Subgroups — C₂×C₄×#13133152 #27382189

Order 13568.

1086 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	C₂²×#13133152	C₂	1	2	yes	maximal
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₄×#13133152	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810387	C₂	1	2	yes	maximal
6	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂×#13133152	C₄	1	16	yes	direct product factor
11	C₂×#13133152	C₂²	1	8	yes	—
12	C₄×#11719917	—	2	2	no	—
13	C₄×#11719917	C₂²	1	4	yes	—
14	C₂×#13133128	—	2	1	no	—
15	C₂×C₄×#88322	—	2	1	no	—
16	C₂×C₄×#88322	C₄	1	4	yes	—
17	C₂×C₄×#88322	C₄	1	4	yes	—
18	C₂×C₄×#88322	C₂²	1	4	yes	—
19	#16810387	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₅₃	—	2	2	no	—
23	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
24	C₂×#97×C₅₃	—	2	1	no	—
25	C₂×C₄×C₈×C₅₃	—	2	1	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	C₂×C₄	1	8	yes	—
29	C₂×#11719917	C₂³	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	4	no	—
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₂²×#88322	C₂×C₄	1	2	yes	—
34	C₂²×#88322	C₂×C₄	1	2	yes	—
35	C₂²×#88322	C₂×C₄	1	2	yes	—
36	C₂²×#88322	C₂×C₄	1	8	yes	—
37	C₂²×#88322	C₂³	1	2	yes	—
38	C₂²×#88322	C₂³	1	2	yes	—
39	#13133152	C₂×C₄	1	32	yes	direct product factor
40	#13133128	—	2	4	no	—
41	C₄×#88322	—	2	4	no	—
42	C₄×#88322	C₂×C₄	1	16	yes	—
43	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	4	no	—
47	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
48	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	2	yes	—
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	#97×C₅₃	—	2	4	no	—
58	C₄×C₈×C₅₃	—	2	4	no	—
59	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	—	2	8	no	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	8	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₅₃	—	2	8	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
80	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₅₃	C₂×D₄	1	2	yes	—
83	C₄²×C₅₃	C₂×D₄	1	2	yes	—
84	#11719917	—	2	8	no	—
85	#11719917	—	2	4	no	—
86	#11719917	C₂²×C₄	1	8	yes	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	8	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	C₄²	1	8	yes	—
93	C₂×#88322	C₄²	1	8	yes	—
94	C₂×#88322	C₂²×C₄	1	4	yes	—
95	C₂×#88322	C₂²×C₄	1	4	yes	—
96	C₂×#88322	C₂²×C₄	1	8	yes	—
97	C₂³×C₅₃	—	2	4	no	—
98	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
101	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	8	no	—
104	C₂×C₄×C₅₃	—	2	8	no	—
105	C₂×C₄×C₅₃	#108	1	8	yes	complemented
106	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₅₃	#117	1	2	yes	—
113	C₂×C₄×C₅₃	#117	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
121	C₈×C₅₃	—	2	8	no	—
122	#88322	—	2	4	no	—
123	#88322	—	2	8	no	—
124	#88322	C₂×C₄²	1	8	yes	—
125	C₂×C₄×#108	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	2	8	no	—
129	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
130	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
131	C₂²×C₅₃	C₂×#108	1	4	yes	—
132	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
134	C₄×C₅₃	—	2	4	no	—
135	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
136	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
137	C₄×C₅₃	C₂×#108	1	8	yes	complemented
138	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
141	C₂²×#108	—	53	2	no	—
142	C₂×C₄×M₁₆	—	53	1	no	—
143	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
144	C₂²×C₄×C₈	—	53	1	no	—
145	C₄×#108	—	53	8	no	—
146	C₂×#341	—	53	2	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	C₄×#108	1	4	yes	complemented
150	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₅₃	C₂²×#108	1	2	yes	—
152	C₂²×M₁₆	—	53	2	no	—
153	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
154	C₂³×C₈	—	53	1	no	—
155	C₂³×C₈	—	53	1	no	—
156	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
157	C₂×#108	—	53	16	no	—
158	C₂×#108	—	53	8	no	—
159	C₄×M₁₆	—	2	2	no	—
160	C₄×M₁₆	—	53	4	no	—
161	C₄×M₁₆	—	106	2	no	—
162	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
163	C₂×#97	—	2	1	no	—
