Subgroups — C₂×#41×#88324 #28329624

Order 13568.

2566 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#88324	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#88324	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#88324	C₂	1	6	yes	maximal, complemented
4	C₂×#16811639	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×#88324	C₂	1	3	yes	maximal, complemented
6	C₂×#41×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88324	C₂²	1	6	yes	—
8	Q₈×#88324	C₂²	1	16	yes	complemented
9	D₄×#88324	C₂²	1	48	yes	complemented
10	#16811639	C₂²	1	24	yes	complemented
11	C₂×C₄⋊#88324	C₂²	1	6	yes	—
12	C₂×C₂²⋊#88324	C₂²	1	6	yes	—
13	C₂×C₄×#88324	C₂²	1	3	yes	—
14	C₂×C₄×#88324	C₂²	1	6	yes	—
15	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88324	C₂²	1	24	yes	complemented
17	#41×Dic₅₃	C₄	1	8	yes	—
18	#41×Dic₅₃	C₂²	1	8	yes	—
19	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×D₄×Dic₅₃	C₄	1	3	yes	—
22	C₂×D₄×Dic₅₃	C₂²	1	3	yes	—
23	C₂×#13133176	C₄	1	3	yes	—
24	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
25	C₂²×#41×C₅₃	C₄	1	1	yes	characteristic
26	C₂²×#88324	—	2	24	no	—
27	C₂²×#88324	C₂³	1	6	yes	—
28	C₂²×#88324	C₂³	1	24	yes	—
29	C₂²×#88324	C₂³	1	2	yes	—
30	C₄⋊#88324	C₂³	1	24	yes	—
31	C₂²⋊#88324	C₂³	1	24	yes	—
32	C₄×#88324	C₂³	1	4	yes	—
33	C₄×#88324	C₂³	1	12	yes	—
34	C₄×#88324	C₂³	1	24	yes	—
35	C₂³×Dic₅₃	C₂×C₄	1	3	yes	—
36	C₂³×Dic₅₃	C₂³	1	3	yes	—
37	Q₈×Dic₅₃	C₂×C₄	1	8	yes	—
38	Q₈×Dic₅₃	C₂×C₄	1	4	yes	—
39	Q₈×Dic₅₃	C₂³	1	4	yes	—
40	D₄×Dic₅₃	C₂×C₄	1	24	yes	—
41	D₄×Dic₅₃	C₂×C₄	1	12	yes	—
42	D₄×Dic₅₃	C₂³	1	12	yes	—
43	#13133176	C₂×C₄	1	24	yes	—
44	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
46	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	6	yes	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
49	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
52	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
53	C₂×#41×C₅₃	C₈	1	16	yes	complemented
54	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
55	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₅₃	C₂×C₄	1	3	yes	—
57	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
58	C₂⁴×C₅₃	C₂²×C₄	1	3	yes	—
59	#41×C₅₃	C₂×C₈	1	64	yes	complemented
60	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	complemented
61	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₅₃	C₂×C₈	1	24	yes	complemented
63	C₂×D₄×C₅₃	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	24	yes	complemented
65	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
68	C₂²×Dic₅₃	—	2	12	no	—
