Subgroups — C₂×#20273474 #27353094

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	#20273474	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810376	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	2	no	—
6	#16810647	—	2	4	no	—
7	#16810647	C₂²	1	8	yes	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	C₂²	1	4	yes	—
10	#16810376	C₂²	1	16	yes	complemented
11	C₂³×C₈×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
18	C₂×#11719917	—	2	8	no	—
19	C₂×#11719917	—	4	8	no	—
20	C₂²×#88322	—	2	4	no	—
21	C₄⋊#88322	—	2	8	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	C₂³	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	8	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	D₈	1	4	yes	—
53	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	16	no	—
62	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
73	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
75	#11719917	—	4	16	no	—
76	C₂×#88322	—	2	8	no	—
77	C₂×#88322	—	4	8	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	#107	1	4	yes	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	#107	1	8	yes	—
90	C₂×C₄×C₅₃	#107	1	4	yes	—
91	C₂×C₄×C₅₃	#108	1	4	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
99	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
102	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
108	#88322	—	4	8	no	—
109	C₂×#1567	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	8	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	#423	1	2	yes	—
114	C₂²×C₅₃	C₂×#107	1	4	yes	—
115	C₂²×C₅₃	C₂×#107	1	2	yes	—
116	C₂²×C₅₃	C₂×#108	1	2	yes	—
117	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	#423	1	2	yes	—
122	C₄×C₅₃	C₂×#107	1	2	yes	—
123	C₄×C₅₃	C₂×#107	1	4	yes	—
124	C₄×C₅₃	C₂×#108	1	4	yes	—
125	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
126	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂×#423	—	53	2	no	—
129	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
130	#1567	—	53	8	no	—
131	C₂×#335	—	53	4	no	—
132	C₂×C₅₃	—	2	4	no	—
133	C₂×C₅₃	#1567	1	4	yes	complemented
134	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
135	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
136	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
137	C₂²×M₁₆	—	106	2	no	—
138	C₂³×C₈	Dic₅₃	1	2	yes	—
139	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	#423	—	53	8	no	—
141	#423	—	106	4	no	—
142	C₂×C₄⋊C₈	—	53	4	no	—
143	C₂×C₄⋊C₈	—	106	2	no	—
144	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
146	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
149	#335	—	53	16	no	—
150	C₅₃	C₂×#1567	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
152	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
153	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂×M₁₆	—	106	8	no	—
155	C₂×M₁₆	—	212	8	no	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	106	4	no	—
158	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
160	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
161	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
162	C₂×C₄²	Dic₁₀₆	1	2	yes	—
163	C₂×C₄²	D₂₁₂	1	2	yes	—
164	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
168	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₄⋊C₈	—	53	8	no	—
170	C₄⋊C₈	—	106	4	no	—
171	C₄⋊C₈	—	106	8	no	—
172	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
173	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
174	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
175	C₄×C₈	C₂²×D₅₃	1	4	yes	—
176	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	D₄₂₄	1	4	yes	—
180	C₂²×C₄	Dic₂₁₂	1	4	yes	—
181	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
183	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
184	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
188	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
189	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
190	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
193	M₁₆	—	212	16	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	16	no	—
196	C₂×C₈	—	106	8	no	—
197	C₂×C₈	—	212	8	no	—
198	C₂×C₈	#11719917	1	8	yes	complemented
199	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
200	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
201	C₄²	C₄⋊Dic₅₃	1	2	yes	—
202	C₄²	C₄⋊Dic₅₃	1	2	yes	—
203	C₄²	C₄⋊Dic₅₃	1	2	yes	—
204	C₄²	C₂×Dic₁₀₆	1	2	yes	—
205	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
206	C₄²	C₂×D₂₁₂	1	2	yes	—
207	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
208	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
209	C₄²	C₂²×Dic₅₃	1	2	yes	—
210	C₄²	C₂³×D₅₃	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#13133151	1	4	yes	—
214	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
215	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	#13133151	1	8	yes	—
223	C₂×C₄	#13133151	1	4	yes	—
224	C₂×C₄	#13133153	1	4	yes	—
225	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
231	C₂×C₄	C₂×#11719917	1	4	yes	—
232	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
233	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
234	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
235	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
236	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
238	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	212	8	no	—
241	C₈	C₂×#11719917	1	8	yes	complemented
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	#16810647	1	2	yes	—
246	C₂²	C₂×#13133151	1	4	yes	—
247	C₂²	C₂×#13133151	1	2	yes	—
248	C₂²	C₂×#13133153	1	2	yes	—
249	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	#16810647	1	2	yes	—
254	C₄	C₂×#13133151	1	4	yes	—
255	C₄	C₂×#13133151	1	2	yes	—
256	C₄	C₂×#13133153	1	4	yes	—
257	C₄	C₂²×#11719917	1	2	yes	—
258	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
259	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
260	C₂	—	2	4	no	—
261	C₂	#20273474	1	4	yes	direct product factor
262	C₂	C₂×#16810647	1	1	yes	characteristic
263	C₂	C₂²×#13133151	1	1	yes	characteristic
264	C₂	C₂²×#13133153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1567
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20273474	C₂	direct (N × H)
3. C₂×#16810376	C₂	semidirect
10. #16810376	C₂²	semidirect
62. C₂×C₈×C₅₃	M₁₆	semidirect
107. C₈×C₅₃	C₂×M₁₆	semidirect
133. C₂×C₅₃	#1567	semidirect
150. C₅₃	C₂×#1567	semidirect
198. C₂×C₈	#11719917	semidirect
241. C₈	C₂×#11719917	semidirect
261. C₂	#20273474	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×#16810647	2
#20273474	2
C₂×#16810376	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1567	53

Subgroups — C2×#20273474 #27353094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273474 #27353094