Subgroups — C₂×#24909277 #35557529

Order 21248.

538 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127658	C₂	1	1	yes	maximal, characteristic
2	#24909277	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19127521	C₂	1	2	yes	maximal, complemented
4	C₂×#19127482	C₂	1	2	yes	maximal
5	C₂×#19127479	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#347×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13050573	C₂²	1	1	yes	characteristic
8	C₂²×C₈×D₈₃	C₄	1	1	yes	characteristic
9	C₂×#14581749	C₄	1	1	yes	characteristic
10	C₂×#14581732	C₂²	1	2	yes	—
11	#19127658	C₄	1	4	yes	—
12	#19127658	C₂²	1	4	yes	—
13	C₂×#14581702	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
16	#19127521	C₂²	1	8	yes	complemented
17	#19127482	C₂²	1	8	yes	—
18	#19127479	C₂²	1	4	yes	complemented
19	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
20	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
21	#347×C₈₃	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#13050573	—	2	2	no	—
24	C₂×#13050573	—	2	2	no	—
25	C₂×#13050573	C₂×C₄	1	2	yes	—
26	C₂×#13050573	C₂³	1	2	yes	—
27	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₈₃	—	2	2	no	—
30	C₂×C₈×D₈₃	—	2	2	no	—
31	C₂×C₈×D₈₃	C₂×C₄	1	4	yes	—
32	#14581749	—	2	2	no	—
33	#14581749	C₂×C₄	1	4	yes	—
34	C₂×Dic₈₃⋊C₄	—	2	1	no	—
35	C₂×(C₂×D₈₃)⋊C₄	—	2	1	no	—
36	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
37	#14581732	—	2	2	no	—
38	#14581732	C₂³	1	4	yes	—
39	C₂×C₈₃⋊C₁₆	—	2	2	no	—
40	#14581702	C₂×C₄	1	4	yes	—
41	C₄×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
42	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
43	C₈×Dic₈₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₈₃	—	2	2	no	—
48	M₃₂×C₈₃	C₂³	1	4	yes	—
49	C₂×C₁₆×C₈₃	—	2	2	no	—
50	#97×C₈₃	C₂×C₄	1	2	yes	—
51	#97×C₈₃	C₂³	1	2	yes	—
52	C₂³×D₈₃	M₁₆	1	1	yes	characteristic
53	C₂²×Dic₈₃	—	2	1	no	—
54	C₂²×Dic₈₃	M₁₆	1	1	yes	characteristic
55	C₂×C₄×D₈₃	—	2	1	no	—
56	C₂×C₄×D₈₃	—	2	1	no	—
57	C₂×C₄×D₈₃	—	2	1	no	—
58	C₂×C₄×D₈₃	—	2	1	no	—
59	C₂×C₄×D₈₃	M₁₆	1	2	yes	—
60	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
61	#13050573	—	2	4	no	—
62	#13050573	—	2	4	no	—
63	C₂×C₈₃⋊C₈	—	2	1	no	—
64	C₂×C₈₃⋊C₈	—	2	1	no	—
65	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
66	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
67	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
69	C₈×D₈₃	—	2	8	no	—
70	Dic₈₃⋊C₄	—	2	4	no	—
71	(C₂×D₈₃)⋊C₄	—	2	4	no	—
72	C₄×Dic₈₃	—	2	1	no	—
73	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
74	C₈₃⋊C₁₆	—	2	4	no	—
75	C₂²×C₄×C₈₃	—	2	1	no	—
76	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	—	2	1	no	—
78	C₂×C₈×C₈₃	—	2	1	no	—
79	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	#41	1	2	yes	—
82	C₂×C₈×C₈₃	#41	1	2	yes	—
83	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
84	C₄²×C₈₃	—	2	1	no	—
85	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
