Subgroups — C₂×#20273534 #27353485

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×Q₈×#88322	C₂	1	2	yes	maximal
2	#20273534	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
5	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	Q₈×#88322	—	2	4	no	—
8	Q₈×#88322	C₂²	1	8	yes	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810374	C₂²	1	4	yes	—
15	Q₈⋊#88322	C₂²	1	8	yes	—
16	#16810344	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₅₃	C₄	1	4	yes	—
20	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
21	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#103×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	1	no	—
26	C₂²×#88322	D₄	1	2	yes	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	8	no	—
29	C₄⋊#88322	C₂³	1	4	yes	—
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	8	no	—
32	C₄×#88322	D₄	1	2	yes	—
33	C₄×#88322	C₂³	1	2	yes	—
34	C₄×#88322	C₂³	1	2	yes	—
35	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×Q₁₆×C₅₃	C₈	1	8	yes	complemented
37	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₅₃	—	2	2	no	—
41	C₄×Q₈×C₅₃	—	2	2	no	—
42	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	C₂×C₄	1	4	yes	—
49	#103×C₅₃	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	2	4	no	—
54	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
55	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	Q₁₆×C₅₃	C₂×C₈	1	16	yes	complemented
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	4	no	—
81	C₂×#88322	C₂×D₄	1	4	yes	—
82	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	4	no	—
84	Q₈×C₅₃	—	2	8	no	—
85	Q₈×C₅₃	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	8	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	#131	1	2	yes	—
96	C₂×C₄×C₅₃	#137	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	C₂²×C₈	1	4	yes	—
102	#88322	—	2	4	no	—
103	#88322	—	4	2	no	—
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	8	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	4	no	—
114	C₄×C₅₃	C₈×D₄	1	2	yes	—
115	C₄×C₅₃	#441	1	2	yes	—
116	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#137	1	2	yes	—
119	C₂×C₈×Q₈	—	53	2	no	—
120	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
121	#1579	—	53	8	no	—
122	C₂×#334	—	53	1	no	—
123	C₂×Q₈⋊C₈	—	53	2	no	—
124	C₂×#322	—	53	1	no	—
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	Dic₅₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
131	C₈×Q₈	—	53	8	no	—
132	C₈×Q₈	—	106	4	no	—
133	C₄×Q₁₆	Dic₅₃	1	4	yes	—
134	C₄×Q₁₆	C₂×D₅₃	1	4	yes	—
135	C₂×#107	Dic₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×C₄⋊C₈	—	106	2	no	—
138	C₂×#103	Dic₅₃	1	2	yes	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	106	2	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#334	—	53	4	no	—
144	Q₈⋊C₈	—	53	8	no	—
145	#322	—	53	4	no	—
146	C₅₃	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
148	C₂×Q₁₆	#88322	1	8	yes	complemented
149	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
150	C₂²×C₈	—	53	2	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	—	106	1	no	—
153	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
158	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	#107	C₂×Dic₅₃	1	4	yes	—
164	C₄⋊C₈	—	53	4	no	—
165	C₄⋊C₈	—	106	8	no	—
166	C₄⋊C₈	—	106	2	no	—
167	#103	C₂×Dic₅₃	1	8	yes	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	53	2	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	106	8	no	—
172	C₄×C₈	—	106	1	no	—
173	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
174	C₄×C₈	C₂²×D₅₃	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	C₂×#88322	1	8	yes	—
178	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	#11719947	1	1	yes	characteristic
183	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
184	Q₁₆	C₂×#88322	1	16	yes	complemented
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	53	4	no	—
187	C₂×C₈	—	106	4	no	—
188	C₂×C₈	—	106	4	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	—	212	4	no	—
192	C₂×C₈	C₂×#88322	1	4	yes	—
193	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
194	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	D₄×D₅₃	1	1	yes	characteristic
202	C₄²	#11719947	1	1	yes	characteristic
203	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₄²	C₂³×D₅₃	1	1	yes	characteristic
205	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	2	8	no	—
208	Q₈	C₂²×#88322	1	8	yes	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×#88322	1	2	yes	—
218	C₂×C₄	#13133224	1	2	yes	—
219	C₂×C₄	#13133249	1	4	yes	—
220	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	106	4	no	—
225	C₈	—	212	2	no	—
226	C₈	—	212	4	no	—
227	C₈	C₂²×#88322	1	4	yes	—
228	C₂²	D₄×#88322	1	2	yes	—
229	C₂²	#16810730	1	2	yes	—
230	C₂²	#16810745	1	2	yes	—
231	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	8	no	—
235	C₄	—	4	4	no	—
236	C₄	D₄×#88322	1	2	yes	—
237	C₄	#16810730	1	2	yes	—
238	C₄	C₂³×#88322	1	1	yes	characteristic
239	C₄	C₂×#13133224	1	1	yes	characteristic
240	C₄	C₂×#13133249	1	2	yes	—
241	C₂	#20273534	1	4	yes	direct product factor
242	C₂	C₂×D₄×#88322	1	1	yes	characteristic
243	C₂	C₂×#16810730	1	1	yes	characteristic
244	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×#1579
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20273534	C₂	direct (N × H)
36. C₂×Q₁₆×C₅₃	C₈	semidirect
61. Q₁₆×C₅₃	C₂×C₈	semidirect
125. C₂×C₅₃	#1579	semidirect
146. C₅₃	C₂×#1579	semidirect
148. C₂×Q₁₆	#88322	semidirect
184. Q₁₆	C₂×#88322	semidirect
241. C₂	#20273534	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×Q₈×#88322	2
#20273534	2
C₂×#16810374	2
C₂×Q₈⋊#88322	2
C₂×#16810344	2
C₂×C₄×Q₁₆×C₅₃	2
C₂×#1579	53

Subgroups — C2×#20273534 #27353485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273534 #27353485