Subgroups — C₄×#16810440 #27299351

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810440	C₂	1	2	yes	maximal
3	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
4	#20272824	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₄×Dic₂₁₂	—	2	1	no	—
9	C₄×Dic₂₁₂	—	2	1	no	—
10	C₄×Dic₂₁₂	C₄	1	2	yes	—
11	C₄×Dic₂₁₂	C₄	1	2	yes	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	C₂×#13133151	C₂²	1	1	yes	characteristic
14	C₂×#13133142	—	2	1	no	—
15	C₄×#11719917	—	2	1	no	—
16	#16810440	C₄	1	8	yes	direct product factor
17	#16810388	—	2	1	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#346	—	2	1	no	—
21	C₄×C₁₆×C₅₃	—	2	1	no	—
22	C₂²×Dic₁₀₆	—	2	1	no	—
23	C₂×Dic₂₁₂	—	2	1	no	—
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
26	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
27	C₂×#11719917	—	2	2	no	—
28	C₄×Dic₁₀₆	—	2	2	no	—
29	C₄×Dic₁₀₆	—	2	2	no	—
30	C₄×Dic₁₀₆	—	4	1	no	—
31	C₄×Dic₁₀₆	—	4	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×#11719919	—	4	1	no	—
34	C₂×C₄×Dic₅₃	—	4	1	no	—
35	#13133153	—	2	2	no	—
36	#13133153	C₂×C₄	1	4	yes	—
37	#13133151	—	2	1	no	—
38	#13133151	C₂×C₄	1	2	yes	—
39	#13133142	—	2	2	no	—
40	#13133142	—	2	2	no	—
41	#13133128	—	4	1	no	—
42	C₄×#88322	—	4	1	no	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	M₃₂×C₅₃	—	2	2	no	—
47	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	1	no	—
57	C₂²×Dic₅₃	—	4	1	no	—
58	C₂×Dic₁₀₆	—	2	2	no	—
59	C₂×Dic₁₀₆	—	2	4	no	—
60	C₂×Dic₁₀₆	—	4	1	no	—
61	C₂×Dic₁₀₆	—	4	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	2	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₃	D₈	1	1	yes	characteristic
72	C₄²×C₅₃	D₈	1	1	yes	characteristic
73	C₄⋊Dic₅₃	—	2	2	no	—
74	C₄⋊Dic₅₃	—	4	1	no	—
75	#11719919	—	4	2	no	—
76	#11719919	—	4	2	no	—
77	C₄×Dic₅₃	—	4	2	no	—
78	C₄×Dic₅₃	—	4	2	no	—
79	#11719917	—	2	4	no	—
80	#11719917	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	Dic₂₁₂	—	2	2	no	—
83	Dic₂₁₂	—	2	4	no	—
84	Dic₂₁₂	C₄²	1	4	yes	—
85	C₁₆×C₅₃	—	2	4	no	—
86	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#135	1	2	yes	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	4	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	Dic₁₀₆	—	2	4	no	—
102	Dic₁₀₆	—	4	2	no	—
103	Dic₁₀₆	—	4	4	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
107	C₈×C₅₃	#117	1	1	yes	characteristic
108	C₈×C₅₃	C₄×D₄	1	2	yes	—
109	#88322	—	4	4	no	—
110	C₄×#362	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
114	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	#362	1	2	yes	complemented
119	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₅₃	#416	1	1	yes	characteristic
121	C₄×C₅₃	C₄×D₈	1	2	yes	—
122	Dic₅₃	—	4	4	no	—
123	Dic₅₃	—	8	2	no	—
124	C₂×C₄×Q₁₆	—	53	1	no	—
125	C₂×#362	—	53	2	no	—
126	C₄×M₃₂	D₅₃	1	1	yes	characteristic
127	C₄×#108	—	53	1	no	—
128	#1387	—	53	2	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
132	C₂×C₅₃	#2148	1	1	yes	characteristic
133	C₂²×Q₁₆	—	53	1	no	—
134	C₂×C₄×Q₈	—	106	1	no	—
135	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
136	C₄×Q₁₆	—	53	2	no	—
137	C₄×Q₁₆	—	53	2	no	—
138	C₄×Q₁₆	—	106	1	no	—
139	C₄×Q₁₆	—	106	1	no	—
140	C₂×#108	—	53	2	no	—
141	C₂×#107	—	53	1	no	—
142	C₂×#103	—	106	1	no	—
143	C₄×M₁₆	—	106	1	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	#362	—	53	8	no	—
146	#346	—	2	1	no	—
147	C₄×C₁₆	—	2	1	no	—
148	#341	—	106	1	no	—
149	C₅₃	C₄×#362	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	—	106	1	no	—
