Subgroups — C₂×#20272698 #27293908

Order 13568.

346 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272698	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂²×#11719908	—	2	1	no	—
7	#16810753	—	2	1	no	—
8	#16810753	—	2	1	no	—
9	#16810753	C₄	1	2	yes	—
10	#16810753	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
12	C₂×#13133146	—	2	1	no	—
13	C₂×#13133127	—	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	C₅₃×#346	—	2	1	no	—
19	C₅₃×#346	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	#13133127	—	2	2	no	—
36	#13133127	—	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	—	4	2	no	—
53	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
67	#11719919	—	2	2	no	—
68	#11719919	—	2	2	no	—
69	C₄×Dic₅₃	—	2	2	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₁₆×C₅₃	—	2	4	no	—
74	#11719908	—	4	2	no	—
75	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	2	no	—
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×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	Dic₁₀₆	—	4	2	no	—
91	Dic₁₀₆	—	4	2	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	#103	1	2	yes	—
95	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
97	#88322	—	4	2	no	—
98	C₂×#1338	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
100	C₂²×C₅₃	#349	1	2	yes	—
101	C₂²×C₅₃	#350	1	2	yes	—
102	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
108	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
110	Dic₅₃	—	4	2	no	—
111	Dic₅₃	—	4	2	no	—
112	C₂×#446	—	53	1	no	—
113	C₂×C₄⋊C₁₆	—	53	1	no	—
114	C₂×#346	D₅₃	1	1	yes	characteristic
115	#1338	—	53	4	no	—
116	C₂×C₅₃	#1338	1	4	yes	complemented
117	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
120	C₂×C₄×Q₈	—	53	1	no	—
121	C₂²×C₁₆	—	2	1	no	—
122	C₂²×C₁₆	—	106	1	no	—
123	#446	—	53	2	no	—
124	#446	—	53	2	no	—
125	#446	—	106	1	no	—
126	#446	—	106	1	no	—
127	C₂×C₄⋊C₈	—	53	1	no	—
128	C₂×C₄⋊C₈	—	106	1	no	—
129	C₂×#97	—	106	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	53	2	no	—
132	C₄⋊C₁₆	—	106	1	no	—
133	#346	—	2	1	no	—
134	#346	C₂×D₅₃	1	2	yes	—
135	C₅₃	C₂×#1338	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	53	1	no	—
137	C₂²×C₈	—	106	1	no	—
138	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄×Q₈	—	53	2	no	—
141	C₄×Q₈	—	53	2	no	—
142	C₄×Q₈	—	106	1	no	—
143	C₄×Q₈	—	106	1	no	—
144	C₂×C₄⋊C₄	—	53	1	no	—
145	C₂×C₄⋊C₄	—	106	1	no	—
146	C₂×C₄²	—	106	1	no	—
147	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₁₆	—	2	4	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	C₂×C₁₆	—	212	2	no	—
152	C₄⋊C₈	—	53	2	no	—
153	C₄⋊C₈	—	106	2	no	—
154	C₄⋊C₈	—	106	2	no	—
155	C₄⋊C₈	—	106	1	no	—
156	#97	—	106	2	no	—
157	#97	—	106	2	no	—
158	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
159	C₄×C₈	D₂₁₂	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×Q₈	—	53	2	no	—
163	C₂×Q₈	—	106	1	no	—
164	C₂×Q₈	—	212	2	no	—
165	C₂²×C₄	—	106	1	no	—
166	C₂²×C₄	—	106	1	no	—
167	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
168	C₂²×C₄	#11719912	1	1	yes	characteristic
169	C₂²×C₄	#11719921	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	#11719913	1	2	yes	—
174	C₂×C₈	Dic₂₁₂	1	2	yes	—
175	C₂×C₈	#11719921	1	1	yes	characteristic
176	C₂×C₈	#11719921	1	1	yes	characteristic
177	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	C₄⋊C₄	—	53	2	no	—
180	C₄⋊C₄	—	106	1	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₄⋊C₄	—	106	2	no	—
183	C₄²	—	106	2	no	—
184	C₄²	—	106	2	no	—
185	C₄²	C₈×D₅₃	1	1	yes	characteristic
186	C₄²	#11719912	1	1	yes	characteristic
187	C₄²	#11719921	1	1	yes	characteristic
188	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
189	C₁₆	—	2	4	no	—
190	C₁₆	—	212	2	no	—
191	C₂³	#13133129	1	1	yes	characteristic
192	Q₈	—	212	2	no	—
193	Q₈	—	212	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	106	2	no	—
198	C₂×C₄	—	212	2	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	#13133129	1	1	yes	characteristic
201	C₂×C₄	#13133129	1	1	yes	characteristic
202	C₂×C₄	#13133129	1	1	yes	characteristic
203	C₂×C₄	#13133142	1	2	yes	—
204	C₂×C₄	#13133145	1	2	yes	—
205	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	212	2	no	—
211	C₈	#13133142	1	2	yes	—
212	C₈	C₂×#11719913	1	1	yes	characteristic
213	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
214	C₂²	#16810355	1	2	yes	—
215	C₂²	#16810404	1	2	yes	—
216	C₂²	#16810408	1	2	yes	—
217	C₂²	C₂×#13133129	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	212	2	no	—
223	C₄	—	212	2	no	—
224	C₄	#16810355	1	2	yes	—
225	C₄	C₂×#13133142	1	1	yes	characteristic
226	C₄	C₂×#13133145	1	1	yes	characteristic
227	C₂	#20272698	1	4	yes	direct product factor
228	C₂	C₂×#16810355	1	1	yes	characteristic
229	C₂	C₂×#16810404	1	1	yes	characteristic
230	C₂	C₂×#16810408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#1338
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #20272698	C₂	direct (N × H)
116. C₂×C₅₃	#1338	semidirect
135. C₅₃	C₂×#1338	semidirect
227. C₂	#20272698	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×#16810753	2
C₂×C₄⋊#11719908	2
#20272698	2
C₂×C₅₃×#346	2
C₂×#1338	53

Subgroups — C2×#20272698 #27293908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272698 #27293908