Subgroups — #28310248

Order 13568.

586 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	C₂²×#13133291	C₂	1	2	yes	maximal
2	#20289762	C₂	1	4	yes	maximal, complemented
3	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133291	—	2	2	no	—
5	C₂×#13133291	—	2	4	no	—
6	C₂×#13133291	C₄	1	8	yes	complemented
7	C₂×#13133291	C₂²	1	4	yes	—
8	C₂²×#11719910	—	2	2	no	—
9	C₂²×#11719917	C₄	1	1	yes	characteristic
10	C₂³×#88322	C₂²	1	1	yes	characteristic
11	#16810579	—	2	2	no	—
12	#16810579	—	2	2	no	—
13	C₂×#13133128	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₄	1	2	yes	—
15	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
16	#13133291	—	2	8	no	—
17	#13133291	—	2	4	no	—
18	#13133291	C₂×C₄	1	8	yes	—
19	C₂×#11719910	—	2	8	no	—
20	C₂×#11719910	—	2	4	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	C₈	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	4	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	D₄	1	2	yes	—
28	C₂²×#88322	D₄	1	2	yes	—
29	C₂²×#88322	D₄	1	2	yes	—
30	C₂²×#88322	Q₈	1	2	yes	—
31	C₂²×#88322	C₂³	1	1	yes	characteristic
32	#13133128	—	2	4	no	—
33	C₄×#88322	—	2	2	no	—
34	C₄×#88322	—	2	2	no	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#117×C₅₃	—	2	2	no	—
37	#117×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
39	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	2	4	no	—
52	C₂²⋊C₄×C₅₃	—	2	4	no	—
53	C₄²×C₅₃	—	2	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	#11719917	—	2	8	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	C₄²	1	2	yes	—
59	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×#88322	C₂²⋊C₄	1	4	yes	—
62	C₂×#88322	C₂²⋊C₄	1	4	yes	—
63	C₂×#88322	C₄⋊C₄	1	2	yes	—
64	C₂×#88322	C₄⋊C₄	1	4	yes	—
65	C₂×#88322	C₄⋊C₄	1	4	yes	—
66	C₂×#88322	C₂×C₈	1	4	yes	—
67	C₂×#88322	C₂²×C₄	1	2	yes	—
68	C₂×#88322	C₂×D₄	1	1	yes	characteristic
69	C₂×#88322	C₂×D₄	1	2	yes	—
70	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
71	#11719910	—	2	8	no	—
72	C₂³×C₅₃	—	4	2	no	—
73	C₂³×C₅₃	C₄×C₈	1	2	yes	—
74	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
75	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	#95	1	4	yes	—
80	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₅₃	#97	1	4	yes	—
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	#88322	—	2	4	no	—
93	#88322	#95	1	2	yes	—
94	#88322	#95	1	2	yes	—
95	#88322	C₂×C₂²⋊C₄	1	2	yes	—
96	#88322	C₂×C₄⋊C₄	1	2	yes	—
97	#12551	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	—	4	2	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	#336	1	4	yes	—
103	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₅₃	—	4	4	no	—
107	C₄×C₅₃	#336	1	4	yes	—
108	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
110	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×M₃₂	—	53	2	no	—
113	C₂×#405	—	53	1	no	—
114	#1377	—	53	4	no	—
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	—	2	1	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
119	C₂×C₅₃	#2115	1	2	yes	—
120	C₂²×M₁₆	—	53	1	no	—
121	C₂³×C₈	—	53	1	no	—
122	C₂×#117	#6157	1	1	yes	characteristic
123	C₂×M₃₂	—	53	4	no	—
124	C₂×M₃₂	—	53	8	no	—
125	C₂×M₃₂	—	106	4	no	—
126	C₂×M₃₂	—	106	2	no	—
127	C₂²×C₁₆	—	106	2	no	—
128	#405	—	106	2	no	—
129	#405	—	106	2	no	—
130	C₂×#97	—	53	2	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	C₅₃	#12551	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
134	C₂×M₁₆	—	53	4	no	—
135	C₂×M₁₆	—	106	2	no	—
136	C₂×M₁₆	—	106	2	no	—
137	C₂²×C₈	—	53	4	no	—
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	—	53	2	no	—
140	C₂²×C₈	—	53	2	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	53	2	no	—
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	#88324	1	2	yes	—
148	C₂×C₂²⋊C₄	#88324	1	2	yes	—
149	C₂×C₄²	C₂×#6157	1	2	yes	—
150	M₃₂	—	53	8	no	—
151	M₃₂	—	106	8	no	—
152	M₃₂	—	106	4	no	—
153	C₂×C₁₆	—	106	8	no	—
154	C₂×C₁₆	—	106	4	no	—
155	#97	—	106	4	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	—	106	2	no	—
158	C₂⁴	C₂×#88324	1	1	yes	characteristic
159	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
160	C₂²×C₄	#11719936	1	2	yes	—
161	C₂²×C₄	C₄×#6157	1	2	yes	—
162	C₂²×C₄	C₄⋊#6157	1	2	yes	—
163	C₂²×C₄	C₂×#88324	1	4	yes	—
164	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
165	C₂²×C₄	#11719940	1	2	yes	—
166	C₂²×C₄	C₂²⋊#6157	1	2	yes	—
167	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
168	M₁₆	—	106	8	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	—	53	4	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	1	no	—
176	C₂×C₈	—	53	1	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	1	no	—
179	C₂×C₈	—	53	2	no	—
180	C₂×C₈	—	53	1	no	—
181	C₂×C₈	—	53	4	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	106	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₁₆	—	106	8	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₄×#88324	1	2	yes	—
191	C₂³	C₂²⋊#88324	1	2	yes	—
192	C₂³	C₄⋊#88324	1	2	yes	—
193	C₂³	C₂²×#88324	1	1	yes	characteristic
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	(C₂×#6157)⋊C₄	1	4	yes	—
197	C₂×C₄	C₄×#88324	1	2	yes	—
198	C₂×C₄	#13133272	1	4	yes	—
199	C₂×C₄	C₂²⋊#88324	1	2	yes	—
200	C₂×C₄	#13133274	1	4	yes	—
201	C₂×C₄	C₄⋊#88324	1	2	yes	—
202	C₂×C₄	#13133284	1	4	yes	—
203	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
206	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
209	C₈	—	53	2	no	—
210	C₈	—	53	2	no	—
211	C₈	—	53	2	no	—
212	C₈	—	53	2	no	—
213	C₈	—	106	4	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	#16811489	1	4	yes	—
219	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
220	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
221	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	#16811489	1	4	yes	—
224	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
225	C₄	C₂×#13133272	1	1	yes	characteristic
226	C₄	C₂×#13133274	1	1	yes	characteristic
227	C₄	C₂×#13133284	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	4	no	—
231	C₂	C₂×#16811489	1	1	yes	characteristic
232	C₂	#20290951	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#117×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×#117	#6157
hypocenter	C₅₃	#12551
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #20289762	C₂	semidirect
6. C₂×#13133291	C₄	semidirect
132. C₅₃	#12551	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
#20289762	2
C₂×#16810579	2
#12551	53