Subgroups — C₄×#19128133 #35557688

Order 21248.

690 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128133	C₂	1	1	yes	maximal, characteristic
2	C₄×#14581814	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₆₆₄	C₂	1	2	yes	maximal, complemented
4	C₄×#14581737	C₂	1	4	yes	maximal, complemented
5	C₄×D₁₃₂₈	C₂	1	2	yes	maximal, complemented
6	#24909247	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909171	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581814	C₂²	1	2	yes	—
10	C₄×#13050590	—	2	2	no	—
11	#19128133	C₄	1	8	yes	direct product factor
12	C₂×Dic₆₆₄	C₂²	1	1	yes	characteristic
13	C₂×#14581737	C₂²	1	2	yes	—
14	C₂×D₁₃₂₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₃₃₂	C₂²	1	4	yes	—
16	C₄×#13050579	—	2	4	no	—
17	C₄×D₆₆₄	C₂²	1	4	yes	—
18	#19127739	C₂²	1	2	yes	—
19	#19127698	—	2	2	no	—
20	#19127529	C₂²	1	2	yes	—
21	#19127527	C₂²	1	2	yes	—
22	#19127506	C₂²	1	4	yes	—
23	#19127503	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
28	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
29	C₂×#13050590	—	2	2	no	—
30	#14581814	—	2	4	no	—
31	#14581814	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₃₃₂	C₂³	1	2	yes	—
33	C₂×#13050579	—	2	2	no	—
34	C₂×D₆₆₄	C₂³	1	2	yes	—
35	C₄×Dic₁₆₆	—	2	4	no	—
36	C₄×D₃₃₂	—	2	4	no	—
37	C₄×C₂²⋊D₈₃	—	4	4	no	—
38	#14581751	—	2	2	no	—
39	#14581749	—	4	2	no	—
40	C₄²×D₈₃	—	4	2	no	—
41	Dic₆₆₄	C₂×C₄	1	4	yes	complemented
42	#14581737	C₂×C₄	1	8	yes	complemented
43	D₁₃₂₈	C₂×C₄	1	4	yes	complemented
44	#14581726	C₂³	1	4	yes	—
45	#14581724	—	2	2	no	—
46	#14581717	—	2	4	no	—
47	#14581714	—	2	4	no	—
48	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₈₃	—	2	2	no	—
52	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
54	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₈₃	D₄	1	2	yes	—
57	C₄×C₈×C₈₃	C₂³	1	2	yes	—
58	#13050590	—	2	8	no	—
59	#13050590	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×D₃₃₂	—	2	2	no	—
62	C₂×C₂²⋊D₈₃	—	4	2	no	—
63	C₂×C₄×D₈₃	—	4	2	no	—
64	C₂×C₄×D₈₃	—	4	4	no	—
65	Dic₃₃₂	—	2	2	no	—
66	Dic₃₃₂	C₂²×C₄	1	4	yes	—
67	#13050579	—	2	8	no	—
68	D₆₆₄	—	2	2	no	—
69	D₆₆₄	C₂²×C₄	1	4	yes	—
70	C₄⋊Dic₈₃	—	2	4	no	—
71	Dic₈₃⋊C₄	—	4	4	no	—
72	C₂²⋊Dic₈₃	—	4	2	no	—
73	(C₂×D₈₃)⋊C₄	—	4	4	no	—
74	C₄×Dic₈₃	—	4	2	no	—
75	C₄×Dic₈₃	—	4	4	no	—
76	C₂²×C₄×C₈₃	D₈	1	2	yes	—
77	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	—	2	2	no	—
79	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₃	#41	1	2	yes	—
83	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
84	C₄²×C₈₃	D₈	1	1	yes	characteristic
85	C₄²×C₈₃	D₈	1	1	yes	characteristic
86	C₄²×C₈₃	C₂×D₄	1	2	yes	—
87	C₁₆×C₈₃	—	2	2	no	—
88	C₁₆×C₈₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₈₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₈₃	—	4	2	no	—
92	C₂×C₄×C₈₃	—	2	4	no	—
93	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₈₃	#135	1	2	yes	—
98	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₈₃	—	4	8	no	—