151	C₂×Q₁₆	—	53	4	no	—
152	C₂×Q₁₆	—	53	2	no	—
153	C₂×Q₁₆	—	106	2	no	—
154	C₂×Q₁₆	—	106	1	no	—
155	C₂×M₁₆	—	106	2	no	—
156	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
158	C₄×Q₈	—	106	2	no	—
159	C₄×Q₈	—	106	2	no	—
160	C₄×Q₈	—	212	1	no	—
161	C₄×Q₈	—	212	1	no	—
162	C₂×C₄⋊C₄	—	106	1	no	—
163	C₂×C₄⋊C₄	—	212	1	no	—
164	C₂×C₄²	—	212	1	no	—
165	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
166	M₃₂	—	2	2	no	—
167	M₃₂	C₄×D₅₃	1	4	yes	—
168	C₂×C₁₆	—	2	2	no	—
169	#108	—	53	4	no	—
170	#108	—	106	2	no	—
171	#107	—	53	2	no	—
172	#107	—	106	1	no	—
173	#103	—	106	2	no	—
174	#103	—	106	2	no	—
175	#97	—	212	1	no	—
176	C₄×C₈	—	212	1	no	—
177	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
178	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×Q₈	—	106	2	no	—
182	C₂×Q₈	—	106	4	no	—
183	C₂×Q₈	—	212	1	no	—
184	C₂×Q₈	—	212	2	no	—
185	C₂²×C₄	—	212	1	no	—
186	C₂²×C₄	—	212	1	no	—
187	C₂²×C₄	#11719913	1	1	yes	characteristic
188	C₂²×C₄	#11719913	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	Q₁₆	—	53	4	no	—
191	Q₁₆	—	106	2	no	—
192	Q₁₆	—	106	4	no	—
193	M₁₆	—	106	4	no	—
194	M₁₆	—	212	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	212	2	no	—
197	C₂×C₈	#11719921	1	1	yes	characteristic
198	C₂×C₈	#11719921	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
201	C₂×C₈	#11719945	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	106	2	no	—
204	C₄⋊C₄	—	212	2	no	—
205	C₄⋊C₄	—	212	2	no	—
206	C₄⋊C₄	—	212	1	no	—
207	C₄²	—	212	2	no	—
208	C₄²	—	212	2	no	—
209	C₄²	D₄₂₄	1	1	yes	characteristic
210	C₄²	D₄₂₄	1	1	yes	characteristic
211	C₄²	#11719921	1	1	yes	characteristic
212	C₄²	#11719921	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₂³	C₂×#11719913	1	1	yes	characteristic
215	Q₈	—	106	4	no	—
216	Q₈	—	212	2	no	—
217	Q₈	—	212	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	424	2	no	—
221	C₂×C₄	—	212	2	no	—
222	C₂×C₄	—	212	2	no	—
223	C₂×C₄	—	212	4	no	—
224	C₂×C₄	#13133139	1	1	yes	characteristic
225	C₂×C₄	#13133139	1	1	yes	characteristic
226	C₂×C₄	#13133139	1	1	yes	characteristic
227	C₂×C₄	#13133139	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
229	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
230	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
231	C₂×C₄	#13133239	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	212	4	no	—
235	C₈	C₄²×D₅₃	1	1	yes	characteristic
236	C₈	#13133174	1	1	yes	characteristic
237	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#13133139	1	1	yes	characteristic
241	C₂²	C₄×#11719913	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	424	2	no	—
246	C₄	—	212	4	no	—
247	C₄	#16810440	1	2	yes	direct product factor
248	C₄	C₄×#11719921	1	1	yes	characteristic
249	C₄	#16810620	1	1	yes	characteristic
250	C₄	C₄×D₄₂₄	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#13133139	1	1	yes	characteristic
253	C₂	C₂×#16810440	1	1	yes	characteristic
254	C₂	#20276465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	C₄×#362
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
16. #16810440	C₄	direct (N × H)
118. C₄×C₅₃	#362	semidirect
149. C₅₃	C₄×#362	semidirect
247. C₄	#16810440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
C₂×#16810440	2
C₄×#13133153	2
#20272824	2
C₄×M₃₂×C₅₃	2
C₄×#362	53

Subgroups — C4×#16810440 #27299351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810440 #27299351