Subgroups — C₂×#24909278 #35557530

Order 21248.

538 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127655	C₂	1	1	yes	maximal, characteristic
2	#24909278	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19127522	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19127482	C₂	1	2	yes	maximal
5	C₂×#19127478	C₂	1	2	yes	maximal, complemented
6	C₂×#364×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	#19127655	C₄	1	4	yes	—
12	#19127655	C₂²	1	4	yes	—
13	C₂×#14581703	C₄	1	1	yes	characteristic
14	C₂×#14581702	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
16	#19127522	C₂²	1	4	yes	complemented
17	#19127482	C₂²	1	8	yes	—
18	#19127478	C₂²	1	8	yes	complemented
19	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	#364×C₈₃	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#13050573	—	2	1	no	—
24	C₂×#13050573	—	2	1	no	—
25	C₂×#13050573	—	2	1	no	—
26	C₂×#13050573	—	2	1	no	—
27	C₂×#13050573	C₂×C₄	1	2	yes	—
28	C₂×#13050573	C₂³	1	2	yes	—
29	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₈₃	—	2	2	no	—
32	C₂×C₈×D₈₃	—	2	2	no	—
33	C₂×C₈×D₈₃	C₂×C₄	1	4	yes	—
34	#14581749	—	2	2	no	—
35	#14581749	C₂×C₄	1	4	yes	—
36	C₂×Dic₈₃⋊C₄	—	2	1	no	—
37	C₂×(C₂×D₈₃)⋊C₄	—	2	1	no	—
38	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
39	#14581732	—	2	2	no	—
40	#14581732	C₂³	1	4	yes	—
41	C₂×C₈₃⋊C₁₆	—	2	2	no	—
42	#14581703	C₂×C₄	1	4	yes	—
43	#14581702	C₂×C₄	1	2	yes	—
44	#14581702	C₂³	1	2	yes	—
45	C₈×Dic₈₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₈₃	—	2	2	no	—
50	M₃₂×C₈₃	C₂³	1	4	yes	—
51	C₂×C₁₆×C₈₃	—	2	2	no	—
52	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₈₃	C₂³	1	2	yes	—
54	C₂³×D₈₃	M₁₆	1	1	yes	characteristic
55	C₂²×Dic₈₃	—	2	1	no	—
56	C₂²×Dic₈₃	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₈₃	—	2	1	no	—
58	C₂×C₄×D₈₃	—	2	1	no	—
59	C₂×C₄×D₈₃	—	2	1	no	—
60	C₂×C₄×D₈₃	—	2	1	no	—
61	C₂×C₄×D₈₃	M₁₆	1	2	yes	—
62	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
63	#13050573	—	2	2	no	—
64	#13050573	—	2	2	no	—
65	#13050573	—	2	2	no	—
66	#13050573	—	2	2	no	—
67	C₂×C₈₃⋊C₈	—	2	1	no	—
68	C₂×C₈₃⋊C₈	—	2	1	no	—
69	C₂×C₈₃⋊C₈	—	2	1	no	—
70	C₂×C₈₃⋊C₈	—	2	1	no	—
71	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
73	C₈×D₈₃	—	2	8	no	—
74	Dic₈₃⋊C₄	—	2	4	no	—
75	(C₂×D₈₃)⋊C₄	—	2	4	no	—
76	C₄×Dic₈₃	—	2	1	no	—
77	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
78	C₈₃⋊C₁₆	—	2	4	no	—
79	C₂²×C₄×C₈₃	—	2	1	no	—
80	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
82	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₈₃	—	2	1	no	—
91	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
92	C₁₆×C₈₃	—	2	4	no	—
93	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂²×D₈₃	—	2	1	no	—
95	C₂²×D₈₃	—	2	1	no	—
96	C₂²×D₈₃	—	4	2	no	—
97	C₂²×D₈₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₈₃	—	2	2	no	—
99	C₂×C₄×C₈₃	—	4	2	no	—
100	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₈₃	—	2	1	no	—
107	C₂×Dic₈₃	—	2	1	no	—
108	C₂×Dic₈₃	—	2	2	no	—
