Subgroups — #28328529

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291099	C₂	1	2	yes	maximal, complemented
2	#20291095	C₂	1	4	yes	maximal, complemented
3	D₄×#11719910	C₂	1	2	yes	maximal, complemented
4	#20291076	C₂	1	1	yes	maximal, characteristic, complemented
5	#20291072	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290945	C₂	1	4	yes	maximal, complemented
7	#20283427	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133291	C₂²	1	4	yes	—
9	C₂²×#11719910	C₂²	1	4	yes	—
10	C₄⋊#11719910	C₂²	1	2	yes	—
11	C₄⋊#11719910	C₂²	1	2	yes	—
12	C₂²⋊#11719910	C₂²	1	2	yes	—
13	C₂²⋊#11719910	C₂²	1	2	yes	—
14	C₂²⋊#11719910	C₂²	1	4	yes	—
15	C₂²⋊#11719910	C₂²	1	4	yes	—
16	#16811511	C₂²	1	2	yes	—
17	C₄×#11719910	C₂²	1	1	yes	characteristic
18	C₄×#11719910	C₂²	1	1	yes	characteristic
19	C₂×#13133224	C₂²	1	2	yes	—
20	#16810701	C₄	1	2	yes	—
21	#16810701	C₄	1	2	yes	—
22	D₄×#88322	C₂²	1	2	yes	—
23	D₄×#88322	C₂²	1	2	yes	—
24	#16810647	C₄	1	1	yes	characteristic
25	#16810590	C₄	1	2	yes	—
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	C₅₃×#532	C₄	1	1	yes	characteristic
28	#13133291	—	4	4	no	—
29	C₂×#11719910	—	4	4	no	—
30	C₂×#11719910	C₂³	1	2	yes	—
31	C₂×#11719910	C₂³	1	4	yes	—
32	C₂×#11719910	C₂³	1	2	yes	—
33	#13133224	—	4	4	no	—
34	C₂×#11719917	C₂×C₄	1	2	yes	—
35	C₂×#11719917	C₂×C₄	1	4	yes	—
36	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂²×#88322	C₂³	1	1	yes	characteristic
38	C₂²×#88322	C₂³	1	4	yes	—
39	C₄⋊#88322	C₂×C₄	1	2	yes	—
40	C₄⋊#88322	C₂×C₄	1	2	yes	—
41	C₂²⋊#88322	C₂×C₄	1	4	yes	—
42	C₂²⋊#88322	C₂×C₄	1	4	yes	—
43	C₄×#88322	C₂×C₄	1	1	yes	characteristic
44	C₄×#88322	C₂×C₄	1	1	yes	characteristic
45	C₄×#88322	C₂³	1	1	yes	characteristic
46	C₄×#88322	C₂³	1	1	yes	characteristic
47	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
49	#127×C₅₃	C₈	1	1	yes	characteristic
50	#124×C₅₃	C₈	1	2	yes	—
51	#121×C₅₃	C₈	1	2	yes	—
52	#121×C₅₃	C₈	1	2	yes	—
53	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	#41×C₅₃	—	4	2	no	—
57	#41×C₅₃	—	4	2	no	—
58	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
62	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
67	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
69	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
70	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
71	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	#11719917	—	4	2	no	—
76	#11719917	—	4	4	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	4	1	no	—
81	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
82	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
83	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
84	C₂×#88322	C₂²×C₄	1	4	yes	—
85	C₂×#88322	C₂⁴	1	1	yes	characteristic
86	#11719910	—	2	2	no	—
87	#11719910	—	2	4	no	—
88	#11719910	—	2	2	no	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
91	Q₈×C₅₃	—	4	2	no	—
92	D₄×C₅₃	—	2	4	no	—
93	D₄×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	4	2	no	—
95	D₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	#88322	—	2	4	no	—
110	#88322	—	2	1	no	—
111	#88322	—	2	1	no	—
112	#88322	—	2	1	no	—
113	#88322	H(5,𝔽₂)	1	2	yes	—
114	#33495	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	4	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	4	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	4	no	—
121	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	#504	1	2	yes	—
128	C₄×C₅₃	#518	1	1	yes	characteristic
129	C₄×C₅₃	#568	1	1	yes	characteristic
130	#2931	—	53	1	no	—
131	#2169	—	53	2	no	—
132	#2168	—	53	4	no	—
133	D₄×C₁₆	—	53	2	no	—
134	#2163	—	53	1	no	—
135	#2162	—	53	1	no	—
136	#2113	—	53	4	no	—
137	C₂×C₅₃	—	4	1	no	—
138	C₂×C₅₃	—	4	4	no	—
139	C₂×C₅₃	#2972	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
141	C₂×C₅₃	#3407	1	1	yes	characteristic