69	C₂²×Dic₅₃	—	2	12	no	—
70	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
71	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
72	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
73	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
74	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
76	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×#88324	—	2	12	no	—
78	C₂×#88324	—	2	48	no	—
79	C₂×#88324	#41	1	8	yes	direct product factor
80	C₂×#88324	C₂⁴	1	12	yes	—
81	C₂×#88324	C₂⁴	1	4	yes	—
82	C₂²⋊Dic₅₃	C₂²×C₄	1	24	yes	—
83	C₄⋊Dic₅₃	C₂²×C₄	1	12	yes	—
84	C₄⋊Dic₅₃	C₂²×C₄	1	12	yes	—
85	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
86	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
87	C₄×Dic₅₃	C₂²×C₄	1	6	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
89	C₄×Dic₅₃	C₂⁴	1	2	yes	—
90	C₄×Dic₅₃	C₂⁴	1	6	yes	—
91	C₂³×C₅₃	—	2	6	no	—
92	C₂³×C₅₃	—	2	6	no	—
93	C₂³×C₅₃	C₂²×C₈	1	12	yes	—
94	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
95	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₅₃	C₂²×C₈	1	16	yes	complemented
97	D₄×C₅₃	C₂²×C₈	1	48	yes	complemented
98	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
100	C₂×C₄×C₅₃	C₂²×C₈	1	48	yes	complemented
101	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
103	C₂×Dic₅₃	—	2	6	no	—
104	C₂×Dic₅₃	—	2	6	no	—
105	C₂×Dic₅₃	—	2	12	no	—
106	C₂×Dic₅₃	—	2	24	no	—
107	C₂×Dic₅₃	—	2	12	no	—
108	C₂×Dic₅₃	#131	1	4	yes	—
109	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
110	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
111	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
113	C₂×Dic₅₃	C₂×#41	1	4	yes	—
114	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
115	#88324	—	2	12	no	—
116	#88324	C₂×#41	1	8	yes	direct product factor
117	C₂×C₈×#41	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	6	no	—
119	C₂²×C₅₃	—	2	24	no	—
120	C₂²×C₅₃	—	2	6	no	—
121	C₂²×C₅₃	C₄×#41	1	2	yes	—
122	C₂²×C₅₃	C₂³×C₈	1	12	yes	—
123	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
124	C₂²×C₅₃	C₂×#131	1	2	yes	—
125	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂³×C₈	1	4	yes	—
127	C₄×C₅₃	C₂³×C₈	1	12	yes	—
128	Dic₅₃	—	2	6	no	—
129	Dic₅₃	—	2	6	no	—
130	Dic₅₃	C₄×#41	1	2	yes	—
131	Dic₅₃	C₂×#131	1	4	yes	—
132	Dic₅₃	C₂²×#41	1	2	yes	—
133	C₂×C₄×#41	—	53	1	no	—
134	C₈×#41	—	53	16	no	—
135	C₂×C₈×Q₈	—	53	2	no	—
136	C₂×C₈×D₄	—	53	6	no	—
137	C₂×#431	—	53	3	no	—
138	C₂²×C₄×C₈	—	53	3	no	—
139	C₂×C₅₃	—	2	12	no	—
140	C₂×C₅₃	C₈×#41	1	4	yes	complemented
141	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