86	C₁₆×C₈₃	—	2	4	no	—
87	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×D₈₃	—	2	1	no	—
89	C₂²×D₈₃	—	2	1	no	—
90	C₂²×D₈₃	—	4	2	no	—
91	C₂²×D₈₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₈₃	—	2	2	no	—
93	C₂×C₄×C₈₃	—	4	2	no	—
94	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₈₃	—	2	1	no	—
101	C₂×Dic₈₃	—	2	1	no	—
102	C₂×Dic₈₃	—	2	2	no	—
103	C₂×Dic₈₃	—	4	2	no	—
104	C₂×Dic₈₃	C₂×M₁₆	1	2	yes	—
105	C₄×D₈₃	—	2	2	no	—
106	C₄×D₈₃	—	2	2	no	—
107	C₄×D₈₃	—	2	2	no	—
108	C₄×D₈₃	—	2	2	no	—
109	C₈×C₈₃	—	2	2	no	—
110	C₈×C₈₃	#117	1	2	yes	—
111	C₈×C₈₃	C₂×#41	1	2	yes	—
112	C₈₃⋊C₈	—	2	2	no	—
113	C₈₃⋊C₈	C₂×M₁₆	1	4	yes	—
114	C₂²×C₈₃	—	2	2	no	—
115	C₂²×C₈₃	#432	1	1	yes	characteristic
116	C₂²×C₈₃	#432	1	1	yes	characteristic
117	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
118	C₂×D₈₃	—	2	2	no	—
119	C₂×D₈₃	—	4	2	no	—
120	C₂×D₈₃	—	4	2	no	—
121	C₂×D₈₃	—	4	2	no	—
122	C₄×C₈₃	—	4	2	no	—
123	C₄×C₈₃	#432	1	1	yes	characteristic
124	C₄×C₈₃	#432	1	1	yes	characteristic
125	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
126	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₈₃	—	2	2	no	—
128	Dic₈₃	—	4	2	no	—
129	C₂×#2166	—	83	1	no	maximal, Sylow
130	C₂×C₈₃	—	2	1	no	—
131	C₂×C₈₃	—	2	1	no	—
132	C₂×C₈₃	#2166	1	2	yes	complemented
133	C₂×C₈₃	C₂×#432	1	1	yes	characteristic
134	D₈₃	—	4	2	no	—
135	C₂×#405	—	83	1	no	—
136	#2166	—	83	8	no	—
137	C₂×#364	—	83	2	no	—
138	C₂×#349	—	83	2	no	—
139	C₂×#347	—	83	1	no	—
140	C₂×#347	D₈₃	1	1	yes	characteristic, complemented
141	C₈₃	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
142	C₂²×M₁₆	—	83	1	no	—
143	C₂³×C₈	—	83	1	no	—
144	C₂×#117	—	83	1	no	—
145	C₂×M₃₂	—	83	2	no	—
146	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
147	#405	—	83	4	no	—
148	#405	—	83	4	no	—
149	C₂×#97	—	83	1	no	—
150	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
151	C₂×C₄×C₈	—	83	1	no	—
152	C₂×C₄×C₈	—	83	1	no	—
153	#364	—	83	8	no	—
154	#349	—	83	8	no	—
155	#347	—	83	4	no	—
156	#347	C₂×D₈₃	1	4	yes	complemented
157	C₂³×C₄	—	83	1	no	—
158	C₂×M₁₆	—	83	2	no	—
159	C₂×M₁₆	—	83	2	no	—
160	C₂×M₁₆	—	166	2	no	—
161	C₂×M₁₆	—	166	2	no	—
162	C₂²×C₈	—	83	1	no	—
163	C₂²×C₈	—	83	1	no	—
164	C₂²×C₈	—	83	4	no	—
165	C₂²×C₈	—	166	2	no	—
166	C₂²×C₈	—	166	2	no	—
167	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
169	#117	—	83	4	no	—
170	#117	—	166	2	no	—
171	C₂×C₄⋊C₄	—	166	1	no	—
172	C₂×C₂²⋊C₄	—	166	1	no	—
173	C₂×C₄²	—	83	1	no	—
174	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	—	83	4	no	—
177	M₃₂	—	166	2	no	—
178	M₃₂	C₂²×D₈₃	1	4	yes	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	166	2	no	—
181	#97	—	83	4	no	—
182	#97	C₄×D₈₃	1	2	yes	—
183	#97	C₂²×D₈₃	1	2	yes	—
184	C₄×C₈	—	83	2	no	—
185	C₄×C₈	—	83	2	no	—
186	C₄×C₈	—	83	4	no	—
187	C₂⁴	—	83	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	83	1	no	—
190	C₂²×C₄	—	83	2	no	—
191	C₂²×C₄	—	83	2	no	—
