Subgroups — #28328334

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811681	C₂	1	2	yes	maximal, complemented
2	#20291099	C₂	1	2	yes	maximal, complemented
3	#20291095	C₂	1	4	yes	maximal, complemented
4	D₄×#11719910	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290945	C₂	1	2	yes	maximal, complemented
7	C₄×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
8	#20283427	C₂	1	1	yes	maximal, characteristic
9	#16811681	—	2	8	no	—
10	C₂×#13133291	C₂²	1	4	yes	—
11	C₂×#13133291	C₂²	1	1	yes	characteristic
12	C₂×#13133291	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	4	yes	—
14	C₂²×#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	1	yes	characteristic
16	C₄⋊#11719910	C₂²	1	2	yes	—
17	C₄⋊#11719910	C₂²	1	1	yes	characteristic
18	C₂²⋊#11719910	C₂²	1	4	yes	—
19	C₂²⋊#11719910	C₂²	1	4	yes	—
20	#16811511	C₂²	1	2	yes	—
21	C₄×#11719910	C₂²	1	2	yes	—
22	C₂×#13133224	C₂²	1	2	yes	—
23	#16810701	C₄	1	2	yes	—
24	#16810701	C₄	1	2	yes	—
25	D₄×#88322	C₂²	1	2	yes	—
26	D₄×#88322	C₂²	1	2	yes	—
27	#16810647	C₄	1	1	yes	characteristic
28	#16810590	C₄	1	2	yes	—
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	C₅₃×#532	C₄	1	1	yes	characteristic
31	#13133291	—	2	2	no	—
32	#13133291	—	2	8	no	—
33	#13133291	D₄	1	4	yes	complemented
34	C₂×#11719910	—	2	4	no	—
35	C₂×#11719910	—	2	8	no	—
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂×#11719910	C₂³	1	2	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	2	yes	—
40	#13133224	—	2	4	no	—
41	#13133224	—	2	4	no	—
42	C₂×#11719917	C₂×C₄	1	2	yes	—
43	C₂×#11719917	C₂×C₄	1	4	yes	—
44	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	4	yes	—
46	C₂²×#88322	C₂³	1	1	yes	characteristic
47	C₄⋊#88322	C₂×C₄	1	2	yes	—
48	C₄⋊#88322	C₂×C₄	1	2	yes	—
49	C₂²⋊#88322	C₂×C₄	1	4	yes	—
50	C₂²⋊#88322	C₂×C₄	1	4	yes	—
51	C₄×#88322	C₂×C₄	1	1	yes	characteristic
52	C₄×#88322	C₂×C₄	1	1	yes	characteristic
53	C₄×#88322	C₂³	1	1	yes	characteristic
54	C₄×#88322	C₂³	1	1	yes	characteristic
55	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
56	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
57	#127×C₅₃	C₈	1	1	yes	characteristic
58	#124×C₅₃	C₈	1	2	yes	—
59	#121×C₅₃	C₈	1	2	yes	—
60	#121×C₅₃	C₈	1	2	yes	—
61	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#41×C₅₃	—	2	4	no	—
65	#41×C₅₃	—	2	4	no	—
66	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
69	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
76	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
77	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
78	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
79	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	#11719917	—	2	4	no	—
84	#11719917	—	2	8	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
89	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
90	C₂×#88322	C₂²×C₄	1	4	yes	—
91	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
92	C₂×#88322	C₂×D₄	1	2	yes	—
93	C₂×#88322	#41	1	2	yes	—
94	C₂×#88322	C₂⁴	1	1	yes	characteristic
95	#11719910	—	2	2	no	—
96	#11719910	—	2	2	no	—
97	#11719910	—	2	2	no	—
98	#11719910	C₂×D₄	1	4	yes	—
99	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
100	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	Q₈×C₅₃	—	2	4	no	—
102	D₄×C₅₃	—	2	8	no	—
103	D₄×C₅₃	—	2	4	no	—
104	D₄×C₅₃	—	2	4	no	—
105	D₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#131	1	2	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	#88322	—	2	4	no	—
120	#88322	—	2	1	no	—
121	#88322	—	2	1	no	—
122	#88322	C₄×D₄	1	2	yes	—
123	#88322	C₂²×D₄	1	1	yes	characteristic
124	#88322	C₂×#41	1	1	yes	characteristic
125	#33438	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	4	no	—
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	C₈×D₄	1	2	yes	—
132	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	4	no	—
137	C₄×C₅₃	C₈×D₄	1	2	yes	—
138	C₄×C₅₃	#504	1	2	yes	complemented
139	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
141	C₂×#504	—	53	2	no	—
142	#2931	—	53	1	no	—
143	#2169	—	53	2	no	—
144	#2168	—	53	4	no	—
145	D₄×C₁₆	—	53	2	no	—
146	C₂×C₄⋊C₁₆	—	53	1	no	—
147	#2113	—	53	2	no	—
148	C₄×M₃₂	—	53	1	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	—	4	4	no	—
