Subgroups — #28311130

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291047	C₂	1	2	yes	maximal, complemented
2	C₂×#16811519	C₂	1	2	yes	maximal
3	#20289762	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₂²	1	4	yes	—
6	C₄⋊#11719910	—	2	4	no	—
7	#16811519	—	2	4	no	—
8	C₂²×#11719917	C₄	1	1	yes	characteristic
9	C₂²×#11719917	C₂²	1	1	yes	characteristic
10	C₄×#11719917	—	2	4	no	—
11	C₂×#13133128	C₄	1	1	yes	characteristic
12	C₂×#13133128	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
16	#13133291	—	2	4	no	—
17	#13133291	—	2	4	no	—
18	C₂×#11719910	—	2	4	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	D₄	1	2	yes	—
25	C₂×#11719917	Q₈	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	1	yes	characteristic
30	#13133128	—	2	2	no	—
31	#13133128	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₄×#88322	D₄	1	4	yes	—
34	C₂³×C₄×C₅₃	C₈	1	2	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	4	4	no	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	2	1	no	—
59	C₄²×C₅₃	—	2	1	no	—
60	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
62	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
63	#11719917	—	2	8	no	—
64	#11719917	—	2	4	no	—
65	#11719917	—	2	4	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	C₂²⋊C₄	1	2	yes	—
73	C₂×#88322	C₄⋊C₄	1	2	yes	—
74	C₂×#88322	M₁₆	1	2	yes	—
75	C₂×#88322	C₂×D₄	1	1	yes	characteristic
76	C₂×#88322	C₂×D₄	1	2	yes	—
77	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
78	C₂×#88322	#41	1	2	yes	—
79	#11719910	—	4	4	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
84	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	#88322	—	2	2	no	—
106	#88322	—	2	2	no	—
107	#88322	—	2	2	no	—
108	#88322	#121	1	2	yes	—
109	#88322	C₂²⋊Q₈	1	2	yes	—
110	#12785	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₅₃	#431	1	1	yes	characteristic
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	#349	1	4	yes	complemented
123	C₄×C₅₃	#380	1	1	yes	characteristic
124	C₄×C₅₃	#407	1	1	yes	characteristic
125	C₄×C₅₃	#423	1	1	yes	characteristic
126	C₄×C₅₃	#432	1	1	yes	characteristic
127	C₂×C₄×M₁₆	—	53	1	no	—
128	#2150	—	53	2	no	—
129	C₂×#349	—	53	2	no	—
130	#1377	—	53	2	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	#1797	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#349	1	2	yes	—
136	C₂²×M₁₆	—	53	1	no	—
137	C₂²×M₁₆	—	53	1	no	—
138	C₂²×C₄²	#6157	1	1	yes	characteristic
139	C₂×M₃₂	—	53	4	no	—
140	C₄×M₁₆	—	106	4	no	—
141	C₂×#97	—	53	1	no	—
142	C₂×#97	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₄⋊C₁₆	—	106	4	no	—
146	#349	—	106	4	no	—
147	C₅₃	#12785	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	#88324	1	2	yes	—
149	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
150	C₂×M₁₆	—	53	2	no	—
151	C₂×M₁₆	—	53	2	no	—
152	C₂×M₁₆	—	106	2	no	—
153	C₂×M₁₆	—	106	2	no	—
154	C₂×M₁₆	—	106	2	no	—
155	C₂×M₁₆	—	106	2	no	—
156	C₂×M₁₆	—	106	2	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	53	1	no	—
159	C₂²×C₈	—	53	1	no	—
160	C₂²×C₈	—	53	1	no	—
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	#88324	1	2	yes	—
165	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
166	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
167	M₃₂	—	106	4	no	—
168	M₃₂	—	106	4	no	—
169	C₂×C₁₆	—	106	4	no	—
170	#97	—	106	2	no	—
171	#97	—	106	2	no	—
172	C₄×C₈	—	53	4	no	—
173	C₄×C₈	—	106	2	no	—
174	C₂⁴	C₂×#88324	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	#11719936	1	2	yes	—
183	C₂²×C₄	C₄⋊#6157	1	2	yes	—
184	C₂²×C₄	C₂×#88324	1	2	yes	—
185	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
186	C₂²×C₄	C₂×#88324	1	2	yes	—
187	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
188	M₁₆	—	106	4	no	—
189	M₁₆	—	106	4	no	—
190	M₁₆	—	106	8	no	—
191	C₂×C₈	—	53	2	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	1	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	2	no	—
196	C₂×C₈	—	53	2	no	—
197	C₂×C₈	—	53	2	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	—	106	1	no	—
200	C₂×C₈	—	106	1	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	106	1	no	—
203	C₂×C₈	—	106	1	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	#11719940	1	1	yes	characteristic
209	C₄²	#11719940	1	1	yes	characteristic
210	C₄²	C₂²⋊#6157	1	2	yes	—
211	C₁₆	—	212	4	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄⋊#88324	1	2	yes	—
216	C₂³	C₂²×#88324	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊#88324	1	2	yes	—
229	C₂×C₄	C₂²⋊#88324	1	2	yes	—
230	C₂×C₄	C₄⋊#88324	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
233	C₂×C₄	#13133296	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
235	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
237	C₈	—	53	2	no	—
238	C₈	—	53	2	no	—
239	C₈	—	106	2	no	—
240	C₈	—	106	2	no	—
241	C₈	—	106	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₂²⋊#88324	1	1	yes	characteristic
248	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
249	C₂²	#16811639	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#16811519	1	4	yes	complemented
254	C₄	#16811565	1	1	yes	characteristic
255	C₄	#16811594	1	1	yes	characteristic
256	C₄	#16811626	1	1	yes	characteristic
257	C₄	#16811644	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#20290569	1	1	yes	characteristic
262	C₂	C₂×#16811519	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂²×C₄²×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂²×C₄²	#6157
hypocenter	C₅₃	#12785
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #20291047	C₂	semidirect
3. #20289762	C₂	semidirect
122. C₄×C₅₃	#349	semidirect
147. C₅₃	#12785	semidirect
253. C₄	#16811519	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×#16811519	2
#20289762	2
C₂×C₄×#11719917	2
#12785	53