Subgroups — C₂×#20272899 #27304543

Order 13568.

294 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272899	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	1	no	—
6	C₂×#13133123	—	2	1	no	—
7	C₂×#13133123	—	2	1	no	—
8	C₂×#13133123	C₂²	1	2	yes	complemented
9	#16810453	—	2	1	no	—
10	#16810453	C₂²	1	2	yes	complemented
11	C₂×#135×C₅₃	—	2	1	no	—
12	C₅₃×#508	—	2	1	no	—
13	C₅₃×#508	—	2	1	no	—
14	C₅₃×#508	C₄	1	2	yes	—
15	C₅₃×#508	C₂²	1	2	yes	—
16	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic
17	C₂×SD₃₂×C₅₃	—	2	1	no	—
18	C₂×D₁₆×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	4	1	no	—
21	#13133153	—	2	2	no	—
22	#13133153	—	4	1	no	—
23	#13133123	—	2	2	no	—
24	C₂×#41×C₅₃	—	4	1	no	—
25	#135×C₅₃	—	2	2	no	—
26	#135×C₅₃	—	2	2	no	—
27	#135×C₅₃	—	4	1	no	—
28	#135×C₅₃	—	4	1	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×SD₁₆×C₅₃	—	4	1	no	—
31	C₂×D₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	Q₃₂×C₅₃	—	2	1	no	—
34	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
35	SD₃₂×C₅₃	—	2	2	no	—
36	SD₃₂×C₅₃	—	2	2	no	—
37	D₁₆×C₅₃	—	2	1	no	—
38	D₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
45	#41×C₅₃	—	4	2	no	—
46	#41×C₅₃	—	4	2	no	—
47	#41×C₅₃	—	8	1	no	—
48	#41×C₅₃	—	8	1	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×D₄×C₅₃	—	4	1	no	—
51	C₂×D₄×C₅₃	—	8	1	no	—
52	C₂²×C₄×C₅₃	—	8	1	no	—
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	Q₁₆×C₅₃	—	2	2	no	—
55	Q₁₆×C₅₃	—	4	1	no	—
56	SD₁₆×C₅₃	—	4	2	no	—
57	SD₁₆×C₅₃	—	4	2	no	—
58	D₈×C₅₃	—	2	2	no	—
59	D₈×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
64	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	#11719917	—	4	2	no	—
67	#11719917	—	8	1	no	—
68	C₂×#88322	—	8	1	no	—
69	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₅₃	—	8	1	no	—
74	C₂³×C₅₃	SD₃₂	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	8	1	no	—
77	D₄×C₅₃	—	4	2	no	—
78	D₄×C₅₃	—	8	2	no	—
79	D₄×C₅₃	—	8	2	no	—
80	D₄×C₅₃	—	8	1	no	—
81	C₂×C₄×C₅₃	—	8	1	no	—
82	C₂×C₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₈×C₅₃	#102	1	1	yes	characteristic
91	C₈×C₅₃	#102	1	1	yes	characteristic
92	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
94	#88322	—	8	2	no	—
95	C₂×#1417	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	8	2	no	—
98	C₂²×C₅₃	—	16	2	no	—
99	C₂²×C₅₃	#357	1	1	yes	characteristic
100	C₂²×C₅₃	#357	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
102	C₄×C₅₃	—	8	2	no	—
103	C₄×C₅₃	#357	1	1	yes	characteristic
104	C₄×C₅₃	#357	1	1	yes	characteristic
105	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
106	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
107	C₂×#508	D₅₃	1	1	yes	characteristic
108	C₂²×C₃₂	—	53	1	no	—
109	C₂×#368	—	53	1	no	—
110	#1417	—	53	4	no	—
111	C₂×C₅₃	—	2	1	no	—
112	C₂×C₅₃	—	2	1	no	—
113	C₂×C₅₃	—	16	2	no	—
114	C₂×C₅₃	#1417	1	2	yes	complemented
115	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
116	C₂×#135	—	2	1	no	—
117	#508	—	2	1	no	—
118	#508	—	2	1	no	—
119	#508	Dic₅₃	1	2	yes	—
120	#508	C₂×D₅₃	1	2	yes	—
121	C₂×Q₃₂	Dic₅₃	1	1	yes	characteristic
122	C₂×SD₃₂	—	2	1	no	—
123	C₂×D₁₆	Dic₅₃	1	1	yes	characteristic
124	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#108	—	106	1	no	—
