	C₂×#14581726	—	2	2	no	—
10	C₂×#14581724	C₄	1	4	yes	complemented
11	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₈₃	—	2	2	no	—
13	C₂×M₃₂×C₈₃	—	2	2	no	—
14	C₂×M₃₂×C₈₃	C₄	1	2	yes	complemented
15	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₈₃	C₄	1	2	yes	complemented
17	#14581751	—	2	4	no	—
18	#14581751	—	2	4	no	—
19	#14581751	—	4	2	no	—
20	#14581751	—	4	2	no	—
21	C₂×C₄⋊Dic₈₃	—	2	4	no	—
22	C₂×C₂²⋊Dic₈₃	—	4	2	no	—
23	C₂×C₄×Dic₈₃	—	4	2	no	—
24	#14581726	—	2	4	no	—
25	#14581726	—	2	4	no	—
26	#14581724	—	2	4	no	—
27	#14581724	C₂×C₄	1	8	yes	complemented
28	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₃	D₄	1	2	yes	—
35	C₂²×C₈×C₈₃	D₄	1	2	yes	—
36	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
39	M₃₂×C₈₃	—	2	4	no	—
40	M₃₂×C₈₃	—	2	4	no	—
41	C₂×C₁₆×C₈₃	—	2	2	no	—
42	C₂×C₁₆×C₈₃	—	2	2	no	—
43	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	complemented
44	C₂²×Dic₈₃	—	4	4	no	—
45	C₄⋊Dic₈₃	—	2	8	no	—
46	C₄⋊Dic₈₃	—	4	4	no	—
47	C₂²⋊Dic₈₃	—	4	4	no	—
48	C₂²⋊Dic₈₃	—	4	4	no	—
49	C₄×Dic₈₃	—	4	4	no	—
50	C₄×Dic₈₃	—	4	4	no	—
51	C₂⁴×C₈₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
58	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₈₃	C₄²	1	2	yes	—
64	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
80	C₁₆×C₈₃	—	2	4	no	—
81	C₂³×C₈₃	—	4	2	no	—
82	C₂³×C₈₃	#102	1	2	yes	—
83	C₂³×C₈₃	#102	1	2	yes	—
84	C₂³×C₈₃	#107	1	1	yes	characteristic
85	C₂³×C₈₃	#107	1	1	yes	characteristic
86	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₈₃	#95	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#95	1	2	yes	—
89	C₂×C₄×C₈₃	#95	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#102	1	2	yes	—
91	C₂×C₄×C₈₃	#102	1	2	yes	—
92	C₂×C₄×C₈₃	#103	1	2	yes	—
93	C₂×C₄×C₈₃	#103	1	2	yes	—
94	C₂×C₄×C₈₃	#103	1	2	yes	—
95	C₂×C₄×C₈₃	#103	1	2	yes	—
96	C₂×C₄×C₈₃	#106	1	2	yes	—
97	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
100	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×Dic₈₃	—	4	8	no	—
106	C₂×Dic₈₃	—	8	8	no	—
107	C₈×C₈₃	#95	1	1	yes	characteristic
108	C₈×C₈₃	#95	1	2	yes	—
109	C₈×C₈₃	#95	1	1	yes	characteristic
110	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
111	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×C₈₃	—	2	4	no	—
114	C₂²×C₈₃	—	2	2	no	—
115	C₂²×C₈₃	—	4	2	no	—
116	C₂²×C₈₃	—	4	1	no	—
117	C₂²×C₈₃	—	4	1	no	—
118	C₂²×C₈₃	#340	1	2	yes	—
119	C₂²×C₈₃	#340	1	1	yes	characteristic
120	C₂²×C₈₃	#340	1	1	yes	characteristic
121	C₂²×C₈₃	C₂×#102	1	2	yes	—
122	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
123	C₄×C₈₃	#340	1	2	yes	—
124	C₄×C₈₃	#340	1	1	yes	characteristic
125	C₄×C₈₃	#340	1	1	yes	characteristic
126	C₄×C₈₃	C₂×#95	1	1	yes	characteristic
127	C₄×C₈₃	C₂×#103	1	2	yes	—
128	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
129	Dic₈₃	—	8	8	no	—
130	C₂×#1384	—	83	1	no	maximal, Sylow
131	C₂×C₈₃	—	2	2	no	—
132	C₂×C₈₃	—	2	2	no	—
133	C₂×C₈₃	—	2	1	no	—
134	C₂×C₈₃	—	2	1	no	—
135	C₂×C₈₃	#1384	1	2	yes	complemented
136	C₂×C₈₃	C₂×#340	1	1	yes	characteristic
137	C₂²×M₃₂	D₈₃	1	1	yes	characteristic
138	C₂×#427	—	83	2	no	—
139	#1384	—	83	4	no	—
140	C₈₃	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
142	C₂×#117	—	166	2	no	—
143	C₂×M₃₂	—	2	2	no	—
144	C₂×M₃₂	—	2	2	no	—
145	C₂×M₃₂	Dic₈₃	1	2	yes	complemented
146	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
147	C₂²×C₁₆	Dic₈₃	1	2	yes	complemented
148	#427	—	83	4	no	—
149	#427	—	83	4	no	—
150	#427	—	166	2	no	—
151	#427	—	166	2	no	—
152	C₂×#107	—	166	2	no	—
153	C₂×#106	—	83	4	no	—
154	C₂³×C₄	D₃₃₂	1	1	yes	characteristic
155	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₈₃	1	2	yes	—
158	C₂²×C₈	C₄×D₈₃	1	2	yes	—
159	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
160	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
164	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
165	#117	—	332	2	no	—
166	#117	—	332	2	no	—
167	#117	—	166	4	no	—
168	#117	—	166	4	no	—
169	C₂×C₄⋊C₄	—	166	4	no	—
170	C₂×C₂²⋊C₄	—	332	2	no	—
171	C₂×C₄²	—	332	2	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	—	2	4	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	complemented
177	#107	—	166	4	no	—