142	C₂×#131	—	53	2	no	—
143	#532	#6157	1	1	yes	characteristic
144	C₂×M₃₂	—	53	4	no	—
145	C₂²×C₁₆	—	53	4	no	—
146	#436	—	53	2	no	—
147	#436	—	53	2	no	—
148	C₈×D₄	—	53	2	no	—
149	C₈×D₄	—	53	2	no	—
150	#423	—	53	1	no	—
151	#408	—	53	2	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₄⋊C₁₆	—	53	2	no	—
154	C₄⋊C₁₆	—	53	2	no	—
155	C₂²⋊C₁₆	—	53	2	no	—
156	C₂²⋊C₁₆	—	53	2	no	—
157	C₂²⋊C₁₆	—	53	4	no	—
158	C₂²⋊C₁₆	—	53	4	no	—
159	#346	—	53	2	no	—
160	C₄×C₁₆	—	53	1	no	—
161	C₄×C₁₆	—	53	1	no	—
162	C₅₃	#33495	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	C₂×#6157	1	2	yes	—
164	#131	—	212	4	no	—
165	C₂×M₁₆	—	53	2	no	—
166	C₂×M₁₆	—	53	4	no	—
167	C₂²×C₈	—	53	1	no	—
168	C₂²×C₈	—	53	4	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₄⋊Q₈	#88324	1	1	yes	characteristic
171	#127	#88324	1	1	yes	characteristic
172	#124	#88324	1	2	yes	—
173	#121	#88324	1	2	yes	—
174	#121	#88324	1	2	yes	—
175	C₄×D₄	C₂×#6157	1	2	yes	—
176	C₄×D₄	C₂×#6157	1	2	yes	—
177	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
178	M₃₂	—	212	4	no	—
179	C₂×C₁₆	—	53	2	no	—
180	C₂×C₁₆	—	53	2	no	—
181	C₂×C₁₆	—	53	4	no	—
182	C₂×C₁₆	—	212	4	no	—
183	C₄⋊C₈	—	53	2	no	—
184	C₄⋊C₈	—	53	2	no	—
185	C₂²⋊C₈	—	53	4	no	—
186	C₂²⋊C₈	—	53	4	no	—
187	C₄×C₈	—	53	1	no	—
188	C₄×C₈	—	53	1	no	—
189	C₄×C₈	—	53	1	no	—
190	C₄×C₈	—	53	1	no	—
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	C₂×#88324	1	2	yes	—
194	C₂×D₄	C₂×#88324	1	2	yes	—
195	C₂×D₄	C₂×#88324	1	2	yes	—
196	C₂×D₄	C₂×#88324	1	4	yes	—
197	C₂×D₄	C₂×#88324	1	2	yes	—
198	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
199	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
200	C₂²×C₄	C₂²×#6157	1	4	yes	—
201	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
202	M₁₆	—	212	2	no	—
203	M₁₆	—	212	4	no	—
204	C₂×C₈	—	53	1	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	53	1	no	—
207	C₂×C₈	—	53	1	no	—
208	C₂×C₈	—	53	4	no	—
209	C₂×C₈	—	106	4	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	212	1	no	—
213	C₄⋊C₄	C₂×#88324	1	2	yes	—
214	C₄⋊C₄	C₂×#88324	1	2	yes	—
215	C₂²⋊C₄	C₂×#88324	1	4	yes	—
216	C₂²⋊C₄	C₂×#88324	1	4	yes	—
217	C₄²	C₂×#88324	1	1	yes	characteristic
218	C₄²	C₂×#88324	1	1	yes	characteristic
219	C₄²	C₂²×#6157	1	1	yes	characteristic
220	C₄²	C₂²×#6157	1	1	yes	characteristic
221	C₁₆	—	106	2	no	—
222	C₁₆	—	106	4	no	—
223	C₁₆	—	106	2	no	—
224	C₂³	C₂²×#88324	1	4	yes	—
225	C₂³	C₂²×#88324	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
238	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
239	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
240	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
241	C₂×C₄	C₂²×#88324	1	4	yes	—
242	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
243	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
244	C₈	—	53	2	no	—
245	C₈	—	106	4	no	—
246	C₈	—	106	1	no	—
247	C₈	—	106	1	no	—
248	C₈	—	106	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₂³×#88324	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#16811681	1	2	yes	—
262	C₄	#16811690	1	1	yes	characteristic
263	C₄	#16811701	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	#20291562	1	1	yes	characteristic
267	C₂	C₂×#16811681	1	1	yes	characteristic
268	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting, prime radical	C₅₃×#532	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#532	#6157
hypocenter	C₅₃	#33495
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20291099	C₂	semidirect
2. #20291095	C₂	semidirect
3. D₄×#11719910	C₂	semidirect
4. #20291076	C₂	semidirect
5. #20291072	C₂	semidirect
6. #20290945	C₂	semidirect
162. C₅₃	#33495	semidirect

Maximal subgroups

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

subgroup	index
#20291099	2
#20291095	2
D₄×#11719910	2
#20291076	2
#20291072	2
#20290945	2
#20283427	2
#33495	53