Subgroups — C₂×#20274835 #27384239

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	2	yes	maximal
2	#20274835	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810378	C₂	1	4	yes	maximal
4	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	4	yes	—
6	C₂×C₄⋊#88322	—	2	8	no	—
7	C₂×C₄⋊#88322	C₂²	1	8	yes	—
8	#16810378	C₂²	1	16	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#376	C₄	1	4	yes	—
12	C₅₃×#376	C₂²	1	4	yes	—
13	C₂×#95×C₅₃	C₄	1	4	yes	—
14	C₂²×#88322	—	2	8	no	—
15	C₂²×#88322	—	2	16	no	—
16	C₂²×#88322	C₂³	1	8	yes	—
17	C₄⋊#88322	—	2	16	no	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	—	2	8	no	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	D₄	1	4	yes	—
25	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
28	#95×C₅₃	C₂×C₄	1	16	yes	—
29	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	—	2	8	no	—
31	C₂²×C₄×C₅₃	—	2	8	no	—
32	C₂²×C₄×C₅₃	—	2	16	no	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₅₃	#41	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₄²×C₅₃	—	2	8	no	—
42	C₄²×C₅₃	—	2	8	no	—
43	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
44	C₄²×C₅₃	C₂×D₄	1	4	yes	—
45	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂×#88322	—	2	8	no	—
47	C₂×#88322	—	2	16	no	—
48	C₂×#88322	—	4	16	no	—
49	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
50	C₂³×C₅₃	#117	1	4	yes	—
51	C₂³×C₅₃	#131	1	8	yes	—
52	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	8	no	—
54	C₂×C₄×C₅₃	—	2	16	no	—
55	C₂×C₄×C₅₃	—	2	8	no	—
56	C₂×C₄×C₅₃	—	2	16	no	—
57	C₂×C₄×C₅₃	—	4	16	no	—
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₅₃	#117	1	4	yes	—
61	C₂×C₄×C₅₃	#124	1	8	yes	—
62	C₂×C₄×C₅₃	#125	1	8	yes	—
63	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
66	#88322	—	4	8	no	—
67	C₂×#1832	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#382	1	4	yes	—
69	C₂²×C₅₃	#424	1	8	yes	—
70	C₂²×C₅₃	#433	1	16	yes	—
71	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×#117	1	2	yes	—
73	C₂²×C₅₃	C₂×#131	1	4	yes	—
74	C₄×C₅₃	—	2	8	no	—
75	C₄×C₅₃	—	4	8	no	—
76	C₄×C₅₃	#382	1	4	yes	—
77	C₄×C₅₃	C₂×#124	1	2	yes	—
78	C₄×C₅₃	C₂×#125	1	2	yes	—
79	C₂²×C₄⋊C₈	—	53	2	no	—
80	C₂×#376	D₅₃	1	1	yes	characteristic
81	#1832	—	53	8	no	—
82	C₂×#336	—	53	4	no	—
83	C₂×C₅₃	#1832	1	8	yes	complemented
84	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#424	1	2	yes	—
86	C₂×C₅₃	C₂×#433	1	4	yes	—
87	C₂³×C₈	—	53	4	no	—
88	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
89	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	53	8	no	—
91	C₂×C₄⋊C₈	—	106	8	no	—
92	#376	Dic₅₃	1	4	yes	—
93	#376	C₂×D₅₃	1	4	yes	—
94	C₂×#95	Dic₅₃	1	4	yes	—
95	#336	—	53	16	no	—
96	C₅₃	C₂×#1832	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
98	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
99	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂²×C₈	—	53	8	no	—
101	C₂²×C₈	—	106	8	no	—
102	C₂²×C₈	—	106	16	no	—
103	C₂×C₄²	—	2	8	no	—
104	C₂×C₄²	Dic₁₀₆	1	4	yes	—
105	C₂×C₄²	D₂₁₂	1	4	yes	—
106	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
107	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
108	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
109	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
110	C₄⋊C₈	—	106	16	no	—
111	#95	C₂×Dic₅₃	1	16	yes	—
112	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	16	no	—
116	C₂²×C₄	C₄⋊Dic₅₃	1	8	yes	—
117	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
118	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
119	C₂²×C₄	#11719945	1	8	yes	—
120	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
121	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
122	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
124	C₂×C₈	—	106	8	no	—
125	C₂×C₈	—	106	16	no	—
126	C₂×C₈	—	212	16	no	—
127	C₄²	—	2	8	no	—
128	C₄²	—	2	8	no	—
129	C₄²	C₄⋊Dic₅₃	1	8	yes	—
130	C₄²	C₂×Dic₁₀₆	1	4	yes	—
131	C₄²	C₂×D₂₁₂	1	4	yes	—
132	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
133	C₂³	#13133176	1	4	yes	—
134	C₂³	#13133224	1	8	yes	—
135	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	16	no	—
140	C₂×C₄	—	4	16	no	—
141	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
143	C₂×C₄	#13133176	1	4	yes	—
144	C₂×C₄	#13133205	1	8	yes	—
145	C₂×C₄	#13133209	1	8	yes	—
146	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
148	C₂×C₄	C₂×#11719945	1	4	yes	—
149	C₈	—	212	8	no	—
150	C₂²	#16810489	1	4	yes	—
151	C₂²	#16810650	1	8	yes	—
152	C₂²	#16810684	1	16	yes	—
153	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₂²	C₂×#13133176	1	2	yes	—
155	C₂²	C₂×#13133224	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	#16810489	1	4	yes	—
159	C₄	C₂×#13133205	1	2	yes	—
160	C₄	C₂×#13133209	1	2	yes	—
161	C₂	#20274835	1	8	yes	direct product factor
162	C₂	C₂×#16810489	1	1	yes	characteristic
163	C₂	C₂×#16810650	1	2	yes	—
164	C₂	C₂×#16810684	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#376	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#376	D₅₃
hypocenter	C₅₃	C₂×#1832
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #20274835	C₂	direct (N × H)
83. C₂×C₅₃	#1832	semidirect
96. C₅₃	C₂×#1832	semidirect
161. C₂	#20274835	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×C₄⋊#88322	2
#20274835	2
C₂×#16810378	2
C₂×C₅₃×#376	2
C₂×#1832	53

Subgroups — C2×#20274835 #27384239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274835 #27384239