Subgroups — C₂×#20272765 #27295333

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20272765	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	—	2	1	no	—
6	C₂²×Dic₂₁₂	C₄	1	2	yes	—
7	C₂×#13133217	—	2	1	no	—
8	#16811003	—	2	1	no	—
9	#16811003	—	2	1	no	—
10	#16811003	C₄	1	2	yes	—
11	#16811003	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	—	2	1	no	—
13	#16810376	—	2	1	no	—
14	#16810376	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	—	2	1	no	—
18	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	4	2	no	—
24	C₂×Dic₂₁₂	—	4	2	no	—
25	C₂×Dic₂₁₂	—	4	4	no	—
26	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
27	C₂²×#88322	—	4	1	no	—
28	#13133217	—	2	2	no	—
29	#13133217	—	2	2	no	—
30	#13133217	—	4	1	no	—
31	#13133217	—	4	1	no	—
32	C₂×C₄⋊Dic₅₃	—	4	1	no	—
33	C₂×C₄⋊Dic₅₃	—	4	1	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	—	4	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	—	4	2	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×Dic₁₀₆	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	4	no	—
51	C₂×Dic₁₀₆	—	4	4	no	—
52	C₂×Dic₁₀₆	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	D₈	1	2	yes	—
57	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	D₈	1	1	yes	characteristic
62	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	8	2	no	—
70	Dic₂₁₂	—	4	4	no	—
71	Dic₂₁₂	—	4	4	no	—
72	Dic₂₁₂	—	4	4	no	—
73	Dic₂₁₂	—	4	2	no	—
74	Dic₂₁₂	—	4	2	no	—
75	C₁₆×C₅₃	—	4	2	no	—
76	C₂³×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	#100	1	2	yes	—
79	C₂×C₄×C₅₃	#102	1	2	yes	—
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
84	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×Dic₅₃	—	4	4	no	—
89	C₂×Dic₅₃	—	8	4	no	—
90	Dic₁₀₆	—	4	4	no	—
91	Dic₁₀₆	—	4	4	no	—
92	Dic₁₀₆	—	8	4	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	#102	1	2	yes	—
96	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
98	#88322	—	8	2	no	—
99	C₂×#1362	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	#331	1	2	yes	—
101	C₂²×C₅₃	#358	1	2	yes	—
102	C₂²×C₅₃	#362	1	2	yes	—
103	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	#331	1	2	yes	—
107	C₄×C₅₃	#358	1	2	yes	—
108	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
110	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
111	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
112	Dic₅₃	—	8	4	no	—
113	C₂×#494	—	53	1	no	—
114	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
115	C₂×#335	—	53	1	no	—
116	#1362	—	53	4	no	—
117	C₂×C₅₃	#1362	1	4	yes	complemented
118	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
121	C₂²×Q₁₆	—	53	2	no	—
122	C₂²×Q₁₆	—	106	1	no	—
123	C₂×C₄⋊Q₈	—	106	1	no	—
124	C₂²×C₁₆	—	2	1	no	—
125	#494	—	53	2	no	—
126	#494	—	53	2	no	—
127	#494	—	106	1	no	—
128	#494	—	106	1	no	—
129	C₂×C₄⋊C₈	—	106	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	1	no	—
132	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
133	#335	—	53	2	no	—
134	#335	—	106	1	no	—
135	C₅₃	C₂×#1362	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	106	2	no	—
137	C₂×Q₁₆	—	53	4	no	—
138	C₂×Q₁₆	—	106	2	no	—
139	C₂×Q₁₆	—	106	2	no	—
140	C₂×Q₁₆	—	106	2	no	—
141	C₂×Q₁₆	—	212	4	no	—
142	C₂×Q₁₆	—	212	2	no	—
143	C₂×Q₁₆	—	212	2	no	—
144	C₂²×C₈	—	212	1	no	—
145	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄⋊Q₈	—	106	2	no	—
148	C₄⋊Q₈	—	106	2	no	—
149	C₄⋊Q₈	—	212	1	no	—
150	C₄⋊Q₈	—	212	1	no	—
151	C₂×C₄⋊C₄	—	212	1	no	—
152	C₂×C₄⋊C₄	—	212	1	no	—
153	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	2	no	—
156	C₄⋊C₈	—	106	2	no	—
157	C₄⋊C₈	—	212	1	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	—	106	4	no	—
163	C₂×Q₈	—	212	4	no	—
164	C₂×Q₈	—	212	4	no	—
165	C₂×Q₈	—	212	2	no	—
166	C₂²×C₄	—	212	2	no	—
167	C₂²×C₄	#11719913	1	1	yes	characteristic
168	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
169	C₂²×C₄	#11719921	1	1	yes	characteristic
170	Q₁₆	—	212	4	no	—
171	Q₁₆	—	212	4	no	—
172	Q₁₆	—	212	4	no	—
173	Q₁₆	—	212	2	no	—
174	Q₁₆	—	212	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	424	2	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	#11719921	1	1	yes	characteristic
179	C₂×C₈	#11719921	1	1	yes	characteristic
180	C₂×C₈	#11719922	1	2	yes	—
181	C₂×C₈	#11719924	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₄⋊C₄	—	212	2	no	—
185	C₄⋊C₄	—	212	2	no	—
186	C₄⋊C₄	—	212	2	no	—
187	C₄⋊C₄	—	212	2	no	—
188	C₄²	#11719913	1	1	yes	characteristic
189	C₄²	D₄₂₄	1	1	yes	characteristic
190	C₄²	#11719921	1	1	yes	characteristic
191	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
192	C₁₆	—	4	2	no	—
193	C₂³	#13133139	1	1	yes	characteristic
194	Q₈	—	424	4	no	—
195	Q₈	—	212	4	no	—
196	Q₈	—	212	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	424	4	no	—
199	C₂×C₄	—	212	4	no	—
200	C₂×C₄	#13133133	1	2	yes	—
201	C₂×C₄	#13133137	1	2	yes	—
202	C₂×C₄	#13133139	1	1	yes	characteristic
203	C₂×C₄	#13133139	1	1	yes	characteristic
204	C₂×C₄	#13133139	1	1	yes	characteristic
205	C₂×C₄	#13133162	1	2	yes	—
206	C₂×C₄	Dic₄₂₄	1	2	yes	—
207	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	424	2	no	—
213	C₈	#13133137	1	2	yes	—
214	C₈	C₂×#11719922	1	1	yes	characteristic
215	C₈	C₂×#11719924	1	1	yes	characteristic
216	C₂²	#16810366	1	2	yes	—
217	C₂²	#16810428	1	2	yes	—
218	C₂²	#16810440	1	2	yes	—
219	C₂²	C₂×#13133139	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	424	4	no	—
223	C₄	#16810366	1	2	yes	—
224	C₄	#16810428	1	2	yes	—
225	C₄	C₂×#13133133	1	1	yes	characteristic
226	C₄	C₂×#13133137	1	1	yes	characteristic
227	C₄	C₂×#13133162	1	1	yes	characteristic
228	C₄	C₂×Dic₄₂₄	1	1	yes	characteristic
229	C₂	#20272765	1	4	yes	direct product factor
230	C₂	C₂×#16810366	1	1	yes	characteristic
231	C₂	C₂×#16810428	1	1	yes	characteristic
232	C₂	C₂×#16810440	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1362
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20272765	C₂	direct (N × H)
117. C₂×C₅₃	#1362	semidirect
135. C₅₃	C₂×#1362	semidirect
229. C₂	#20272765	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×#16811003	2
C₂×#16810376	2
#20272765	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#1362	53

Subgroups — C2×#20272765 #27295333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272765 #27295333