Subgroups — C₂×#20290938 #28327928

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133291	C₂	1	2	yes	maximal
2	#20290938	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811516	C₂	1	4	yes	maximal, complemented
4	C₂×#16810581	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₄	1	4	yes	complemented
6	C₂×#13133291	C₄	1	8	yes	complemented
7	C₂×#13133291	C₂²	1	4	yes	—
8	C₂×#13133291	C₂²	1	8	yes	—
9	C₂²×#11719910	C₄	1	4	yes	complemented
10	#16811516	C₂²	1	16	yes	complemented
11	C₂³×#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133176	C₄	1	1	yes	characteristic
13	#16810581	C₄	1	4	yes	—
14	#16810581	C₂²	1	4	yes	—
15	C₂×#13133127	C₂²	1	2	yes	—
16	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
17	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
18	#13133291	C₂×C₄	1	8	yes	—
19	#13133291	C₂×C₄	1	16	yes	complemented
20	#13133291	C₂³	1	8	yes	—
21	C₂×#11719910	C₂×C₄	1	8	yes	—
22	C₂×#11719910	C₂×C₄	1	16	yes	complemented
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	2	yes	—
30	#13133176	—	2	2	no	—
31	#13133176	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	#13133127	C₂×C₄	1	4	yes	—
36	#13133127	C₂³	1	4	yes	—
37	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
38	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×Dic₅₃	—	2	2	no	—
53	M₁₆×C₅₃	—	2	8	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²⋊Dic₅₃	—	2	4	no	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
61	C₂×#88322	C₄²	1	4	yes	—
62	C₂×#88322	C₄²	1	2	yes	—
63	C₂×#88322	C₄²	1	4	yes	—
64	C₂×#88322	C₄²	1	1	yes	characteristic
65	C₂×#88322	C₄²	1	1	yes	characteristic
66	C₂×#88322	C₂²×C₄	1	2	yes	—
67	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
68	C₂×#88322	C₂²×C₄	1	2	yes	—
69	C₂×#88322	C₂²×C₄	1	4	yes	—
70	C₂×#88322	C₂²×C₄	1	2	yes	—
71	C₂×#88322	C₂²×C₄	1	4	yes	—
72	C₂×#88322	C₂⁴	1	1	yes	characteristic
73	#11719910	C₂²×C₄	1	16	yes	—
74	C₂³×C₅₃	—	4	2	no	—
75	C₂³×C₅₃	#97	1	2	yes	—
76	C₂³×C₅₃	#97	1	2	yes	—
77	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	#97	1	2	yes	—
80	C₂×C₄×C₅₃	#97	1	4	yes	—
81	C₂×C₄×C₅₃	#97	1	4	yes	—
82	C₂×C₄×C₅₃	#97	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₅₃	—	2	4	no	—
93	C₂×Dic₅₃	—	4	4	no	—
94	C₈×C₅₃	—	2	4	no	—
95	#88322	C₂×C₄²	1	2	yes	—
96	#88322	C₂×C₄²	1	2	yes	—
97	#88322	C₂×C₄²	1	2	yes	—
98	#88322	C₂³×C₄	1	2	yes	—
99	C₂×#2110	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	4	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	—	4	2	no	—
104	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×#97	1	4	yes	—
106	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
107	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
108	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#97	1	4	yes	—
110	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
113	Dic₅₃	—	4	4	no	—
114	C₂²×M₃₂	—	53	2	no	—
115	C₂×#405	—	53	1	no	—
116	#2110	—	53	8	no	—
117	C₂×#347	—	53	4	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	2	4	no	—
121	C₂×C₅₃	#2110	1	2	yes	complemented
122	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
123	C₂²×M₁₆	#6157	1	1	yes	characteristic
124	C₂³×C₈	—	53	1	no	—
125	C₂×#117	—	53	1	no	—
126	C₂×M₃₂	—	53	4	no	—
127	C₂×M₃₂	—	53	8	no	—
128	C₂×M₃₂	—	53	4	no	—
129	C₂×M₃₂	—	53	8	no	—
130	C₂²×C₁₆	—	53	4	no	—
131	#405	—	53	4	no	—
132	#405	—	53	4	no	—