151	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
153	C₂×C₅₃	#3407	1	1	yes	characteristic
154	C₂×#131	—	53	2	no	—
155	#532	#6157	1	1	yes	characteristic
156	#504	—	106	8	no	—
157	C₂×M₃₂	—	53	1	no	—
158	C₂×M₃₂	—	53	1	no	—
159	C₂×M₃₂	—	53	4	no	—
160	C₂²×C₁₆	—	53	2	no	—
161	C₂²×C₁₆	—	53	4	no	—
162	#436	—	53	2	no	—
163	#436	—	53	2	no	—
164	C₈×D₄	—	53	2	no	—
165	C₈×D₄	—	53	2	no	—
166	#423	—	53	1	no	—
167	#408	—	53	2	no	—
168	C₂×C₄×C₈	—	53	1	no	—
169	C₄⋊C₁₆	—	53	1	no	—
170	C₄⋊C₁₆	—	53	2	no	—
171	C₄⋊C₁₆	—	53	1	no	—
172	C₂²⋊C₁₆	—	53	4	no	—
173	C₂²⋊C₁₆	—	53	4	no	—
174	#346	—	53	2	no	—
175	C₄×C₁₆	—	53	2	no	—
176	C₅₃	#33438	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	C₂×#6157	1	2	yes	—
178	#131	—	106	4	no	—
179	#131	—	106	4	no	—
180	C₂×M₁₆	—	53	2	no	—
181	C₂×M₁₆	—	53	4	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	4	no	—
184	C₂²×C₈	—	53	1	no	—
185	C₄⋊Q₈	#88324	1	1	yes	characteristic
186	#127	#88324	1	1	yes	characteristic
187	#124	#88324	1	2	yes	—
188	#121	#88324	1	2	yes	—
189	#121	#88324	1	2	yes	—
190	C₄×D₄	C₂×#6157	1	2	yes	—
191	C₄×D₄	C₂×#6157	1	2	yes	—
192	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
193	M₃₂	—	53	4	no	—
194	M₃₂	—	106	8	no	—
195	M₃₂	—	106	2	no	—
196	C₂×C₁₆	—	53	2	no	—
197	C₂×C₁₆	—	53	2	no	—
198	C₂×C₁₆	—	53	2	no	—
199	C₂×C₁₆	—	53	2	no	—
200	C₂×C₁₆	—	106	8	no	—
201	C₂×C₁₆	—	106	4	no	—
202	C₄⋊C₈	—	53	2	no	—
203	C₄⋊C₈	—	53	2	no	—
204	C₂²⋊C₈	—	53	4	no	—
205	C₂²⋊C₈	—	53	4	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	53	1	no	—
209	C₄×C₈	—	53	1	no	—
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	C₂×#88324	1	2	yes	—
213	C₂×D₄	C₂×#88324	1	2	yes	—
214	C₂×D₄	C₂×#88324	1	4	yes	—
215	C₂×D₄	C₂×#88324	1	2	yes	—
216	C₂×D₄	C₂×#88324	1	2	yes	—
217	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
218	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
219	C₂²×C₄	C₂²×#6157	1	4	yes	—
220	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
221	M₁₆	—	106	8	no	—
222	M₁₆	—	106	4	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	1	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	2	no	—
233	C₄⋊C₄	C₂×#88324	1	2	yes	—
234	C₄⋊C₄	C₂×#88324	1	2	yes	—
235	C₂²⋊C₄	C₂×#88324	1	4	yes	—
236	C₂²⋊C₄	C₂×#88324	1	4	yes	—
237	C₄²	C₂×#88324	1	1	yes	characteristic
238	C₄²	C₂×#88324	1	1	yes	characteristic
239	C₄²	C₂²×#6157	1	1	yes	characteristic
240	C₄²	C₂²×#6157	1	1	yes	characteristic
241	C₁₆	—	53	4	no	—
242	C₁₆	—	106	2	no	—
243	C₁₆	—	106	2	no	—
244	C₁₆	—	106	2	no	—
245	C₂³	C₂²×#88324	1	4	yes	—
246	C₂³	C₂²×#88324	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	D₄	—	2	8	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	D₄×#6157	1	2	yes	—
257	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
258	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
259	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
260	C₂×C₄	C₂²×#88324	1	4	yes	—
261	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
262	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
263	C₂×C₄	#13133304	1	2	yes	—
264	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
265	C₈	—	53	2	no	—
266	C₈	—	53	1	no	—
267	C₈	—	53	1	no	—
268	C₈	—	106	4	no	—
269	C₈	—	106	1	no	—
270	C₈	—	106	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	D₄×#88324	1	2	yes	—
277	C₂²	C₂³×#88324	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	D₄×#88324	1	2	yes	—
283	C₄	#16811681	1	2	yes	complemented
284	C₄	C₂×D₄×#6157	1	1	yes	characteristic
285	C₄	C₂×#13133304	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×D₄×#88324	1	1	yes	characteristic
289	C₂	C₂×#16811681	1	1	yes	characteristic
290	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	#33438
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. C₂×#16811681	C₂	semidirect
2. #20291099	C₂	semidirect
3. #20291095	C₂	semidirect
4. D₄×#11719910	C₂	semidirect
5. C₂×C₄⋊#11719910	C₂	semidirect
6. #20290945	C₂	semidirect
7. C₄×#13133291	C₂	semidirect
33. #13133291	D₄	semidirect
138. C₄×C₅₃	#504	semidirect
176. C₅₃	#33438	semidirect
283. C₄	#16811681	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811681	2
#20291099	2
#20291095	2
D₄×#11719910	2
C₂×C₄⋊#11719910	2
#20290945	2
C₄×#13133291	2
#20283427	2
#33438	53