Subgroups — #27388871

Order 13568.

1030 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20275434	C₂	1	4	yes	maximal, complemented
4	#20274895	C₂	1	4	yes	maximal
5	C₄×C₄⋊#88322	C₂	1	4	yes	maximal, complemented
6	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	2	yes	—
8	#16810676	—	2	4	no	—
9	C₂×C₄⋊#88322	C₂²	1	8	yes	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	—	2	4	no	—
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	#16810378	C₂²	1	8	yes	—
17	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#376	C₄	1	2	yes	—
25	C₄³×C₅₃	C₂²	1	2	yes	—
26	C₂²×#88322	—	2	8	no	—
27	C₂²×#88322	—	2	4	no	—
28	C₂²×#88322	C₂³	1	4	yes	—
29	C₂²×#88322	C₂³	1	4	yes	—
30	C₄⋊#88322	—	2	4	no	—
31	C₄⋊#88322	—	2	16	no	—
32	C₂²⋊#88322	—	2	4	no	—
33	C₄×#88322	—	2	4	no	—
34	C₄×#88322	—	2	4	no	—
35	C₄×#88322	—	2	4	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	#117×C₅₃	—	2	4	no	—
40	#117×C₅₃	C₈	1	8	yes	complemented
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	D₄	1	4	yes	—
51	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#95×C₅₃	C₂×C₄	1	4	yes	—
56	#95×C₅₃	C₂×C₄	1	4	yes	—
57	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
76	C₂²⋊C₄×C₅₃	—	2	4	no	—
77	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
78	C₄²×C₅₃	—	2	4	no	—
79	C₄²×C₅₃	—	2	8	no	—
80	C₄²×C₅₃	—	2	8	no	—
81	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
83	C₄²×C₅₃	C₂×C₈	1	8	yes	complemented
84	C₄²×C₅₃	C₂×D₄	1	2	yes	—
85	C₄²×C₅₃	C₂×D₄	1	2	yes	—
86	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
87	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
88	C₂×#88322	—	2	16	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	8	no	—
93	C₂×#88322	#41	1	8	yes	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	—	2	2	no	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
98	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
99	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	complemented
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₅₃	#131	1	4	yes	—
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
125	#88322	—	2	8	no	—
126	#88322	—	4	4	no	—
127	#21349	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	2	4	no	—
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
133	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₄×#41	1	2	yes	—
135	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
136	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#131	1	2	yes	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	8	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	4	4	no	—
142	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
143	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
144	C₄×C₅₃	#424	1	4	yes	—
145	C₄×C₅₃	#525	1	2	yes	—
146	C₄×C₅₃	#533	1	2	yes	—
147	C₂×#431	—	53	1	no	—
148	C₂²×C₄×C₈	—	53	1	no	—
149	C₄×#117	D₅₃	1	1	yes	characteristic
150	#1939	—	53	4	no	—
151	#1850	—	53	4	no	—
152	C₄×C₄⋊C₈	—	53	4	no	—
153	C₂×C₅₃	—	2	4	no	—
154	C₂×C₅₃	#2296	1	1	yes	characteristic
155	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
157	C₂×C₅₃	C₈×#41	1	2	yes	—
158	C₂×C₅₃	#2965	1	2	yes	—
159	C₂³×C₈	—	53	2	no	—
160	C₂×#117	Dic₅₃	1	1	yes	characteristic
161	C₂×#117	Dic₅₃	1	1	yes	characteristic
162	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
163	#431	—	106	4	no	—
164	C₂×C₄⋊C₈	—	53	8	no	—
165	C₂×C₄⋊C₈	—	53	2	no	—
166	C₂×C₂²⋊C₈	—	53	2	no	—
167	C₂×C₄×C₈	—	53	2	no	—
168	C₂×C₄×C₈	—	53	2	no	—
169	C₂×C₄×C₈	—	53	2	no	—
170	C₂×C₄×C₈	—	106	4	no	—
171	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
172	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
173	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
