Subgroups — #28310217

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133291	C₂	1	2	yes	maximal
2	#20289753	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133291	—	2	4	no	—
5	C₂×#13133291	—	2	4	no	—
6	C₂×#13133291	C₄	1	8	yes	complemented
7	C₂²×#11719910	C₂²	1	4	yes	—
8	C₂³×#88322	C₄	1	1	yes	characteristic
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×#88322	—	2	4	no	—
11	C₂×C₄×#88322	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
14	#13133291	—	2	16	no	—
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	—	2	4	no	—
17	C₂×#11719910	C₂×C₄	1	8	yes	complemented
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	C₈	1	4	yes	—
21	C₂²×#88322	C₂×C₄	1	4	yes	—
22	C₂²×#88322	C₂×C₄	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	D₄	1	4	yes	—
25	C₂²×#88322	D₄	1	2	yes	—
26	C₂²×#88322	Q₈	1	2	yes	—
27	C₂²×#88322	C₂³	1	1	yes	characteristic
28	C₄×#88322	—	2	4	no	—
29	C₄×#88322	—	2	4	no	—
30	C₂³×C₄×C₅₃	C₈	1	2	yes	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	—	2	8	no	—
39	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
46	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₂×#88322	—	2	8	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	C₄²	1	2	yes	—
55	C₂×#88322	C₂²⋊C₄	1	2	yes	—
56	C₂×#88322	C₂²⋊C₄	1	8	yes	—
57	C₂×#88322	C₄⋊C₄	1	2	yes	—
58	C₂×#88322	C₄⋊C₄	1	8	yes	—
59	C₂×#88322	C₂×C₈	1	4	yes	—
60	C₂×#88322	C₂²×C₄	1	2	yes	—
61	C₂×#88322	C₂×D₄	1	1	yes	characteristic
62	C₂×#88322	C₂×D₄	1	2	yes	—
63	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
64	#11719910	—	2	8	no	—
65	C₂³×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₄×C₈	1	2	yes	—
67	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	16	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	#95	1	4	yes	—
74	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₅₃	#97	1	4	yes	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₅₃	M₃₂	1	8	yes	complemented
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₄⋊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×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	#88322	—	2	4	no	—
88	#88322	#95	1	4	yes	—
89	#88322	C₂×C₂²⋊C₄	1	2	yes	—
90	#88322	C₂×C₄⋊C₄	1	2	yes	—
91	#12544	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	4	no	—
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	#336	1	4	yes	—
96	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
100	C₄×C₅₃	—	2	8	no	—
101	C₄×C₅₃	#336	1	4	yes	—
102	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
103	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
104	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×M₃₂	—	53	2	no	—
107	C₂²×C₄×C₈	—	53	1	no	—
108	#1374	—	53	4	no	—
109	C₂×C₅₃	—	2	4	no	—
110	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
111	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
112	C₂×C₅₃	#2112	1	2	yes	—
113	C₂×C₅₃	#2150	1	2	yes	—
114	C₂³×C₈	—	53	1	no	—
115	C₂³×C₈	—	53	1	no	—
116	C₂²×C₄²	#6157	1	1	yes	characteristic
117	C₂×M₃₂	—	53	8	no	—
118	C₂×M₃₂	—	106	4	no	—
119	C₂×M₃₂	—	106	4	no	—
120	C₂²×C₁₆	—	53	4	no	—
121	C₂×C₄×C₈	—	53	2	no	—
122	C₂×C₄×C₈	—	53	2	no	—
123	C₂×C₄×C₈	—	106	4	no	—
124	C₅₃	#12544	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₄	#88324	1	2	yes	—
126	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
127	C₂²×C₈	—	53	4	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	—	53	4	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	2	no	—
134	C₂²×C₈	—	53	4	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₂²×C₈	—	106	2	no	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	#88324	1	2	yes	—
140	C₂×C₄²	C₂×#6157	1	2	yes	—
141	M₃₂	—	106	16	no	—
142	C₂×C₁₆	—	53	8	no	—
143	C₂×C₁₆	—	106	4	no	—
144	C₂×C₁₆	—	106	4	no	—
145	C₄×C₈	—	106	4	no	—
146	C₄×C₈	—	106	4	no	—
147	C₂⁴	C₂×#88324	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
151	C₂²×C₄	#11719936	1	2	yes	—
152	C₂²×C₄	C₄×#6157	1	2	yes	—
153	C₂²×C₄	C₄⋊#6157	1	2	yes	—
154	C₂²×C₄	C₂×#88324	1	4	yes	—
155	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
156	C₂²×C₄	#11719940	1	2	yes	—
157	C₂²×C₄	C₂²⋊#6157	1	2	yes	—
158	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
159	C₂×C₈	—	53	8	no	—
160	C₂×C₈	—	53	8	no	—
161	C₂×C₈	—	53	2	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	106	8	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₁₆	—	106	8	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄×#88324	1	2	yes	—
178	C₂³	C₂²⋊#88324	1	2	yes	—
179	C₂³	C₄⋊#88324	1	2	yes	—
180	C₂³	C₂²×#88324	1	1	yes	characteristic
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	(C₂×#6157)⋊C₄	1	4	yes	—
185	C₂×C₄	C₄×#88324	1	2	yes	—
186	C₂×C₄	#13133272	1	4	yes	—
187	C₂×C₄	C₂²⋊#88324	1	2	yes	—
188	C₂×C₄	#13133274	1	4	yes	—
189	C₂×C₄	C₄⋊#88324	1	2	yes	—
190	C₂×C₄	#13133284	1	4	yes	—
191	C₂×C₄	#13133291	1	8	yes	complemented
192	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
195	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
197	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
198	C₈	—	53	4	no	—
199	C₈	—	53	2	no	—
200	C₈	—	53	2	no	—
201	C₈	—	106	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#16811489	1	4	yes	—
206	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
207	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
208	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
209	C₂²	C₂×#13133291	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#16811489	1	4	yes	—
212	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
213	C₄	C₂×#13133272	1	1	yes	characteristic
214	C₄	C₂×#13133274	1	1	yes	characteristic
215	C₄	C₂×#13133284	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₂×#16811489	1	1	yes	characteristic
218	C₂	C₄×#13133291	1	2	yes	—
219	C₂	#20290942	1	2	yes	—
220	C₂	#20291047	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂²×C₄²×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂²×C₄²	#6157
hypocenter	C₅₃	#12544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #20289753	C₂	semidirect
6. C₂×#13133291	C₄	semidirect
17. C₂×#11719910	C₂×C₄	semidirect
80. C₂×C₄×C₅₃	M₃₂	semidirect
124. C₅₃	#12544	semidirect
191. C₂×C₄	#13133291	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133291	2
#20289753	2
C₂²×C₄×#88322	2
#12544	53