142	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
144	C₂²×#41	#6157	1	1	yes	characteristic
145	C₂³×C₈	—	53	6	no	—
146	C₄×#41	—	53	8	no	—
147	C₄×#41	—	53	8	no	—
148	C₂×C₄×Q₈	—	53	1	no	—
149	C₂×C₄×Q₈	—	53	1	no	—
150	C₂×C₄×D₄	—	53	3	no	—
151	C₂×C₄×D₄	—	53	3	no	—
152	C₂×#117	—	53	3	no	—
153	C₂²×C₄²	—	53	3	no	—
154	C₈×Q₈	—	53	16	no	—
155	C₈×D₄	—	53	48	no	—
156	#431	—	53	24	no	—
157	C₂×C₄⋊C₈	—	53	6	no	—
158	C₂×C₂²⋊C₈	—	53	6	no	—
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	—	53	3	no	—
161	C₂×C₄×C₈	—	53	6	no	—
162	C₂×C₄×C₈	—	53	24	no	—
163	C₅₃	C₂×C₈×#41	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	#88324	1	16	yes	direct product factor
165	C₂×#41	C₂×#6157	1	8	yes	—
166	C₂²×Q₈	C₂×#6157	1	1	yes	characteristic
167	C₂²×D₄	C₂×#6157	1	3	yes	—
168	C₂³×C₄	—	53	3	no	—
169	C₂³×C₄	—	53	3	no	—
170	C₂³×C₄	C₂×#6157	1	3	yes	—
171	C₂²×C₈	—	53	2	no	—
172	C₂²×C₈	—	53	6	no	—
173	C₂²×C₈	—	53	24	no	—
174	C₂²×C₈	—	106	24	no	—
175	C₄×Q₈	—	53	8	no	—
176	C₄×Q₈	—	53	4	no	—
177	C₄×Q₈	—	53	4	no	—
178	C₄×D₄	—	53	24	no	—
179	C₄×D₄	—	53	12	no	—
180	C₄×D₄	—	53	12	no	—
181	#117	—	53	24	no	—
182	C₂×C₄⋊C₄	—	53	3	no	—
183	C₂×C₄⋊C₄	—	53	3	no	—
184	C₂×C₂²⋊C₄	—	53	6	no	—
185	C₂×C₄²	—	53	3	no	—
186	C₂×C₄²	—	53	3	no	—
187	C₂×C₄²	—	53	12	no	—
188	C₂×C₄²	—	53	12	no	—
189	C₂×C₄²	—	53	3	no	—
190	C₂×C₄²	—	53	1	no	—
191	C₄⋊C₈	—	53	24	no	—
192	C₂²⋊C₈	—	53	24	no	—
193	C₄×C₈	—	53	4	no	—
194	C₄×C₈	—	53	12	no	—
195	C₄×C₈	—	53	24	no	—
196	C₂⁴	C₂²×#6157	1	3	yes	—
197	#41	C₂×#88324	1	64	yes	direct product factor
198	C₂×Q₈	C₂×#88324	1	8	yes	complemented
199	C₂×Q₈	C₂²×#6157	1	4	yes	—
200	C₂×D₄	C₂×#88324	1	24	yes	complemented
201	C₂×D₄	C₂²×#6157	1	12	yes	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	3	no	—
205	C₂²×C₄	—	53	3	no	—
206	C₂²×C₄	—	53	6	no	—
207	C₂²×C₄	—	53	12	no	—
208	C₂²×C₄	—	53	6	no	—
209	C₂²×C₄	—	106	12	no	—
210	C₂²×C₄	—	106	12	no	—
211	C₂²×C₄	C₂×#88324	1	24	yes	complemented
212	C₂²×C₄	C₂²×#6157	1	3	yes	—
213	C₂²×C₄	C₂²×#6157	1	12	yes	—
214	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
215	C₂×C₈	—	53	8	no	—
216	C₂×C₈	—	53	4	no	—
217	C₂×C₈	—	53	12	no	—
218	C₂×C₈	—	106	48	no	—
219	C₂×C₈	—	106	12	no	—
220	C₄⋊C₄	—	53	12	no	—
221	C₄⋊C₄	—	53	12	no	—
222	C₂²⋊C₄	—	53	24	no	—
223	C₄²	—	53	2	no	—
224	C₄²	—	53	6	no	—
225	C₄²	—	53	2	no	—
226	C₄²	—	53	12	no	—
227	C₄²	—	53	12	no	—
228	C₄²	—	53	6	no	—
229	C₂³	—	2	6	no	—
230	C₂³	—	2	6	no	—
231	C₂³	C₂²×#88324	1	12	yes	—
232	C₂³	C₂³×#6157	1	6	yes	—
233	C₂³	C₂³×#6157	1	1	yes	characteristic