164	C₂×#97	—	106	1	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	—	53	4	no	—
167	C₂×C₄×C₈	—	53	4	no	—
168	C₂×C₄×C₈	—	53	4	no	—
169	C₂×C₄×C₈	—	106	1	no	—
170	#341	—	53	8	no	—
171	C₅₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	Dic₁₀₆	1	2	yes	—
173	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	53	8	no	—
177	C₂×M₁₆	—	53	4	no	—
178	C₂×M₁₆	—	106	4	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	C₄×D₅₃	1	8	yes	—
181	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₂²×C₈	—	53	8	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	53	2	no	—
188	C₂²×C₈	—	53	2	no	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	106	4	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂×C₄²	Dic₁₀₆	1	2	yes	—
193	C₂×C₄²	C₄×D₅₃	1	2	yes	—
194	C₂×C₄²	C₄×D₅₃	1	2	yes	—
195	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
199	#108	—	53	32	no	—
200	#97	—	2	4	no	—
201	#97	—	106	4	no	—
202	C₄×C₈	—	2	4	no	—
203	C₄×C₈	—	53	16	no	—
204	C₄×C₈	—	106	4	no	—
205	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	#11719919	1	2	yes	—
208	C₂²×C₄	#11719919	1	4	yes	—
209	C₂²×C₄	#11719919	1	2	yes	—
210	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
211	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
215	C₂²×C₄	#11719945	1	4	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
219	M₁₆	—	2	8	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	53	8	no	—
222	M₁₆	—	106	8	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	C₂×C₄×D₅₃	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	53	8	no	—
228	C₂×C₈	—	53	8	no	—
229	C₂×C₈	—	53	4	no	—
230	C₂×C₈	—	53	8	no	—
231	C₂×C₈	—	53	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	8	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	106	4	no	—
237	C₄²	#11719919	1	2	yes	—
238	C₄²	#11719919	1	4	yes	—
239	C₄²	#11719919	1	2	yes	—
240	C₄²	C₂×C₄×D₅₃	1	4	yes	—
241	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
242	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×#11719919	1	1	yes	characteristic
245	C₂³	C₂×#11719919	1	1	yes	characteristic
246	C₂³	C₄×Dic₁₀₆	1	4	yes	—
247	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	#13133152	1	8	yes	direct product factor
252	C₂×C₄	C₄²×D₅₃	1	2	yes	—
253	C₂×C₄	C₄²×D₅₃	1	2	yes	—
254	C₂×C₄	C₂×#11719919	1	4	yes	—
255	C₂×C₄	C₂×#11719919	1	4	yes	—
256	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
258	C₂×C₄	#13133174	1	2	yes	—
259	C₂×C₄	#13133174	1	2	yes	—
260	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
261	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
262	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719945	1	2	yes	—
267	C₈	—	2	8	no	—
268	C₈	—	53	8	no	—
269	C₈	—	106	4	no	—
270	C₈	—	106	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	8	no	—
274	C₂²	C₄×#11719919	1	2	yes	—
275	C₂²	C₄×#11719919	1	2	yes	—
276	C₂²	C₂×#13133152	1	4	yes	—
277	C₂²	C₂²×#11719919	1	1	yes	characteristic
278	C₂²	C₂×C₄×Dic₁₀₆	1	2	yes	—
279	C₄	—	2	4	no	—
280	C₄	C₄×#11719919	1	2	yes	—
281	C₄	C₄×#11719919	1	2	yes	—
282	C₄	C₂×#13133152	1	8	yes	direct product factor
283	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
284	C₄	C₂×#13133174	1	1	yes	characteristic
285	C₄	C₂×C₄×C₂²⋊D₅₃	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	2	2	no	—
288	C₂	C₄×#13133152	1	4	yes	direct product factor
289	C₂	C₂×C₄×#11719919	1	1	yes	characteristic
290	C₂	C₂²×#13133152	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₄×#13133152	C₂	direct (N × H)
10. C₂×#13133152	C₄	direct (N × H)
39. #13133152	C₂×C₄	direct (N × H)
105. C₂×C₄×C₅₃	#108	semidirect
137. C₄×C₅₃	C₂×#108	semidirect
149. C₂×C₅₃	C₄×#108	semidirect
171. C₅₃	C₂×C₄×#108	semidirect
251. C₂×C₄	#13133152	direct (N × H)
282. C₄	C₂×#13133152	direct (N × H)
288. C₂	C₄×#13133152	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133152	2
C₂×C₄×#11719917	2
C₂²×C₄×#88322	2
C₄×#13133152	2
C₂×#16810387	2
C₂×C₄×M₁₆×C₅₃	2
C₂×C₄×#108	53

Subgroups — C2×C4×#13133152 #27382189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133152 #27382189