Subgroups — C₂×#20273535 #27353486

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273535	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810355	C₂	1	2	yes	maximal
6	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
9	C₈×Dic₁₀₆	—	2	4	no	—
10	C₈×Dic₁₀₆	C₂²	1	8	yes	—
11	C₄×Dic₂₁₂	C₄	1	4	yes	—
12	C₄×Dic₂₁₂	C₂²	1	4	yes	—
13	C₂×#13133151	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×#13133146	—	2	2	no	—
16	C₂×#13133142	C₄	1	2	yes	—
17	C₂×C₈×Dic₅₃	—	2	2	no	—
18	#16810374	C₂²	1	4	yes	—
19	#16810355	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#322	C₂²	1	4	yes	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₁₂	C₈	1	8	yes	complemented
26	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
27	C₂²×#88322	—	2	2	no	—
28	C₄×Dic₁₀₆	—	2	2	no	—
29	C₄×Dic₁₀₆	—	2	2	no	—
30	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₀₆	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
33	C₂×#11719919	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	2	2	no	—
35	#13133151	C₂×C₄	1	4	yes	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	C₂³	1	4	yes	—
38	#13133146	—	2	8	no	—
39	#13133142	C₂×C₄	1	8	yes	—
40	C₈×Dic₅₃	—	2	8	no	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	C₂³	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	2	no	—
56	C₂²×Dic₅₃	—	2	2	no	—
57	C₂×Dic₁₀₆	—	2	2	no	—
58	C₂×Dic₁₀₆	—	2	4	no	—
59	C₂×Dic₁₀₆	C₂×C₈	1	8	yes	—
60	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊Dic₅₃	—	2	2	no	—
74	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
75	#11719919	—	2	4	no	—
76	#11719919	—	2	4	no	—
77	C₄×Dic₅₃	—	2	4	no	—
78	C₄×Dic₅₃	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	4	4	no	—
81	Dic₂₁₂	C₂×C₈	1	16	yes	complemented
82	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	#131	1	2	yes	—
89	C₂×C₄×C₅₃	#137	1	4	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₅₃	—	2	4	no	—
94	C₂×Dic₅₃	—	2	8	no	—
95	C₂×Dic₅₃	—	2	4	no	—
96	C₂×Dic₅₃	—	4	4	no	—
97	Dic₁₀₆	—	2	4	no	—
98	Dic₁₀₆	—	2	8	no	—
99	Dic₁₀₆	C₂²×C₈	1	8	yes	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	4	no	—
102	C₈×C₅₃	—	4	2	no	—
103	C₈×C₅₃	C₂²×C₈	1	4	yes	—
104	#88322	—	4	4	no	—
105	C₂×#1579	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	C₈×D₄	1	2	yes	—
107	C₂²×C₅₃	#441	1	2	yes	—
108	C₂²×C₅₃	#444	1	2	yes	—
109	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	C₈×D₄	1	2	yes	—
113	C₄×C₅₃	#441	1	2	yes	—
114	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
116	C₄×C₅₃	C₂×#137	1	2	yes	—
117	Dic₅₃	—	2	8	no	—
118	Dic₅₃	—	4	4	no	—
119	C₂×C₈×Q₈	—	53	2	no	—
120	C₂×C₄×Q₁₆	—	53	1	no	—
121	#1579	—	53	8	no	—
122	C₂×#334	—	53	1	no	—
123	C₂×Q₈⋊C₈	—	53	2	no	—
124	C₂×#322	D₅₃	1	1	yes	characteristic
125	C₂×C₅₃	#1579	1	4	yes	complemented
126	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	—	53	1	no	—
130	C₂×C₄×Q₈	—	53	2	no	—
131	C₈×Q₈	—	53	8	no	—
132	C₈×Q₈	—	106	4	no	—
133	C₄×Q₁₆	—	53	4	no	—
134	C₄×Q₁₆	—	53	4	no	—
135	C₂×#107	—	53	1	no	—
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×C₄⋊C₈	—	106	2	no	—
138	C₂×#103	—	53	2	no	—
139	C₂×C₄×C₈	—	106	2	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#334	—	53	4	no	—
144	Q₈⋊C₈	—	53	8	no	—
145	#322	C₂×D₅₃	1	4	yes	—