109	C₂×Dic₈₃	—	4	2	no	—
110	C₂×Dic₈₃	C₂×M₁₆	1	2	yes	—
111	C₄×D₈₃	—	2	2	no	—
112	C₄×D₈₃	—	2	2	no	—
113	C₄×D₈₃	—	2	2	no	—
114	C₄×D₈₃	—	2	2	no	—
115	C₈×C₈₃	#117	1	1	yes	characteristic
116	C₈×C₈₃	#117	1	1	yes	characteristic
117	C₈×C₈₃	C₂×M₁₆	1	4	yes	—
118	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
119	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
120	C₈₃⋊C₈	—	2	2	no	—
121	C₈₃⋊C₈	—	2	2	no	—
122	C₂²×C₈₃	—	2	2	no	—
123	C₂²×C₈₃	#432	1	1	yes	characteristic
124	C₂²×C₈₃	#432	1	1	yes	characteristic
125	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
126	C₂×D₈₃	—	2	2	no	—
127	C₂×D₈₃	—	4	2	no	—
128	C₂×D₈₃	—	4	2	no	—
129	C₂×D₈₃	—	4	2	no	—
130	C₄×C₈₃	—	4	2	no	—
131	C₄×C₈₃	#432	1	1	yes	characteristic
132	C₄×C₈₃	#432	1	1	yes	characteristic
133	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
134	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
135	Dic₈₃	—	2	2	no	—
136	Dic₈₃	—	4	2	no	—
137	C₂×#2166	—	83	1	no	maximal, Sylow
138	C₂×C₈₃	—	2	1	no	—
139	C₂×C₈₃	—	2	1	no	—
140	C₂×C₈₃	#2166	1	2	yes	complemented
141	C₂×C₈₃	C₂×#432	1	1	yes	characteristic
142	D₈₃	—	4	2	no	—
143	C₂×#405	—	83	1	no	—
144	#2166	—	83	8	no	—
145	C₂×#364	—	83	1	no	—
146	C₂×#364	D₈₃	1	1	yes	characteristic, complemented
147	C₂×#349	—	83	2	no	—
148	C₂×#347	—	83	2	no	—
149	C₈₃	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	—	83	1	no	—
151	C₂³×C₈	—	83	1	no	—
152	C₂×#117	—	83	1	no	—
153	C₂×M₃₂	—	83	2	no	—
154	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
155	#405	—	83	4	no	—
156	#405	—	83	4	no	—
157	C₂×#97	—	83	1	no	—
158	C₂×#97	—	83	1	no	—
159	C₂×C₄×C₈	—	83	1	no	—
160	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
161	#364	—	83	4	no	—
162	#364	C₂×D₈₃	1	4	yes	complemented
163	#349	—	83	8	no	—
164	#347	—	83	8	no	—
165	C₂³×C₄	—	83	1	no	—
166	C₂×M₁₆	—	83	2	no	—
167	C₂×M₁₆	—	83	2	no	—
168	C₂×M₁₆	—	166	1	no	—
169	C₂×M₁₆	—	166	1	no	—
170	C₂×M₁₆	—	166	1	no	—
171	C₂×M₁₆	—	166	1	no	—
172	C₂²×C₈	—	83	1	no	—
173	C₂²×C₈	—	83	1	no	—
174	C₂²×C₈	—	83	4	no	—
175	C₂²×C₈	—	166	2	no	—
176	C₂²×C₈	—	166	2	no	—
177	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
179	#117	—	83	4	no	—
180	#117	—	166	2	no	—
181	C₂×C₄⋊C₄	—	166	1	no	—
182	C₂×C₂²⋊C₄	—	166	1	no	—
183	C₂×C₄²	—	83	1	no	—
184	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
185	M₃₂	—	2	2	no	—
186	M₃₂	—	83	4	no	—
187	M₃₂	—	166	2	no	—
188	M₃₂	C₂²×D₈₃	1	4	yes	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	166	2	no	—
191	#97	—	83	2	no	—
192	#97	—	83	2	no	—
193	#97	—	83	4	no	—
194	C₄×C₈	—	83	4	no	—
195	C₄×C₈	C₄×D₈₃	1	2	yes	—
196	C₄×C₈	C₂²×D₈₃	1	2	yes	—
197	C₂⁴	—	83	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	83	1	no	—
200	C₂²×C₄	—	83	2	no	—
201	C₂²×C₄	—	83	2	no	—
202	C₂²×C₄	—	166	1	no	—
203	C₂²×C₄	—	166	1	no	—
204	C₂²×C₄	—	166	1	no	—
205	C₂²×C₄	—	166	1	no	—
206	C₂²×C₄	—	166	1	no	—
207	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
208	M₁₆	—	166	2	no	—
209	M₁₆	—	166	2	no	—
210	M₁₆	—	166	2	no	—
211	M₁₆	—	166	2	no	—
