Subgroups — #28328568

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	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	2	yes	maximal, complemented
5	#20291072	C₂	1	1	yes	maximal, characteristic, complemented
6	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal, complemented
8	#41×#88322	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133291	C₂²	1	2	yes	—
10	C₂²×#11719910	C₂²	1	4	yes	—
11	C₄⋊#11719910	C₂²	1	1	yes	characteristic
12	C₄⋊#11719910	C₂²	1	1	yes	characteristic
13	C₄⋊#11719910	C₂²	1	2	yes	—
14	C₄⋊#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	4	yes	—
16	C₂²⋊#11719910	C₂²	1	2	yes	—
17	C₂²⋊#11719910	C₂²	1	4	yes	—
18	#16811511	C₂²	1	4	yes	—
19	C₄×#11719910	C₂²	1	2	yes	—
20	Q₈×#88322	C₄	1	1	yes	characteristic
21	Q₈×#88322	C₂²	1	1	yes	characteristic
22	D₄×#88322	C₄	1	2	yes	—
23	D₄×#88322	C₄	1	1	yes	characteristic
24	D₄×#88322	C₂²	1	1	yes	characteristic
25	D₄×#88322	C₂²	1	2	yes	—
26	#16810676	C₄	1	1	yes	characteristic
27	#16810676	C₄	1	2	yes	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₄×#88322	C₂²	1	2	yes	—
30	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
31	#13133291	—	4	2	no	—
32	C₂×#11719910	—	4	4	no	—
33	C₂×#11719910	C₂³	1	2	yes	—
34	C₂×#11719910	C₂³	1	4	yes	—
35	C₂×#11719910	C₂³	1	2	yes	—
36	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂²×#88322	C₂×C₄	1	2	yes	—
38	C₂²×#88322	C₂³	1	2	yes	—
39	C₂²×#88322	C₂³	1	1	yes	characteristic
40	C₄⋊#88322	C₂×C₄	1	2	yes	—
41	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
42	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	C₂²⋊#88322	C₂×C₄	1	2	yes	—
45	C₂²⋊#88322	C₂×C₄	1	2	yes	—
46	C₂²⋊#88322	C₂×C₄	1	2	yes	—
47	C₄×#88322	C₂×C₄	1	1	yes	characteristic
48	C₄×#88322	C₂×C₄	1	2	yes	—
49	C₄×#88322	C₂×C₄	1	1	yes	characteristic
50	C₄×#88322	C₂×C₄	1	2	yes	—
51	C₄×#88322	C₂³	1	1	yes	characteristic
52	C₄×#88322	C₂³	1	1	yes	characteristic
53	C₄×#88322	C₂³	1	2	yes	—
54	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
56	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×D₄×C₅₃	C₈	1	2	yes	—
58	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
59	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	#117×C₅₃	C₈	1	2	yes	—
62	#117×C₅₃	C₈	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
65	#41×C₅₃	—	2	4	no	—
66	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
76	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
81	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×C₈	1	2	yes	—
84	C₄²×C₅₃	C₂×C₈	1	2	yes	—
85	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	2	no	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	1	no	—
95	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
96	C₂×#88322	C₂²×C₄	1	2	yes	—
97	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
98	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
99	C₂×#88322	C₂²×C₄	1	2	yes	—
100	C₂×#88322	C₂⁴	1	1	yes	characteristic
101	#11719910	—	2	2	no	—
102	#11719910	—	2	4	no	—
103	#11719910	—	2	2	no	—
104	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
106	Q₈×C₅₃	—	2	2	no	—
107	D₄×C₅₃	—	2	4	no	—
108	D₄×C₅₃	M₃₂	1	4	yes	complemented
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	#88322	—	2	2	no	—
127	#88322	—	2	1	no	—
128	#88322	—	2	1	no	—
129	#88322	—	2	1	no	—
130	#88322	—	2	2	no	—
131	#88322	H(5,𝔽₂)	1	1	yes	characteristic
132	#88322	2₋¹⁺⁴	1	1	yes	characteristic
133	#33504	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
140	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
148	C₄×C₅₃	#504	1	2	yes	—
149	C₄×C₅₃	#520	1	1	yes	characteristic
150	C₄×C₅₃	#568	1	1	yes	characteristic
151	C₈×#41	—	53	1	no	—
152	#2183	—	53	2	no	—
153	#2168	—	53	4	no	—
154	D₄×C₁₆	—	53	2	no	—
155	#2163	—	53	2	no	—
156	#2162	—	53	1	no	—
157	#2150	—	53	1	no	—
158	C₂×C₄⋊C₁₆	—	53	2	no	—
159	C₂×C₅₃	—	2	4	no	—
160	C₂×C₅₃	—	4	1	no	—
161	C₂×C₅₃	#2988	1	1	yes	characteristic
162	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
163	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
164	C₄×#41	#6157	1	1	yes	characteristic
165	C₂×M₃₂	—	53	2	no	—
166	C₂²×C₁₆	—	53	4	no	—
167	C₈×Q₈	—	53	1	no	—
168	C₈×Q₈	—	53	1	no	—
169	C₈×D₄	—	53	1	no	—
170	C₈×D₄	—	53	2	no	—
171	C₈×D₄	—	53	2	no	—
172	C₈×D₄	—	53	1	no	—
173	#431	—	53	1	no	—
174	#431	—	53	2	no	—
175	C₂×C₄×C₈	—	53	2	no	—
176	C₂×C₄×C₈	—	53	1	no	—