178	#107	—	166	4	no	—
179	#106	—	83	8	no	—
180	#106	—	166	4	no	—
181	C₂⁴	D₆₆₄	1	1	yes	characteristic
182	C₂²×C₄	—	332	4	no	—
183	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
184	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
187	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
188	C₂²×C₄	#13050579	1	2	yes	—
189	C₂²×C₄	#13050579	1	2	yes	—
190	C₂²×C₄	Dic₃₃₂	1	1	yes	characteristic
191	C₂²×C₄	Dic₃₃₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
193	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
194	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
195	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
196	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
197	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
201	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
202	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
203	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
204	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
205	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
206	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
208	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
210	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
211	C₄⋊C₄	—	332	4	no	—
212	C₄⋊C₄	—	166	8	no	—
213	C₂²⋊C₄	—	332	4	no	—
214	C₂²⋊C₄	—	332	4	no	—
215	C₄²	—	332	4	no	—
216	C₄²	—	332	4	no	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	#14581714	1	2	yes	—
220	C₂³	#14581714	1	2	yes	—
221	C₂³	#14581726	1	1	yes	characteristic
222	C₂³	#14581726	1	1	yes	characteristic
223	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
224	C₂×C₄	—	332	8	no	—
225	C₂×C₄	—	664	8	no	—
226	C₂×C₄	C₈₃⋊#95	1	2	yes	—
227	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
228	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
229	C₂×C₄	#14581714	1	2	yes	—
230	C₂×C₄	#14581714	1	2	yes	—
231	C₂×C₄	#14581717	1	2	yes	—
232	C₂×C₄	#14581717	1	2	yes	—
233	C₂×C₄	#14581717	1	2	yes	—
234	C₂×C₄	#14581717	1	2	yes	—
235	C₂×C₄	#14581724	1	2	yes	—
236	C₂×C₄	#14581724	1	1	yes	characteristic
237	C₂×C₄	#14581724	1	1	yes	characteristic
238	C₂×C₄	#14581726	1	1	yes	characteristic
239	C₂×C₄	#14581726	1	1	yes	characteristic
240	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
242	C₂×C₄	C₂×#13050579	1	2	yes	—
243	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
244	C₈	C₈₃⋊#95	1	1	yes	characteristic
245	C₈	C₈₃⋊#95	1	2	yes	—
246	C₈	C₈₃⋊#95	1	1	yes	characteristic
247	C₈	C₂×C₄×Dic₈₃	1	1	yes	characteristic
248	C₈	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
249	C₈	C₂×Dic₈₃⋊C₄	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	#19127464	1	2	yes	—
256	C₂²	#19127464	1	1	yes	characteristic
257	C₂²	#19127464	1	1	yes	characteristic
258	C₂²	C₂×#14581714	1	2	yes	—
259	C₂²	C₂×#14581726	1	1	yes	characteristic
260	C₄	—	664	8	no	—
261	C₄	#19127464	1	2	yes	—
262	C₄	#19127464	1	1	yes	characteristic
263	C₄	#19127464	1	1	yes	characteristic
264	C₄	C₂×C₈₃⋊#95	1	1	yes	characteristic
265	C₄	C₂×#14581717	1	2	yes	—
266	C₄	C₂×#14581724	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#24905565	1	2	yes	direct product factor
272	C₂	C₂×#19127464	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581724
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂²×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	#340
prime radical	C₂⁴×C₈₃	D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂²×M₃₂	D₈₃
hypocenter	C₈₃	C₂×#1384
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #24905565	C₂	direct (N × H)
7. #19127739	C₄	semidirect
10. C₂×#14581724	C₄	semidirect
14. C₂×M₃₂×C₈₃	C₄	semidirect
16. C₂²×C₁₆×C₈₃	C₄	semidirect
27. #14581724	C₂×C₄	semidirect
43. C₂×C₁₆×C₈₃	C₂×C₄	semidirect
135. C₂×C₈₃	#1384	semidirect
140. C₈₃	C₂×#1384	semidirect
145. C₂×M₃₂	Dic₈₃	semidirect
147. C₂²×C₁₆	Dic₈₃	semidirect
176. C₂×C₁₆	C₂×Dic₈₃	semidirect
271. C₂	#24905565	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×#19127739	2
#24905565	2
C₂²×M₃₂×C₈₃	2
C₂×#1384	83

Subgroups — C2×#24905565 #35319313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24905565 #35319313