100	C₂×Dic₈₃	—	4	2	no	—
101	C₂×Dic₈₃	—	4	4	no	—
102	D₃₃₂	—	2	4	no	—
103	D₃₃₂	—	4	2	no	—
104	C₄×D₈₃	—	4	8	no	—
105	C₄×D₈₃	—	8	8	no	—
106	Dic₁₆₆	—	2	4	no	—
107	Dic₁₆₆	—	4	2	no	—
108	C₈×C₈₃	—	2	2	no	—
109	C₈×C₈₃	C₄×D₄	1	2	yes	—
110	C₈×C₈₃	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
112	C₂²×C₈₃	—	2	2	no	—
113	C₂²×C₈₃	C₄×D₈	1	2	yes	—
114	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
115	C₂×D₈₃	—	4	4	no	—
116	C₂×D₈₃	—	8	4	no	—
117	C₄×C₈₃	—	2	2	no	—
118	C₄×C₈₃	C₄×D₈	1	2	yes	—
119	C₄×C₈₃	#508	1	4	yes	complemented
120	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
122	Dic₈₃	—	4	4	no	—
123	Dic₈₃	—	8	4	no	—
124	C₄×#508	—	83	1	no	maximal, Sylow
125	C₂×C₈₃	—	2	2	no	—
126	C₂×C₈₃	C₂×C₄×D₈	1	1	yes	characteristic
127	C₂×C₈₃	C₂×#508	1	2	yes	—
128	D₈₃	—	8	4	no	—
129	C₂×#508	—	83	1	no	—
130	C₄×#135	—	83	2	no	—
131	C₄×Q₃₂	—	83	2	no	—
132	C₄×SD₃₂	—	83	4	no	—
133	C₄×D₁₆	—	83	2	no	—
134	#2158	—	83	1	no	—
135	#2138	—	83	2	no	—
136	C₂×C₄×C₁₆	D₈₃	1	1	yes	characteristic, complemented
137	C₈₃	C₄×#508	1	1	yes	characteristic, complemented, Sylow
138	C₂×#135	—	83	2	no	—
139	C₄×#41	—	166	2	no	—
140	#508	—	83	8	no	—
141	C₂×Q₃₂	—	83	1	no	—
142	C₂×SD₃₂	—	83	2	no	—
143	C₂×D₁₆	—	83	1	no	—
144	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
145	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
146	C₄×Q₁₆	—	83	4	no	—
147	C₄×SD₁₆	—	166	4	no	—
148	C₄×D₈	—	83	4	no	—
149	#427	—	83	2	no	—
150	#416	—	166	2	no	—
151	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
152	#367	—	83	2	no	—
153	#366	—	83	2	no	—
154	#358	—	83	4	no	—
155	#357	—	83	4	no	—
156	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
157	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
158	C₂×#41	—	166	2	no	—
159	#135	—	83	8	no	—
160	#135	—	166	4	no	—
161	C₂×Q₁₆	—	83	2	no	—
162	C₂×SD₁₆	—	166	2	no	—
163	C₂×D₈	—	83	2	no	—
164	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
166	C₄×Q₈	—	166	4	no	—
167	C₄×D₄	—	332	4	no	—
168	C₄×D₄	—	166	4	no	—
169	#117	—	332	2	no	—
170	#117	—	166	2	no	—
171	C₂×C₄²	—	332	2	no	—
172	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
173	Q₃₂	—	83	4	no	—
174	SD₃₂	—	83	8	no	—
175	D₁₆	—	83	4	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	C₄×D₈₃	1	4	yes	complemented
178	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
179	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
180	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
181	#107	—	83	4	no	—
182	#106	—	166	2	no	—
183	#103	—	166	4	no	—
184	#102	—	166	4	no	—
185	C₄×C₈	D₃₃₂	1	2	yes	—
186	C₄×C₈	C₂²×D₈₃	1	2	yes	—
187	#41	—	332	4	no	—
188	#41	—	166	8	no	—
189	C₂×Q₈	—	166	2	no	—
190	C₂×D₄	—	332	2	no	—
191	C₂×D₄	—	166	2	no	—
192	C₂²×C₄	—	332	2	no	—
193	C₂²×C₄	—	332	4	no	—
194	C₂²×C₄	D₆₆₄	1	2	yes	—
195	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
196	Q₁₆	—	83	4	no	—
197	Q₁₆	—	166	2	no	—
198	SD₁₆	—	166	8	no	—
199	D₈	—	83	4	no	—