133	C₂×#97	—	53	2	no	—
134	C₂×C₄×C₈	—	53	2	no	—
135	#347	—	53	16	no	—
136	C₅₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	C₂×#6157	1	4	yes	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	—	53	4	no	—
143	C₂²×C₈	—	53	2	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	—	53	2	no	—
147	C₂²×C₈	—	53	2	no	—
148	C₂²×C₈	C₂×#6157	1	2	yes	—
149	#117	—	53	4	no	—
150	#117	—	106	2	no	—
151	C₂×C₄⋊C₄	—	106	1	no	—
152	C₂×C₂²⋊C₄	—	106	1	no	—
153	C₂×C₄²	—	53	2	no	—
154	M₃₂	—	53	8	no	—
155	M₃₂	—	53	8	no	—
156	M₃₂	—	53	16	no	—
157	C₂×C₁₆	—	53	8	no	—
158	C₂×C₁₆	—	53	16	no	—
159	#97	—	53	4	no	—
160	#97	—	53	4	no	—
161	C₄×C₈	—	53	8	no	—
162	C₂⁴	#11719936	1	1	yes	characteristic
163	C₂²×C₄	—	106	2	no	—
164	C₂²×C₄	#11719936	1	1	yes	characteristic
165	C₂²×C₄	#11719936	1	2	yes	—
166	C₂²×C₄	C₄×#6157	1	2	yes	—
167	C₂²×C₄	C₄×#6157	1	2	yes	—
168	C₂²×C₄	#11719940	1	2	yes	—
169	C₂²×C₄	#11719940	1	2	yes	—
170	C₂²×C₄	C₂²×#6157	1	2	yes	—
171	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	4	no	—
178	C₂×C₈	—	53	4	no	—
179	C₂×C₈	—	53	2	no	—
180	C₂×C₈	—	53	4	no	—
181	C₂×C₈	—	53	2	no	—
182	C₂×C₈	—	53	1	no	—
183	C₂×C₈	—	53	1	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	—	53	1	no	—
186	C₂×C₈	—	53	1	no	—
187	C₂×C₈	C₂²×#6157	1	4	yes	—
188	C₄⋊C₄	—	106	4	no	—
189	C₂²⋊C₄	—	106	4	no	—
190	C₄²	—	53	4	no	—
191	C₄²	—	106	2	no	—
192	C₁₆	—	53	16	no	—
193	C₂³	—	4	2	no	—
194	C₂³	#13133272	1	2	yes	—
195	C₂³	#13133272	1	2	yes	—
196	C₂³	C₂×#11719936	1	1	yes	characteristic
197	C₂³	C₂×#11719936	1	2	yes	—
198	C₂×C₄	—	106	4	no	—
199	C₂×C₄	—	212	4	no	—
200	C₂×C₄	#13133272	1	2	yes	—
201	C₂×C₄	#13133272	1	4	yes	—
202	C₂×C₄	#13133272	1	2	yes	—
203	C₂×C₄	#13133272	1	4	yes	—
204	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
205	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719940	1	4	yes	—
210	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
211	C₂×C₄	C₂×#11719936	1	2	yes	—
212	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	—	53	2	no	—
215	C₈	—	53	2	no	—
216	C₈	—	53	2	no	—
217	C₈	—	53	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#13133272	1	1	yes	characteristic
223	C₂²	C₂×#13133272	1	4	yes	—
224	C₂²	C₂×#13133272	1	1	yes	characteristic
225	C₂²	C₂²×#11719936	1	1	yes	characteristic
226	C₄	—	212	4	no	—
227	C₄	C₂×#13133272	1	1	yes	characteristic
228	C₄	C₂×#13133272	1	4	yes	—
229	C₄	C₂×#13133272	1	1	yes	characteristic
230	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
231	C₄	C₂²×#11719940	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	#20290938	1	2	yes	direct product factor
236	C₂	C₂²×#13133272	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂²×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂²×M₁₆	#6157
hypocenter	C₅₃	C₂×#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20290938	C₂	direct (N × H)
3. C₂×#16811516	C₂	semidirect
5. C₂×#13133291	C₄	semidirect
6. C₂×#13133291	C₄	semidirect
9. C₂²×#11719910	C₄	semidirect
10. #16811516	C₂²	semidirect
19. #13133291	C₂×C₄	semidirect
22. C₂×#11719910	C₂×C₄	semidirect
121. C₂×C₅₃	#2110	semidirect
136. C₅₃	C₂×#2110	semidirect
235. C₂	#20290938	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133291	2
#20290938	2
C₂×#16811516	2
C₂×#16810581	2
C₂×#2110	53

Subgroups — C2×#20290938 #28327928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290938 #28327928