126	C₂×C₃₂	—	53	2	no	—
127	C₂×C₃₂	—	106	1	no	—
128	C₂×C₃₂	—	106	1	no	—
129	#368	—	53	2	no	—
130	#368	—	106	1	no	—
131	C₅₃	C₂×#1417	1	1	yes	characteristic, complemented, Sylow
132	C₂×#41	—	4	1	no	—
133	#135	—	2	2	no	—
134	#135	—	2	2	no	—
135	#135	—	4	1	no	—
136	#135	—	4	1	no	—
137	C₂×Q₁₆	—	2	1	no	—
138	C₂×SD₁₆	—	4	1	no	—
139	C₂×D₈	—	2	1	no	—
140	C₂×M₁₆	—	212	1	no	—
141	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	Q₃₂	—	2	1	no	—
143	Q₃₂	C₂×Dic₅₃	1	2	yes	—
144	SD₃₂	—	2	2	no	—
145	SD₃₂	—	2	2	no	—
146	D₁₆	—	2	1	no	—
147	D₁₆	C₂×Dic₅₃	1	2	yes	—
148	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
154	#108	—	106	2	no	—
155	#108	—	212	1	no	—
156	C₃₂	—	106	2	no	—
157	#41	—	4	2	no	—
158	#41	—	4	2	no	—
159	#41	—	8	1	no	—
160	#41	—	8	1	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂×D₄	—	4	1	no	—
163	C₂×D₄	—	8	1	no	—
164	C₂²×C₄	—	8	1	no	—
165	C₂²×C₄	#11719922	1	1	yes	characteristic
166	Q₁₆	—	2	2	no	—
167	Q₁₆	—	4	1	no	—
168	SD₁₆	—	4	2	no	—
169	SD₁₆	—	4	2	no	—
170	D₈	—	2	2	no	—
171	D₈	—	4	1	no	—
172	M₁₆	—	424	1	no	—
173	M₁₆	—	212	2	no	—
174	C₂×C₈	—	424	1	no	—
175	C₂×C₈	#11719922	1	1	yes	characteristic
176	C₂×C₈	#11719923	1	1	yes	characteristic
177	C₂×C₈	#11719923	1	1	yes	characteristic
178	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
184	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂³	—	8	1	no	—
186	C₂³	#13133161	1	1	yes	characteristic
187	Q₈	—	4	2	no	—
188	Q₈	—	8	1	no	—
189	D₄	—	4	2	no	—
190	D₄	—	8	2	no	—
191	D₄	—	8	2	no	—
192	D₄	—	8	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	#13133138	1	1	yes	characteristic
197	C₂×C₄	#13133138	1	1	yes	characteristic
198	C₂×C₄	#13133158	1	1	yes	characteristic
199	C₂×C₄	#13133158	1	1	yes	characteristic
200	C₂×C₄	#13133161	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
202	C₈	—	424	2	no	—
203	C₈	#13133138	1	1	yes	characteristic
204	C₈	#13133138	1	1	yes	characteristic
205	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₈	C₂×#11719923	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	16	2	no	—
210	C₂²	#16810424	1	1	yes	characteristic
211	C₂²	#16810424	1	1	yes	characteristic
212	C₂²	C₂×#13133161	1	1	yes	characteristic
213	C₄	—	8	2	no	—
214	C₄	#16810424	1	1	yes	characteristic
215	C₄	#16810424	1	1	yes	characteristic
216	C₄	C₂×#13133138	1	1	yes	characteristic
217	C₄	C₂×#13133158	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	—	16	2	no	—
221	C₂	#20272899	1	2	yes	direct product factor
222	C₂	C₂×#16810424	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133158
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#508	C₂
socle	C₂²×C₅₃	#357
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	C₂×#1417
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133123	C₂	semidirect
2. C₂×#16810453	C₂	semidirect
3. #20272899	C₂	direct (N × H)
8. C₂×#13133123	C₂²	semidirect
10. #16810453	C₂²	semidirect
114. C₂×C₅₃	#1417	semidirect
131. C₅₃	C₂×#1417	semidirect
221. C₂	#20272899	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×#13133123	2
C₂×#16810453	2
#20272899	2
C₂×C₅₃×#508	2
C₂×#1417	53

Subgroups — C2×#20272899 #27304543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272899 #27304543