234	Q₈	C₂²×#88324	1	16	yes	complemented
235	D₄	C₂²×#88324	1	48	yes	complemented
236	C₂×C₄	—	53	4	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	1	no	—
241	C₂×C₄	—	53	6	no	—
242	C₂×C₄	—	53	6	no	—
243	C₂×C₄	—	106	12	no	—
244	C₂×C₄	—	106	12	no	—
245	C₂×C₄	—	106	24	no	—
246	C₂×C₄	—	106	6	no	—
247	C₂×C₄	—	106	6	no	—
248	C₂×C₄	C₂²×#88324	1	4	yes	—
249	C₂×C₄	C₂²×#88324	1	12	yes	—
250	C₂×C₄	C₂²×#88324	1	48	yes	complemented
251	C₂×C₄	C₂³×#6157	1	2	yes	—
252	C₂×C₄	C₂³×#6157	1	6	yes	—
253	C₈	—	53	8	no	—
254	C₈	—	106	12	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	2	6	no	—
257	C₂²	—	2	24	no	—
258	C₂²	#41×#6157	1	2	yes	—
259	C₂²	C₂³×#88324	1	12	yes	—
260	C₂²	C₂³×#88324	1	2	yes	—
261	C₂²	D₅₃⋊#131	1	2	yes	—
262	C₂²	C₂⁴×#6157	1	1	yes	characteristic
263	C₄	—	53	2	no	—
264	C₄	—	53	2	no	—
265	C₄	—	53	4	no	—
266	C₄	—	106	6	no	—
267	C₄	—	106	6	no	—
268	C₄	C₂³×#88324	1	4	yes	—
269	C₄	C₂³×#88324	1	12	yes	—
270	C₂	—	2	12	no	—
271	C₂	#41×#88324	1	4	yes	direct product factor
272	C₂	C₂×#41×#6157	1	1	yes	characteristic
273	C₂	C₂⁴×#88324	1	1	yes	characteristic
274	C₂	C₂×D₅₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#6157
Fitting, prime radical	C₂²×#41×C₅₃	C₄
socle	C₂³×C₅₃	C₂³×C₄
cosocle	C₂×Dic₅₃	C₂⁵
hypercenter	C₂²×#41	#6157
hypocenter	C₅₃	C₂×C₈×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #41×#88324	C₂	direct (N × H)
2. C₂×Q₈×#88324	C₂	semidirect
3. C₂×D₄×#88324	C₂	semidirect
4. C₂×#16811639	C₂	semidirect
5. C₂²×C₄×#88324	C₂	semidirect
8. Q₈×#88324	C₂²	semidirect
9. D₄×#88324	C₂²	semidirect
10. #16811639	C₂²	semidirect
16. C₂×C₄×#88324	C₂²	semidirect
53. C₂×#41×C₅₃	C₈	semidirect
59. #41×C₅₃	C₂×C₈	semidirect
60. C₂×Q₈×C₅₃	C₂×C₈	semidirect
62. C₂×D₄×C₅₃	C₂×C₈	semidirect
64. C₂²×C₄×C₅₃	C₂×C₈	semidirect
79. C₂×#88324	#41	direct (N × H)
96. Q₈×C₅₃	C₂²×C₈	semidirect
97. D₄×C₅₃	C₂²×C₈	semidirect
100. C₂×C₄×C₅₃	C₂²×C₈	semidirect
116. #88324	C₂×#41	direct (N × H)
140. C₂×C₅₃	C₈×#41	semidirect
163. C₅₃	C₂×C₈×#41	semidirect
164. C₂×#41	#88324	direct (N × H)
197. #41	C₂×#88324	direct (N × H)
198. C₂×Q₈	C₂×#88324	semidirect
200. C₂×D₄	C₂×#88324	semidirect
211. C₂²×C₄	C₂×#88324	semidirect
234. Q₈	C₂²×#88324	semidirect
235. D₄	C₂²×#88324	semidirect
250. C₂×C₄	C₂²×#88324	semidirect
271. C₂	#41×#88324	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×#88324	2
C₂×Q₈×#88324	2
C₂×D₄×#88324	2
C₂×#16811639	2
C₂²×C₄×#88324	2
C₂×#41×Dic₅₃	2
C₂×C₈×#41	53

Subgroups — C2×#41×#88324 #28329624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×#88324 #28329624