Subgroups — C₂×#20291682 #28330791

Order 13568.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291682	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16811681	C₂	1	8	yes	maximal, complemented
3	C₂²×#13133291	C₂	1	6	yes	maximal, complemented
4	C₂²×#13133224	C₂	1	1	yes	maximal, characteristic
5	#16811681	C₂²	1	64	yes	complemented
6	C₂×#13133291	C₂²	1	4	yes	—
7	C₂×#13133291	C₂²	1	12	yes	complemented
8	C₂×#13133291	C₂²	1	48	yes	complemented
9	C₂²×#11719910	C₂²	1	12	yes	complemented
10	C₂×#13133224	C₄	1	4	yes	—
11	C₂×#13133224	C₄	1	8	yes	—
12	C₂×#13133224	C₂²	1	4	yes	—
13	C₂×#13133224	C₂²	1	8	yes	—
14	C₂²×#11719917	C₄	1	3	yes	—
15	C₂³×#88322	C₂²	1	3	yes	—
16	C₂²×#41×C₅₃	C₄	1	1	yes	characteristic
17	#13133291	C₂³	1	16	yes	—
18	#13133291	C₂³	1	48	yes	complemented
19	C₂×#11719910	C₂³	1	8	yes	—
20	C₂×#11719910	C₂³	1	48	yes	complemented
21	#13133224	C₂×C₄	1	16	yes	—
22	#13133224	C₂×C₄	1	32	yes	—
23	#13133224	C₂³	1	16	yes	—
24	C₂×#11719917	C₂×C₄	1	12	yes	—
25	C₂×#11719917	C₂×C₄	1	24	yes	—
26	C₂²×#88322	C₂×C₄	1	3	yes	—
27	C₂²×#88322	C₂×C₄	1	12	yes	—
28	C₂²×#88322	C₂×C₄	1	6	yes	—
29	C₂²×#88322	C₂³	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	6	yes	—
31	C₂²×#88322	C₂³	1	12	yes	—
32	C₂×#41×C₅₃	C₈	1	4	yes	—
33	C₂×#41×C₅₃	C₈	1	8	yes	—
34	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
37	C₂²×D₄×C₅₃	C₈	1	3	yes	—
38	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
39	C₂⁴×C₅₃	C₂×C₈	1	3	yes	—
40	#41×C₅₃	C₂×C₈	1	16	yes	—
41	#41×C₅₃	C₂×C₈	1	32	yes	—
42	#41×C₅₃	C₂²×C₄	1	16	yes	—
43	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
44	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×D₄×C₅₃	C₂×C₈	1	12	yes	—
46	C₂×D₄×C₅₃	C₂×C₈	1	24	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
53	#11719917	C₂²×C₄	1	48	yes	—
54	C₂×#88322	C₂²×C₄	1	2	yes	—
55	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₂²×C₄	1	12	yes	—
57	C₂×#88322	C₂²×C₄	1	6	yes	—
58	C₂×#88322	C₂²×C₄	1	12	yes	—
59	C₂×#88322	C₂²×C₄	1	24	yes	—
60	C₂×#88322	C₂⁴	1	1	yes	characteristic
61	C₂×#88322	C₂⁴	1	2	yes	—
62	C₂×#88322	C₂⁴	1	12	yes	—
63	#11719910	C₂⁴	1	16	yes	—
64	C₂³×C₅₃	—	4	6	no	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂³×C₅₃	C₂²×C₈	1	6	yes	—
67	C₂³×C₅₃	C₂²×C₈	1	12	yes	—
68	Q₈×C₅₃	C₂²×C₈	1	16	yes	—
69	D₄×C₅₃	C₂²×C₈	1	48	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
78	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
79	#88322	C₂³×C₄	1	1	yes	characteristic
80	#88322	C₂³×C₄	1	2	yes	—
81	#88322	C₂³×C₄	1	12	yes	—
82	#88322	C₂⁵	1	1	yes	characteristic
83	C₂×#3407	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	2	12	no	—
86	C₂²×C₅₃	—	4	12	no	—
87	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
88	C₂²×C₅₃	C₂³×C₈	1	12	yes	—
89	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₅₃	C₂³×C₈	1	2	yes	—
91	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
92	C₄×C₅₃	C₂³×C₈	1	12	yes	—
93	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
94	C₂²×#131	—	53	1	no	—
95	#3407	—	53	16	no	—
96	C₂×#504	—	53	8	no	—
97	C₂²×M₃₂	—	53	6	no	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	—	2	12	no	—
100	C₂×C₅₃	#3407	1	2	yes	complemented
101	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
102	C₂²×#41	#6157	1	1	yes	characteristic
103	C₂×#131	—	53	4	no	—
104	C₂×#131	—	53	8	no	—
105	C₂×#131	—	53	8	no	—
106	C₂×#131	—	53	4	no	—
