Subgroups — #28310803

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291047	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811515	C₂	1	2	yes	maximal
3	C₄×#13133291	C₂	1	1	yes	maximal, characteristic
4	#20289756	C₂	1	2	yes	maximal
5	#20274584	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	1	yes	characteristic
7	C₂×#13133291	C₂²	1	2	yes	—
8	C₂×#13133291	C₂²	1	1	yes	characteristic
9	C₄⋊#11719910	C₄	1	2	yes	complemented
10	C₄⋊#11719910	C₄	1	2	yes	complemented
11	#16811515	—	2	4	no	—
12	#16811511	C₄	1	2	yes	complemented
13	C₄×#11719910	C₄	1	2	yes	complemented
14	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
15	C₂×#13133128	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810378	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
19	#13133291	—	2	2	no	—
20	#13133291	—	2	1	no	—
21	#13133291	—	2	1	no	—
22	#13133291	—	2	2	no	—
23	#13133291	D₄	1	2	yes	—
24	#13133291	Q₈	1	2	yes	—
25	C₂×#11719910	C₂×C₄	1	4	yes	—
26	C₂×#11719910	C₂×C₄	1	2	yes	—
27	C₂×#11719910	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	1	yes	characteristic
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	#13133128	—	2	2	no	—
34	#13133128	—	2	2	no	—
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₄×#88322	C₂×C₄	1	1	yes	characteristic
37	C₄×#88322	C₂×C₄	1	1	yes	characteristic
38	C₄×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#95×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	C₄²	1	1	yes	characteristic
51	C₄²×C₅₃	C₄²	1	1	yes	characteristic
52	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
53	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	—	2	1	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	C₄²	1	1	yes	characteristic
60	C₂×#88322	C₄²	1	1	yes	characteristic
61	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
62	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
63	C₂×#88322	C₂×D₄	1	1	yes	characteristic
64	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
65	C₂×#88322	#41	1	1	yes	characteristic
66	C₂×#88322	#41	1	1	yes	characteristic
67	#11719910	—	2	2	no	—
68	#11719910	—	2	4	no	—
69	#11719910	C₄⋊C₄	1	4	yes	complemented
70	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
86	#88322	—	2	1	no	—
87	#88322	—	2	1	no	—
88	#88322	—	4	2	no	—
89	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
90	#88322	#117	1	1	yes	characteristic
91	#88322	C₄×D₄	1	1	yes	characteristic
92	#88322	C₄×Q₈	1	1	yes	characteristic
93	#12666	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
97	C₂²×C₅₃	#436	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	8	2	no	—
103	C₄×C₅₃	#347	1	2	yes	complemented
104	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₅₃	#405	1	1	yes	characteristic
106	C₄×C₅₃	#436	1	1	yes	characteristic
107	C₄×C₅₃	#446	1	1	yes	characteristic
108	#2150	—	53	1	no	—
109	C₂×#347	—	53	2	no	—
110	C₄×M₃₂	—	53	1	no	—
111	#1770	—	53	1	no	—
112	#1375	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1770	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
117	C₂×C₅₃	#2110	1	1	yes	characteristic
118	C₂×M₃₂	—	53	1	no	—
119	C₂×M₃₂	—	53	1	no	—
120	C₂×M₃₂	—	53	2	no	—
121	C₂×C₄⋊C₈	—	53	1	no	—
122	C₂×#97	—	53	2	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₄×C₄⋊C₄	#6157	1	1	yes	characteristic
125	C₄⋊C₁₆	—	53	2	no	—
126	C₄⋊C₁₆	—	53	2	no	—
127	#347	—	106	4	no	—
128	#346	—	53	2	no	—
129	C₄×C₁₆	—	53	2	no	—
130	#336	—	53	2	no	—
131	C₅₃	#12666	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂×#6157	1	2	yes	—
137	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
138	M₃₂	—	53	2	no	—
139	M₃₂	—	53	2	no	—
140	M₃₂	—	106	2	no	—
141	M₃₂	—	106	1	no	—
142	M₃₂	—	106	1	no	—
143	M₃₂	—	106	2	no	—
144	C₂×C₁₆	—	53	2	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	53	4	no	—
147	C₄⋊C₈	—	106	1	no	—
148	C₄⋊C₈	—	106	1	no	—
149	#97	—	106	2	no	—
150	#97	—	106	2	no	—
151	C₄×C₈	—	53	1	no	—
152	C₄×C₈	—	53	1	no	—
153	C₄×C₈	—	53	1	no	—
154	C₄×C₈	—	53	1	no	—
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#11719940	1	1	yes	characteristic
158	C₂²×C₄	#11719940	1	1	yes	characteristic
159	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
160	C₂×C₈	—	53	1	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	106	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	#11719936	1	1	yes	characteristic
177	C₄²	#11719936	1	1	yes	characteristic
178	C₄²	C₄×#6157	1	1	yes	characteristic
179	C₄²	C₄×#6157	1	1	yes	characteristic
180	C₁₆	—	53	4	no	—
181	C₁₆	—	106	2	no	—
182	C₁₆	—	106	4	no	—
183	C₂³	C₂×#11719940	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#13133272	1	1	yes	characteristic
190	C₂×C₄	#13133272	1	1	yes	characteristic
191	C₂×C₄	#13133272	1	1	yes	characteristic
192	C₂×C₄	#13133272	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
194	C₂×C₄	D₄×#6157	1	1	yes	characteristic
195	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
196	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
197	C₂×C₄	#13133304	1	1	yes	characteristic
198	C₂×C₄	#13133305	1	1	yes	characteristic
199	C₈	—	53	1	no	—
200	C₈	—	53	1	no	—
201	C₈	—	53	1	no	—
202	C₈	—	53	1	no	—
203	C₈	—	106	1	no	—
204	C₈	—	106	1	no	—
205	C₈	—	212	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#13133272	1	1	yes	characteristic
209	C₂²	#16811658	1	1	yes	characteristic
210	C₂²	#16811674	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	#16811515	1	2	yes	complemented
216	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
217	C₄	#16811588	1	1	yes	characteristic
218	C₄	#16811657	1	1	yes	characteristic
219	C₄	#16811673	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#20290511	1	1	yes	characteristic
223	C₂	C₂×#16811515	1	1	yes	characteristic
224	C₂	#20290936	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₄×C₄⋊C₄×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×C₄⋊C₄	#6157
hypocenter	C₅₃	#12666
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₄⋊#11719910	C₄	semidirect
10. C₄⋊#11719910	C₄	semidirect
12. #16811511	C₄	semidirect
13. C₄×#11719910	C₄	semidirect
69. #11719910	C₄⋊C₄	semidirect
103. C₄×C₅₃	#347	semidirect
131. C₅₃	#12666	semidirect
215. C₄	#16811515	semidirect

Maximal subgroups

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

subgroup	index
#20291047	2
C₂×#16811515	2
C₄×#13133291	2
#20289756	2
#20274584	2
#12666	53