Subgroups — C₂×C₅₃×#398 #20271328

Order 6784.

692 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
3	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#398	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95×C₅₃	C₂	1	2	yes	maximal
6	C₂×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
18	#95×C₅₃	C₂²	1	8	yes	—
19	#95×C₅₃	C₂²	1	4	yes	—
20	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	—	2	4	no	—
23	C₂²×C₄×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	—	2	4	no	—
25	C₂²×C₄×C₅₃	—	2	4	no	—
26	C₂²×C₄×C₅₃	—	2	4	no	—
27	C₂²×C₄×C₅₃	—	2	4	no	—
28	C₂²×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	—	2	4	no	—
30	C₂²×C₄×C₅₃	—	2	4	no	—
31	C₂²×C₄×C₅₃	D₄	1	4	yes	—
32	C₂²×C₄×C₅₃	D₄	1	4	yes	—
33	C₂²×C₄×C₅₃	Q₈	1	8	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
39	C₄⋊C₄×C₅₃	—	2	8	no	—
40	C₄⋊C₄×C₅₃	—	2	16	no	—
41	C₄⋊C₄×C₅₃	—	2	8	no	—
42	C₂³×C₅₃	C₂×D₄	1	2	yes	—
43	C₂³×C₅₃	C₂×D₄	1	2	yes	—
44	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
45	C₂³×C₅₃	#41	1	2	yes	—
46	C₂³×C₅₃	#41	1	4	yes	—
47	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	8	no	—
49	C₂×C₄×C₅₃	—	2	16	no	—
50	C₂×C₄×C₅₃	—	2	8	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	4	8	no	—
62	C₂×C₄×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
66	C₂²×C₅₃	#121	1	4	yes	—
67	C₂²×C₅₃	C₂²⋊Q₈	1	8	yes	—
68	C₂²×C₅₃	#123	1	4	yes	—
69	C₂²×C₅₃	#125	1	8	yes	—
70	C₂²×C₅₃	C₄⋊Q₈	1	4	yes	—
71	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
74	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×#41	1	2	yes	—
76	C₄×C₅₃	—	2	4	no	—
77	C₄×C₅₃	—	2	8	no	—
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	4	4	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₂×#398	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₅₃	#398	1	8	yes	direct product factor
83	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
84	C₂×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
85	C₂×C₅₃	C₂×#123	1	1	yes	characteristic
86	C₂×C₅₃	C₂×#125	1	2	yes	—
87	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
89	C₂²×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
90	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
91	#398	C₂×C₅₃	1	8	yes	direct product factor
92	C₂×#95	C₂×C₅₃	1	2	yes	—
93	C₂×#95	C₂×C₅₃	1	1	yes	characteristic
94	C₅₃	C₂×#398	1	1	yes	characteristic, direct product factor, Sylow
95	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
96	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
97	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
98	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	4	no	—
101	C₂×C₄⋊C₄	—	2	8	no	—
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
104	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
105	C₂×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
106	#95	C₂²×C₅₃	1	8	yes	—
107	#95	C₂²×C₅₃	1	4	yes	—
108	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	D₄×C₅₃	1	4	yes	—
120	C₂²×C₄	D₄×C₅₃	1	4	yes	—
121	C₂²×C₄	Q₈×C₅₃	1	8	yes	—
122	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
124	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
125	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
126	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	2	16	no	—
129	C₄⋊C₄	—	2	8	no	—
130	C₂³	C₂×D₄×C₅₃	1	2	yes	—
131	C₂³	C₂×D₄×C₅₃	1	2	yes	—
132	C₂³	C₂×Q₈×C₅₃	1	4	yes	—
133	C₂³	#41×C₅₃	1	2	yes	—
134	C₂³	#41×C₅₃	1	4	yes	—
135	C₂³	C₂⁴×C₅₃	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	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
152	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
153	C₂×C₄	C₂×Q₈×C₅₃	1	8	yes	—
154	C₂²	#121×C₅₃	1	4	yes	—
155	C₂²	C₂²⋊Q₈×C₅₃	1	8	yes	—
156	C₂²	#123×C₅₃	1	4	yes	—
157	C₂²	#125×C₅₃	1	8	yes	—
158	C₂²	C₄⋊Q₈×C₅₃	1	4	yes	—
159	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
160	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
161	C₂²	C₂²×Q₈×C₅₃	1	2	yes	—
162	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
163	C₂²	C₂×#41×C₅₃	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	2	8	no	—
166	C₄	—	2	4	no	—
167	C₄	—	4	4	no	—
168	C₄	—	4	2	no	—
169	C₂	C₅₃×#398	1	8	yes	direct product factor
170	C₂	C₂×#121×C₅₃	1	1	yes	characteristic
171	C₂	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
172	C₂	C₂×#123×C₅₃	1	1	yes	characteristic
173	C₂	C₂×#125×C₅₃	1	2	yes	—
174	C₂	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#398	C₂	direct (N × H)
81. C₂×#398	C₅₃	direct (N × H)
82. C₂×C₅₃	#398	direct (N × H)
91. #398	C₂×C₅₃	direct (N × H)
94. C₅₃	C₂×#398	direct (N × H)
169. C₂	C₅₃×#398	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⋊C₄×C₅₃	2
C₅₃×#398	2
C₂×#95×C₅₃	2
C₂×#398	53

Subgroups — C2×C53×#398 #20271328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#398 #20271328