212	C₂×C₈	—	83	2	no	—
213	C₂×C₈	—	83	2	no	—
214	C₂×C₈	—	166	1	no	—
215	C₂×C₈	—	166	1	no	—
216	C₂×C₈	—	166	1	no	—
217	C₂×C₈	—	166	1	no	—
218	C₂×C₈	—	166	8	no	—
219	C₂×C₈	#13050573	1	2	yes	—
220	C₂×C₈	#13050573	1	2	yes	—
221	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
223	C₂×C₈	#13050589	1	1	yes	characteristic
224	C₂×C₈	#13050589	1	1	yes	characteristic
225	C₂×C₈	#13050591	1	1	yes	characteristic
226	C₂×C₈	#13050591	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
228	C₄⋊C₄	—	166	4	no	—
229	C₂²⋊C₄	—	166	4	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	83	2	no	—
232	C₄²	—	166	1	no	—
233	C₄²	C₂×C₄×D₈₃	1	2	yes	—
234	C₁₆	—	2	4	no	—
235	C₁₆	—	166	4	no	—
236	C₂³	—	332	2	no	—
237	C₂³	—	83	2	no	—
238	C₂³	—	166	1	no	—
239	C₂³	—	166	1	no	—
240	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	332	2	no	—
244	C₂×C₄	—	83	2	no	—
245	C₂×C₄	—	166	2	no	—
246	C₂×C₄	—	166	1	no	—
247	C₂×C₄	—	166	1	no	—
248	C₂×C₄	—	166	2	no	—
249	C₂×C₄	—	166	2	no	—
250	C₂×C₄	—	166	2	no	—
251	C₂×C₄	—	166	2	no	—
252	C₂×C₄	#14581750	1	1	yes	characteristic
253	C₂×C₄	#14581750	1	1	yes	characteristic
254	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
256	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
258	C₈	—	166	2	no	—
259	C₈	—	166	2	no	—
260	C₈	#14581750	1	1	yes	characteristic
261	C₈	#14581750	1	1	yes	characteristic
262	C₈	C₂×#13050573	1	4	yes	—
263	C₈	C₂×#13050589	1	1	yes	characteristic
264	C₈	C₂×#13050591	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	332	2	no	—
267	C₂²	—	332	2	no	—
268	C₂²	—	332	2	no	—
269	C₂²	—	166	2	no	—
270	C₂²	#19127757	1	1	yes	characteristic
271	C₂²	#19127757	1	1	yes	characteristic
272	C₂²	C₂×M₁₆×D₈₃	1	1	yes	characteristic
273	C₄	—	4	2	no	—
274	C₄	—	332	2	no	—
275	C₄	—	166	2	no	—
276	C₄	#19127757	1	1	yes	characteristic
277	C₄	#19127757	1	1	yes	characteristic
278	C₄	C₂×#14581750	1	1	yes	characteristic
279	C₄	C₂²×#13050573	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	332	2	no	—
283	C₂	#24909278	1	2	yes	direct product factor
284	C₂	C₂×#19127757	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581750
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#364×C₈₃	C₂
socle	C₂²×C₈₃	#432
prime radical	C₂³×D₈₃	M₁₆
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#364	D₈₃
hypocenter	C₈₃	C₂×#2166
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #24909278	C₂	direct (N × H)
3. C₂×#19127522	C₂	semidirect
5. C₂×#19127478	C₂	semidirect
6. C₂×#364×C₈₃	C₂	semidirect
16. #19127522	C₂²	semidirect
18. #19127478	C₂²	semidirect
21. #364×C₈₃	C₂²	semidirect
140. C₂×C₈₃	#2166	semidirect
146. C₂×#364	D₈₃	semidirect
149. C₈₃	C₂×#2166	semidirect
162. #364	C₂×D₈₃	semidirect
283. C₂	#24909278	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×#19127655	2
#24909278	2
C₂×#19127522	2
C₂×#19127482	2
C₂×#19127478	2
C₂×#364×C₈₃	2
C₂×#2166	83

Subgroups — C2×#24909278 #35557530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909278 #35557530