107	C₂²×M₁₆	—	53	3	no	—
108	C₂³×C₈	—	53	3	no	—
109	#504	—	53	64	no	—
110	C₂×M₃₂	—	53	4	no	—
111	C₂×M₃₂	—	53	12	no	—
112	C₂×M₃₂	—	53	48	no	—
113	C₂²×C₁₆	—	53	12	no	—
114	C₅₃	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
115	C₂×#41	#88324	1	4	yes	—
116	C₂×#41	#88324	1	8	yes	—
117	C₂×#41	C₂×#6157	1	8	yes	—
118	C₂×#41	C₂×#6157	1	4	yes	—
119	C₂²×Q₈	#88324	1	1	yes	characteristic
120	C₂²×D₄	#88324	1	3	yes	—
121	C₂³×C₄	C₂×#6157	1	3	yes	—
122	#131	—	53	16	no	—
123	#131	—	53	32	no	—
124	#131	—	53	16	no	—
125	C₂×M₁₆	—	53	24	no	—
126	C₂×M₁₆	—	53	12	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	53	3	no	—
129	C₂²×C₈	—	53	12	no	—
130	C₂²×C₈	—	53	6	no	—
131	C₂²×C₈	—	53	6	no	—
132	C₂²×C₈	—	53	12	no	—
133	M₃₂	—	53	16	no	—
134	M₃₂	—	53	48	no	—
135	C₂×C₁₆	—	53	8	no	—
136	C₂×C₁₆	—	53	48	no	—
137	C₂⁴	C₂×#88324	1	3	yes	—
138	#41	C₂×#88324	1	32	yes	—
139	#41	C₂×#88324	1	16	yes	—
140	#41	C₂²×#6157	1	16	yes	—
141	C₂×Q₈	C₂×#88324	1	8	yes	—
142	C₂×Q₈	C₂×#88324	1	4	yes	—
143	C₂×D₄	C₂×#88324	1	12	yes	—
144	C₂×D₄	C₂×#88324	1	24	yes	—
145	C₂²×C₄	C₂×#88324	1	3	yes	—
146	C₂²×C₄	C₂×#88324	1	6	yes	—
147	C₂²×C₄	C₂×#88324	1	12	yes	—
148	C₂²×C₄	C₂²×#6157	1	12	yes	—
149	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
150	C₂²×C₄	C₂²×#6157	1	6	yes	—
151	M₁₆	—	53	48	no	—
152	C₂×C₈	—	53	12	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	12	no	—
156	C₂×C₈	—	53	12	no	—
157	C₂×C₈	—	53	24	no	—
158	C₂×C₈	—	53	1	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	6	no	—
161	C₁₆	—	53	16	no	—
162	C₂³	—	4	6	no	—
163	C₂³	C₂²×#88324	1	1	yes	characteristic
164	C₂³	C₂²×#88324	1	6	yes	—
165	C₂³	C₂²×#88324	1	12	yes	—
166	Q₈	C₂²×#88324	1	16	yes	—
167	D₄	C₂²×#88324	1	48	yes	—
168	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
169	C₂×C₄	C₂²×#88324	1	2	yes	—
170	C₂×C₄	C₂²×#88324	1	12	yes	—
171	C₂×C₄	C₂²×#88324	1	6	yes	—
172	C₂×C₄	C₂²×#88324	1	24	yes	—
173	C₂×C₄	C₂²×#88324	1	12	yes	—
174	C₂×C₄	C₂³×#6157	1	2	yes	—
175	C₂×C₄	C₂³×#6157	1	12	yes	—
176	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
177	C₈	—	53	2	no	—
178	C₈	—	53	1	no	—
179	C₈	—	53	12	no	—
180	C₈	—	53	1	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×#88324	1	2	yes	—
185	C₂²	C₂³×#88324	1	12	yes	—
186	C₂²	C₂³×#88324	1	1	yes	characteristic
187	C₄	C₂³×#88324	1	2	yes	—
188	C₄	C₂³×#88324	1	1	yes	characteristic
189	C₄	C₂³×#88324	1	12	yes	—
190	C₄	C₂⁴×#6157	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	#20291682	1	2	yes	direct product factor
194	C₂	C₂⁴×#88324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#6157
Fitting, prime radical	C₂²×#41×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
cosocle	#88322	C₂⁵
hypercenter	C₂²×#41	#6157
hypocenter	C₅₃	C₂×#3407
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #20291682	C₂	direct (N × H)
2. C₂×#16811681	C₂	semidirect
3. C₂²×#13133291	C₂	semidirect
5. #16811681	C₂²	semidirect
7. C₂×#13133291	C₂²	semidirect
8. C₂×#13133291	C₂²	semidirect
9. C₂²×#11719910	C₂²	semidirect
18. #13133291	C₂³	semidirect
20. C₂×#11719910	C₂³	semidirect
100. C₂×C₅₃	#3407	semidirect
114. C₅₃	C₂×#3407	semidirect
193. C₂	#20291682	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291682	2
C₂×#16811681	2
C₂²×#13133291	2
C₂²×#13133224	2
C₂×#3407	53

Subgroups — C2×#20291682 #28330791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291682 #28330791