192	C₂²×C₄	—	166	1	no	—
193	C₂²×C₄	—	166	1	no	—
194	C₂²×C₄	—	166	1	no	—
195	C₂²×C₄	—	166	1	no	—
196	C₂²×C₄	—	166	1	no	—
197	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
198	M₁₆	—	166	4	no	—
199	M₁₆	—	166	4	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	83	2	no	—
203	C₂×C₈	—	83	2	no	—
204	C₂×C₈	—	83	2	no	—
205	C₂×C₈	—	83	2	no	—
206	C₂×C₈	—	166	8	no	—
207	C₂×C₈	—	166	1	no	—
208	C₂×C₈	—	166	1	no	—
209	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
211	C₂×C₈	#13050590	1	2	yes	—
212	C₂×C₈	#13050590	1	2	yes	—
213	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
214	C₄⋊C₄	—	166	4	no	—
215	C₂²⋊C₄	—	166	4	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	83	2	no	—
218	C₄²	—	166	1	no	—
219	C₄²	C₂×C₄×D₈₃	1	2	yes	—
220	C₁₆	—	2	4	no	—
221	C₁₆	—	166	4	no	—
222	C₂³	—	332	2	no	—
223	C₂³	—	83	2	no	—
224	C₂³	—	166	1	no	—
225	C₂³	—	166	1	no	—
226	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	332	2	no	—
230	C₂×C₄	—	83	2	no	—
231	C₂×C₄	—	166	2	no	—
232	C₂×C₄	—	166	1	no	—
233	C₂×C₄	—	166	1	no	—
234	C₂×C₄	—	166	2	no	—
235	C₂×C₄	—	166	2	no	—
236	C₂×C₄	—	166	2	no	—
237	C₂×C₄	—	166	2	no	—
238	C₂×C₄	#14581749	1	1	yes	characteristic
239	C₂×C₄	#14581749	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
242	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	83	4	no	—
246	C₈	—	166	2	no	—
247	C₈	#14581749	1	2	yes	—
248	C₈	C₂×#13050590	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	332	2	no	—
251	C₂²	—	332	2	no	—
252	C₂²	—	332	2	no	—
253	C₂²	—	166	2	no	—
254	C₂²	#19127756	1	1	yes	characteristic
255	C₂²	#19127756	1	1	yes	characteristic
256	C₂²	C₂×M₁₆×D₈₃	1	1	yes	characteristic
257	C₄	—	4	2	no	—
258	C₄	—	332	2	no	—
259	C₄	—	166	2	no	—
260	C₄	#19127756	1	1	yes	characteristic
261	C₄	#19127756	1	1	yes	characteristic
262	C₄	C₂×#14581749	1	1	yes	characteristic
263	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	332	2	no	—
267	C₂	#24909277	1	2	yes	direct product factor
268	C₂	C₂×#19127756	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581749
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#347×C₈₃	C₂
socle	C₂²×C₈₃	#432
prime radical	C₂³×D₈₃	M₁₆
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#347	D₈₃
hypocenter	C₈₃	C₂×#2166
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
2. #24909277	C₂	direct (N × H)
3. C₂×#19127521	C₂	semidirect
5. C₂×#19127479	C₂	semidirect
6. C₂×#347×C₈₃	C₂	semidirect
16. #19127521	C₂²	semidirect
18. #19127479	C₂²	semidirect
21. #347×C₈₃	C₂²	semidirect
132. C₂×C₈₃	#2166	semidirect
140. C₂×#347	D₈₃	semidirect
141. C₈₃	C₂×#2166	semidirect
156. #347	C₂×D₈₃	semidirect
267. C₂	#24909277	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×#19127658	2
#24909277	2
C₂×#19127521	2
C₂×#19127482	2
C₂×#19127479	2
C₂×#347×C₈₃	2
C₂×#2166	83

Subgroups — C2×#24909277 #35557529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909277 #35557529