177	C₄⋊C₁₆	—	53	1	no	—
178	C₄⋊C₁₆	—	53	2	no	—
179	C₄⋊C₁₆	—	53	4	no	—
180	C₄⋊C₁₆	—	53	2	no	—
181	C₄⋊C₁₆	—	53	1	no	—
182	C₂²⋊C₁₆	—	53	2	no	—
183	C₂²⋊C₁₆	—	53	4	no	—
184	#346	—	53	4	no	—
185	C₄×C₁₆	—	53	2	no	—
186	C₅₃	#33504	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	#88324	1	1	yes	characteristic
188	C₂²×C₈	—	53	2	no	—
189	C₂²×C₈	—	53	1	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₄×Q₈	#88324	1	1	yes	characteristic
193	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
194	C₄×D₄	#88324	1	2	yes	—
195	C₄×D₄	#88324	1	1	yes	characteristic
196	C₄×D₄	C₂×#6157	1	2	yes	—
197	C₄×D₄	C₂×#6157	1	1	yes	characteristic
198	#117	#88324	1	1	yes	characteristic
199	#117	#88324	1	2	yes	—
200	C₂×C₄²	C₂×#6157	1	2	yes	—
201	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
202	M₃₂	—	212	2	no	—
203	C₂×C₁₆	—	53	2	no	—
204	C₂×C₁₆	—	53	2	no	—
205	C₂×C₁₆	—	53	4	no	—
206	C₂×C₁₆	—	212	4	no	—
207	C₄⋊C₈	—	53	2	no	—
208	C₄⋊C₈	—	53	1	no	—
209	C₄⋊C₈	—	53	1	no	—
210	C₄⋊C₈	—	53	2	no	—
211	C₂²⋊C₈	—	53	2	no	—
212	C₂²⋊C₈	—	53	2	no	—
213	C₂²⋊C₈	—	53	2	no	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	53	1	no	—
216	C₄×C₈	—	53	2	no	—
217	C₄×C₈	—	53	1	no	—
218	C₄×C₈	—	53	1	no	—
219	C₄×C₈	—	53	2	no	—
220	C₄×C₈	—	53	1	no	—
221	#41	—	2	4	no	—
222	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
223	C₂×D₄	C₂×#88324	1	2	yes	—
224	C₂×D₄	C₂×#88324	1	1	yes	characteristic
225	C₂²×C₄	C₂×#88324	1	2	yes	—
226	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
227	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
228	C₂²×C₄	C₂×#88324	1	2	yes	—
229	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
230	C₂²×C₄	C₂²×#6157	1	2	yes	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	53	1	no	—
235	C₂×C₈	—	53	1	no	—
236	C₂×C₈	—	53	2	no	—
237	C₂×C₈	—	106	2	no	—
238	C₂×C₈	—	106	1	no	—
239	C₂×C₈	—	106	2	no	—
240	C₂×C₈	—	106	1	no	—
241	C₂×C₈	—	212	2	no	—
242	C₂×C₈	—	212	1	no	—
243	C₄⋊C₄	C₂×#88324	1	2	yes	—
244	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
245	C₄⋊C₄	C₂×#88324	1	2	yes	—
246	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
247	C₂²⋊C₄	C₂×#88324	1	2	yes	—
248	C₂²⋊C₄	C₂×#88324	1	2	yes	—
249	C₂²⋊C₄	C₂×#88324	1	2	yes	—
250	C₄²	C₂×#88324	1	2	yes	—
251	C₄²	C₂×#88324	1	1	yes	characteristic
252	C₄²	C₂×#88324	1	2	yes	—
253	C₄²	C₂×#88324	1	1	yes	characteristic
254	C₄²	C₂²×#6157	1	1	yes	characteristic
255	C₄²	C₂²×#6157	1	2	yes	—
256	C₄²	C₂²×#6157	1	1	yes	characteristic
257	C₁₆	—	106	2	no	—
258	C₁₆	—	106	4	no	—
259	C₁₆	—	106	2	no	—
260	C₂³	C₂²×#88324	1	1	yes	characteristic
261	C₂³	C₂²×#88324	1	2	yes	—
262	Q₈	—	2	2	no	—
263	D₄	—	2	4	no	—
264	D₄	#13133291	1	4	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
274	C₂×C₄	C₂²×#88324	1	2	yes	—
275	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
276	C₂×C₄	C₂²×#88324	1	2	yes	—
277	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
278	C₂×C₄	C₂²×#88324	1	2	yes	—
279	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
280	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
281	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
282	C₈	—	53	1	no	—
283	C₈	—	53	1	no	—
284	C₈	—	106	2	no	—
285	C₈	—	106	1	no	—
286	C₈	—	106	1	no	—
287	C₈	—	106	2	no	—
288	C₈	—	106	1	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₂×#13133291	1	4	yes	—
295	C₂²	C₂³×#88324	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	C₂×#13133291	1	2	yes	—
303	C₄	#16811681	1	2	yes	—
304	C₄	#16811692	1	1	yes	characteristic
305	C₄	#16811702	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	—	4	1	no	—
308	C₂	#20291630	1	1	yes	characteristic
309	C₂	C₂²×#13133291	1	1	yes	characteristic
310	C₂	C₂×#16811681	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×#41×C₅₃	C₄
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₄×#41	#6157
hypocenter	C₅₃	#33504
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #20291130	C₂	semidirect
2. #20291095	C₂	semidirect
3. D₄×#11719910	C₂	semidirect
4. #20291076	C₂	semidirect
5. #20291072	C₂	semidirect
6. #20291047	C₂	semidirect
7. C₂×C₄⋊#11719910	C₂	semidirect
108. D₄×C₅₃	M₃₂	semidirect
186. C₅₃	#33504	semidirect
264. D₄	#13133291	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
#20291095	2
D₄×#11719910	2
#20291076	2
#20291072	2
#20291047	2
C₂×C₄⋊#11719910	2
#41×#88322	2
#33504	53