200	D₈	—	166	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
203	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
204	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
205	C₂×C₈	#13050590	1	2	yes	—
206	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
207	C₄⋊C₄	—	332	4	no	—
208	C₄⋊C₄	—	166	4	no	—
209	C₂²⋊C₄	—	332	2	no	—
210	C₂²⋊C₄	—	332	4	no	—
211	C₄²	—	332	2	no	—
212	C₄²	—	332	4	no	—
213	C₄²	D₆₆₄	1	1	yes	characteristic
214	C₄²	D₆₆₄	1	1	yes	characteristic
215	C₄²	C₂×D₃₃₂	1	2	yes	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×C₄×D₈₃	1	2	yes	—
218	C₁₆	C₂×C₄×D₈₃	1	2	yes	—
219	C₂³	—	332	2	no	—
220	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
221	Q₈	—	332	2	no	—
222	Q₈	—	166	4	no	—
223	D₄	—	332	8	no	—
224	D₄	—	332	2	no	—
225	D₄	—	166	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	332	2	no	—
228	C₂×C₄	—	332	8	no	—
229	C₂×C₄	—	332	4	no	—
230	C₂×C₄	—	664	8	no	—
231	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
232	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
233	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
234	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
235	C₂×C₄	#14581814	1	2	yes	—
236	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	C₄×D₃₃₂	1	2	yes	—
239	C₈	C₂²×C₄×D₈₃	1	1	yes	characteristic
240	C₈	C₂×#13050590	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	332	4	no	—
243	C₂²	—	664	4	no	—
244	C₂²	C₄×D₆₆₄	1	2	yes	—
245	C₂²	C₂²×D₆₆₄	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	332	4	no	—
248	C₄	—	664	4	no	—
249	C₄	C₄×D₆₆₄	1	2	yes	—
250	C₄	#19128133	1	4	yes	direct product factor
251	C₄	C₂×C₄×D₃₃₂	1	1	yes	characteristic
252	C₄	C₂×#14581814	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	664	4	no	—
255	C₂	C₂×C₄×D₆₆₄	1	1	yes	characteristic
256	C₂	C₂×#19128133	1	2	yes	—

	subgroup	G/H
center	C₄²	D₆₆₄
commutator	C₈×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×C₄×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂×#19128133	C₂
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₃
hypocenter	C₈₃	C₄×#508
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14581814	C₂	semidirect
3. C₄×Dic₆₆₄	C₂	semidirect
4. C₄×#14581737	C₂	semidirect
5. C₄×D₁₃₂₈	C₂	semidirect
6. #24909247	C₂	semidirect
7. #24909171	C₂	semidirect
8. C₂×C₄×C₁₆×C₈₃	C₂	semidirect
11. #19128133	C₄	direct (N × H)
31. #14581814	C₂×C₄	semidirect
41. Dic₆₆₄	C₂×C₄	semidirect
42. #14581737	C₂×C₄	semidirect
43. D₁₃₂₈	C₂×C₄	semidirect
52. C₂×C₁₆×C₈₃	C₂×C₄	semidirect
119. C₄×C₈₃	#508	semidirect
136. C₂×C₄×C₁₆	D₈₃	semidirect
137. C₈₃	C₄×#508	semidirect
177. C₂×C₁₆	C₄×D₈₃	semidirect
250. C₄	#19128133	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×#19128133	2
C₄×#14581814	2
C₄×Dic₆₆₄	2
C₄×#14581737	2
C₄×D₁₃₂₈	2
#24909247	2
#24909171	2
C₂×C₄×C₁₆×C₈₃	2
C₄×#508	83

Subgroups — C4×#19128133 #35557688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19128133 #35557688