146	C₅₃	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	53	2	no	—
148	C₂×Q₁₆	—	53	4	no	—
149	C₂×Q₁₆	—	53	8	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₈	D₂₁₂	1	2	yes	—
155	C₄×Q₈	—	53	4	no	—
156	C₄×Q₈	—	53	4	no	—
157	C₄×Q₈	—	106	2	no	—
158	C₄×Q₈	—	106	2	no	—
159	C₂×C₄⋊C₄	—	53	2	no	—
160	C₂×C₄⋊C₄	—	106	2	no	—
161	C₂×C₄²	—	106	2	no	—
162	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	#107	—	53	4	no	—
164	C₄⋊C₈	—	53	4	no	—
165	C₄⋊C₈	—	106	8	no	—
166	C₄⋊C₈	—	106	2	no	—
167	#103	—	53	8	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	106	8	no	—
170	C₄×C₈	C₄×D₅₃	1	2	yes	—
171	C₄×C₈	D₂₁₂	1	2	yes	—
172	C₄×C₈	C₂²×D₅₃	1	2	yes	—
173	C₄×C₈	C₂²×D₅₃	1	2	yes	—
174	C₄×C₈	C₂²×D₅₃	1	2	yes	—
175	C₂×Q₈	—	53	8	no	—
176	C₂×Q₈	—	53	4	no	—
177	C₂×Q₈	—	106	2	no	—
178	C₂×Q₈	—	106	4	no	—
179	C₂²×C₄	—	106	2	no	—
180	C₂²×C₄	—	106	2	no	—
181	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
183	C₂²×C₄	#11719945	1	1	yes	characteristic
184	Q₁₆	—	53	16	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	106	4	no	—
190	C₂×C₈	—	212	4	no	—
191	C₂×C₈	C₈×D₅₃	1	4	yes	—
192	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
194	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
195	C₄⋊C₄	—	53	4	no	—
196	C₄⋊C₄	—	106	2	no	—
197	C₄⋊C₄	—	106	4	no	—
198	C₄⋊C₄	—	106	4	no	—
199	C₄²	—	106	4	no	—
200	C₄²	—	106	4	no	—
201	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
203	C₄²	#11719945	1	1	yes	characteristic
204	C₄²	C₂³×D₅₃	1	1	yes	characteristic
205	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
206	Q₈	—	53	8	no	—
207	Q₈	—	106	4	no	—
208	Q₈	—	106	8	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	106	4	no	—
211	C₂×C₄	—	106	8	no	—
212	C₂×C₄	—	106	4	no	—
213	C₂×C₄	—	212	4	no	—
214	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
216	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
217	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
218	C₂×C₄	#13133223	1	2	yes	—
219	C₂×C₄	#13133250	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
222	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	4	no	—
225	C₈	—	4	2	no	—
226	C₈	—	212	4	no	—
227	C₈	C₂×C₈×D₅₃	1	4	yes	—
228	C₂²	C₈×D₂₁₂	1	2	yes	—
229	C₂²	#16810731	1	2	yes	—
230	C₂²	#16810746	1	2	yes	—
231	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	106	8	no	—
235	C₄	—	212	4	no	—
236	C₄	C₈×D₂₁₂	1	2	yes	—
237	C₄	#16810731	1	2	yes	—
238	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
239	C₄	C₂×#13133223	1	1	yes	characteristic
240	C₄	C₂×#13133250	1	2	yes	—
241	C₂	#20273535	1	4	yes	direct product factor
242	C₂	C₂×C₈×D₂₁₂	1	1	yes	characteristic
243	C₂	C₂×#16810731	1	1	yes	characteristic
244	C₂	C₂×#16810746	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	C₂×#1579
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #20273535	C₂	direct (N × H)
25. C₂×Dic₂₁₂	C₈	semidirect
81. Dic₂₁₂	C₂×C₈	semidirect
125. C₂×C₅₃	#1579	semidirect
146. C₅₃	C₂×#1579	semidirect
241. C₂	#20273535	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
C₂×C₄×Dic₂₁₂	2
#20273535	2
C₂×#16810374	2
C₂×#16810355	2
C₂×C₅₃×#322	2
C₂×#1579	53

Subgroups — C2×#20273535 #27353486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273535 #27353486