174	C₄×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
175	#376	Dic₅₃	1	2	yes	—
176	C₄³	C₂×D₅₃	1	2	yes	—
177	#336	—	53	8	no	—
178	C₅₃	#21349	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
182	C₂²×C₈	—	53	4	no	—
183	C₂²×C₈	—	53	4	no	—
184	C₂²×C₈	—	106	8	no	—
185	C₂²×C₈	—	106	4	no	—
186	#117	—	2	4	no	—
187	#117	#88322	1	8	yes	complemented
188	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
192	C₂×C₄²	Dic₁₀₆	1	4	yes	—
193	C₂×C₄²	D₂₁₂	1	4	yes	—
194	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
195	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
196	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
197	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
202	C₄⋊C₈	—	106	16	no	—
203	C₄⋊C₈	—	106	4	no	—
204	C₂²⋊C₈	—	106	4	no	—
205	C₄×C₈	—	106	4	no	—
206	C₄×C₈	—	106	4	no	—
207	C₄×C₈	—	106	4	no	—
208	#95	C₂×Dic₅₃	1	4	yes	—
209	#95	C₂×Dic₅₃	1	4	yes	—
210	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₂×#88322	1	4	yes	—
214	C₂²×C₄	#11719917	1	4	yes	—
215	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
216	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
217	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
218	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
227	C₂×C₈	—	53	8	no	—
228	C₂×C₈	—	106	16	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₂×#88322	1	8	yes	complemented
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	C₂×#88322	1	8	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	8	no	—
240	C₄²	C₂×#88322	1	8	yes	complemented
241	C₄²	C₄⋊Dic₅₃	1	4	yes	—
242	C₄²	C₄⋊Dic₅₃	1	4	yes	—
243	C₄²	C₂×Dic₁₀₆	1	2	yes	—
244	C₄²	C₂×Dic₁₀₆	1	2	yes	—
245	C₄²	C₂×D₂₁₂	1	2	yes	—
246	C₄²	C₂×D₂₁₂	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
250	C₂³	C₂²×#88322	1	2	yes	—
251	C₂³	C₂×#11719917	1	2	yes	—
252	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	C₄⋊#88322	1	16	yes	complemented
264	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
269	C₂×C₄	C₂²×#88322	1	8	yes	—
270	C₂×C₄	C₂²×#88322	1	2	yes	—
271	C₂×C₄	C₂²×#88322	1	2	yes	—
272	C₂×C₄	C₂×#11719917	1	2	yes	—
273	C₂×C₄	C₂×#11719917	1	2	yes	—
274	C₂×C₄	#13133224	1	4	yes	—
275	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
276	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
277	C₂×C₄	#41×D₅₃	1	4	yes	—
278	C₈	—	106	8	no	—
279	C₈	—	212	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	C₂×C₄⋊#88322	1	4	yes	—
285	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
286	C₂²	#41×Dic₅₃	1	2	yes	—
287	C₂²	C₂³×#88322	1	1	yes	characteristic
288	C₂²	C₂²×#11719917	1	1	yes	characteristic
289	C₂²	C₂×#13133224	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₄⋊#88322	1	4	yes	—
295	C₄	C₂×C₄⋊#88322	1	4	yes	—
296	C₄	#16810650	1	4	yes	—
297	C₄	#16811135	1	2	yes	—
298	C₄	#16811176	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	#20277199	1	1	yes	characteristic
301	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
302	C₂	C₂×#16810650	1	1	yes	characteristic
303	C₂	#41×#88322	1	2	yes	—
304	C₂	#20283682	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#21349
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20275434	C₂	semidirect
5. C₄×C₄⋊#88322	C₂	semidirect
40. #117×C₅₃	C₈	semidirect
75. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
83. C₄²×C₅₃	C₂×C₈	semidirect
110. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
178. C₅₃	#21349	semidirect
187. #117	#88322	semidirect
234. C₄⋊C₄	C₂×#88322	semidirect
240. C₄²	C₂×#88322	semidirect
263. C₂×C₄	C₄⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄×#88322	2
#20275434	2
#20274895	2
C₄×C₄⋊#88322	2
C₄×#117×C₅₃	2
#21349	53