Subgroups — C₂×#20272695 #27293892

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 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	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272695	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂²×#11719908	—	2	1	no	—
7	C₈×Dic₁₀₆	—	2	1	no	—
8	C₈×Dic₁₀₆	—	2	1	no	—
9	C₈×Dic₁₀₆	C₄	1	2	yes	—
10	C₈×Dic₁₀₆	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
12	C₂×#13133146	—	2	1	no	—
13	C₂×C₈×Dic₅₃	—	2	1	no	—
14	C₄⋊#11719908	—	2	1	no	—
15	C₄⋊#11719908	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	—	2	1	no	—
19	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
21	C₂²×#88322	—	2	1	no	—
22	C₄×Dic₁₀₆	—	2	1	no	—
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	C₈	1	2	yes	—
25	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
27	C₂×#11719919	—	2	1	no	—
28	C₂×C₄×Dic₅₃	—	2	1	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₂×#11719908	—	4	2	no	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	C₂×C₄	1	2	yes	—
33	#13133146	—	2	2	no	—
34	#13133146	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	—	2	4	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	1	no	—
50	C₂²×Dic₅₃	—	2	1	no	—
51	C₂×Dic₁₀₆	—	2	1	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	C₁₆	1	4	yes	complemented
54	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄⋊Dic₅₃	—	2	1	no	—
67	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
68	#11719919	—	2	2	no	—
69	#11719919	—	2	2	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₁₆×C₅₃	—	2	4	no	—
75	#11719908	—	4	2	no	—
76	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#103	1	2	yes	—
82	C₂×C₄×C₅₃	#104	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₅₃	—	2	4	no	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	Dic₁₀₆	—	2	4	no	—
93	Dic₁₀₆	—	2	2	no	—
94	Dic₁₀₆	C₂×C₁₆	1	4	yes	complemented
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	#103	1	2	yes	—
98	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
100	#88322	—	4	2	no	—
101	C₂×Q₈⋊C₁₆	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
104	C₂²×C₅₃	#350	1	2	yes	—
105	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
109	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
110	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
111	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
112	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
113	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
114	Dic₅₃	—	2	4	no	—
115	Dic₅₃	—	4	2	no	—
116	C₂×C₈×Q₈	—	53	1	no	—
117	C₂×C₄⋊C₁₆	—	53	1	no	—
118	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
119	Q₈⋊C₁₆	—	53	4	no	—
120	C₂×C₅₃	Q₈⋊C₁₆	1	4	yes	complemented
121	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
122	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
124	C₂×C₄×Q₈	—	53	1	no	—
125	C₂²×C₁₆	—	2	1	no	—
126	C₂²×C₁₆	—	106	1	no	—
127	C₈×Q₈	—	53	2	no	—
128	C₈×Q₈	—	53	2	no	—
129	C₈×Q₈	—	106	1	no	—
130	C₈×Q₈	—	106	1	no	—
131	C₂×C₄⋊C₈	—	53	1	no	—
132	C₂×C₄⋊C₈	—	106	1	no	—
133	C₂×C₄×C₈	—	106	1	no	—
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₄⋊C₁₆	—	53	2	no	—
136	C₄⋊C₁₆	—	106	1	no	—
137	C₄×C₁₆	—	2	1	no	—
138	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
139	C₅₃	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	—	53	1	no	—
141	C₂²×C₈	—	106	1	no	—
142	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
144	C₄×Q₈	—	53	2	no	—
145	C₄×Q₈	—	53	2	no	—
146	C₄×Q₈	—	106	1	no	—
147	C₄×Q₈	—	106	1	no	—
148	C₂×C₄⋊C₄	—	53	1	no	—
149	C₂×C₄⋊C₄	—	106	1	no	—
150	C₂×C₄²	—	106	1	no	—
151	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	106	2	no	—
155	C₂×C₁₆	—	212	2	no	—
156	C₄⋊C₈	—	53	2	no	—
157	C₄⋊C₈	—	106	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	106	1	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	106	2	no	—
162	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
163	C₄×C₈	D₂₁₂	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₂×Q₈	—	53	4	no	—
167	C₂×Q₈	—	53	2	no	—
168	C₂×Q₈	—	106	1	no	—
169	C₂×Q₈	—	106	2	no	—
170	C₂²×C₄	—	106	1	no	—
171	C₂²×C₄	—	106	1	no	—
172	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	#11719912	1	1	yes	characteristic
174	C₂²×C₄	#11719921	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	106	2	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	#11719913	1	2	yes	—
179	C₂×C₈	Dic₂₁₂	1	2	yes	—
180	C₂×C₈	#11719921	1	1	yes	characteristic
181	C₂×C₈	#11719921	1	1	yes	characteristic
182	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₄⋊C₄	—	53	2	no	—
185	C₄⋊C₄	—	106	1	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₄⋊C₄	—	106	2	no	—
188	C₄²	—	106	2	no	—
189	C₄²	—	106	2	no	—
190	C₄²	C₈×D₅₃	1	1	yes	characteristic
191	C₄²	#11719912	1	1	yes	characteristic
192	C₄²	#11719921	1	1	yes	characteristic
193	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
194	C₁₆	—	2	4	no	—
195	C₁₆	—	212	2	no	—
196	C₂³	#13133129	1	1	yes	characteristic
197	Q₈	—	53	4	no	—
198	Q₈	—	106	2	no	—
199	Q₈	—	106	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	106	2	no	—
202	C₂×C₄	—	106	4	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	212	2	no	—
205	C₂×C₄	#13133129	1	1	yes	characteristic
206	C₂×C₄	#13133129	1	1	yes	characteristic
207	C₂×C₄	#13133129	1	1	yes	characteristic
208	C₂×C₄	#13133142	1	2	yes	—
209	C₂×C₄	#13133145	1	2	yes	—
210	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
211	C₂×C₄	#13133156	1	2	yes	—
212	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
213	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	212	2	no	—
218	C₈	#13133142	1	2	yes	—
219	C₈	C₂×#11719913	1	1	yes	characteristic
220	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
221	C₂²	#16810355	1	2	yes	—
222	C₂²	#16810402	1	2	yes	—
223	C₂²	#16810406	1	2	yes	—
224	C₂²	C₂×#13133129	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	106	4	no	—
228	C₄	—	212	2	no	—
229	C₄	#16810355	1	2	yes	—
230	C₄	#16810402	1	2	yes	—
231	C₄	C₂×#13133142	1	1	yes	characteristic
232	C₄	C₂×#13133145	1	1	yes	characteristic
233	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
234	C₄	C₂×#13133156	1	1	yes	characteristic
235	C₂	#20272695	1	4	yes	direct product factor
236	C₂	C₂×#16810355	1	1	yes	characteristic
237	C₂	C₂×#16810402	1	1	yes	characteristic
238	C₂	C₂×#16810406	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20272695	C₂	direct (N × H)
53. C₂×Dic₁₀₆	C₁₆	semidirect
94. Dic₁₀₆	C₂×C₁₆	semidirect
120. C₂×C₅₃	Q₈⋊C₁₆	semidirect
139. C₅₃	C₂×Q₈⋊C₁₆	semidirect
235. C₂	#20272695	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⋊#11719908	2
#20272695	2
C₂×C₄×C₁₆×C₅₃	2
C₂×Q₈⋊C₁₆	53

Subgroups — C2×#20272695 #27293892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272695 #27293892