Subgroups — C₄×#16810754 #27382719

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	2	yes	maximal
2	#20274590	C₂	1	4	yes	maximal
3	#20274584	C₂	1	2	yes	maximal
4	C₄×C₄⋊#88322	C₂	1	2	yes	maximal
5	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
6	C₄×#13133128	C₂	1	2	yes	maximal
7	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810754	C₄	1	16	yes	direct product factor
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	4	yes	—
12	C₂×#13133128	C₂²	1	4	yes	—
13	C₂×#13133128	C₂²	1	4	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	#16810378	C₂²	1	4	yes	—
20	#16810378	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₃	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₄³×C₅₃	C₂²	1	1	yes	characteristic
28	C₄³×C₅₃	C₂²	1	2	yes	—
29	C₂²×#88322	C₂³	1	2	yes	—
30	C₂²×#88322	C₂³	1	4	yes	—
31	C₂²×#88322	C₂³	1	2	yes	—
32	C₄⋊#88322	C₂×C₄	1	16	yes	—
33	C₄⋊#88322	C₂×C₄	1	8	yes	—
34	#13133128	—	2	8	no	—
35	#13133128	C₂×C₄	1	16	yes	—
36	C₄×#88322	—	2	2	no	—
37	C₄×#88322	—	2	4	no	—
38	C₄×#88322	—	2	4	no	—
39	C₄×#88322	C₂×C₄	1	8	yes	—
40	C₄×#88322	Q₈	1	4	yes	complemented
41	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₅₃	—	2	8	no	—
43	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#95×C₅₃	C₂×C₄	1	2	yes	—
58	#95×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₅₃	—	2	4	no	—
60	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	8	no	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
72	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
73	C₄²×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	2	8	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	M₁₆	1	8	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
81	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	C₂²×C₄	1	4	yes	—
88	C₂×#88322	C₂²×C₄	1	8	yes	—
89	C₂×#88322	C₂²×C₄	1	4	yes	—
90	C₂×#88322	C₂×Q₈	1	4	yes	—
91	C₂×#88322	#41	1	4	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₅₃	—	2	8	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
110	C₂×C₄×C₅₃	#131	1	4	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	#88322	—	2	4	no	—
116	#88322	—	2	8	no	—
117	#88322	C₄×Q₈	1	8	yes	complemented
118	C₄×#446	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
122	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
123	C₂²×C₅₃	C₂×#131	1	2	yes	—
124	C₄×C₅₃	—	2	4	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	4	4	no	—
129	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
130	C₄×C₅₃	#446	1	8	yes	complemented
131	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
132	C₄×C₅₃	C₄×#41	1	2	yes	—
133	C₂×#446	—	53	2	no	—
134	C₄²×Q₈	D₅₃	1	1	yes	characteristic
135	#1771	—	53	4	no	—
136	#1770	—	53	2	no	—
137	C₄×C₄⋊C₈	—	53	2	no	—
138	C₄×C₄⋊C₈	—	53	1	no	—
139	C₄×#97	—	53	2	no	—
140	C₄²×C₈	—	53	1	no	—
141	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
142	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
143	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
144	C₂×C₅₃	C₂×#446	1	2	yes	—
145	C₂×C₅₃	#2965	1	2	yes	—
146	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
147	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
148	#446	—	53	16	no	—
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	—	53	4	no	—
151	C₂×#97	—	53	4	no	—
152	C₂×#97	—	53	4	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	2	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	2	no	—
158	C₄×C₄⋊C₄	Dic₅₃	1	4	yes	—
159	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
160	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
161	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
162	C₄³	C₂×D₅₃	1	1	yes	characteristic
163	C₄³	C₂×D₅₃	1	2	yes	—
164	#336	—	53	4	no	—
165	#336	—	53	2	no	—
166	C₅₃	C₄×#446	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₄×Q₈	—	2	8	no	—
172	C₄×Q₈	C₄×D₅₃	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
178	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
179	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
180	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
181	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	53	8	no	—
187	C₄⋊C₈	—	53	16	no	—
188	#97	—	53	16	no	—
189	#97	—	106	8	no	—
190	C₄×C₈	—	53	8	no	—
191	C₄×C₈	—	53	4	no	—
192	C₄×C₈	—	106	4	no	—
193	C₄×C₈	—	106	2	no	—
194	C₄×C₈	—	106	4	no	—
195	#95	C₂×Dic₅₃	1	2	yes	—
196	#95	C₂×Dic₅₃	1	4	yes	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
206	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
207	C₂×C₈	—	53	4	no	—
208	C₂×C₈	—	53	4	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	53	8	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	106	4	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₄×Dic₅₃	1	8	yes	—
219	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	4	no	—
225	C₄²	#11719917	1	8	yes	—
226	C₄²	C₂×C₄×D₅₃	1	4	yes	—
227	C₄²	C₂×C₄×D₅₃	1	8	yes	—
228	C₄²	Q₈×D₅₃	1	2	yes	—
229	C₄²	#11719949	1	2	yes	—
230	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
244	C₂×C₄	Q₈×Dic₅₃	1	4	yes	—
245	C₂×C₄	C₂×#11719917	1	4	yes	—
246	C₂×C₄	C₂×#11719917	1	4	yes	—
247	C₂×C₄	C₂×#11719917	1	4	yes	—
248	C₂×C₄	#13133224	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
250	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
251	C₂×C₄	#41×D₅₃	1	2	yes	—
252	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
253	C₈	—	53	8	no	—
254	C₈	—	106	4	no	—
255	C₈	—	106	8	no	—
256	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
257	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
258	C₂²	#41×Dic₅₃	1	1	yes	characteristic
259	C₂²	C₂²×#11719917	1	2	yes	—
260	C₂²	C₂×#13133224	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	4	no	—
266	C₄	C₄×#11719917	1	4	yes	—
267	C₄	#16810754	1	8	yes	direct product factor
268	C₄	C₄×Q₈×D₅₃	1	2	yes	—
269	C₄	C₄×#11719949	1	2	yes	—
270	C₂	C₄×Q₈×Dic₅₃	1	1	yes	characteristic
271	C₂	C₂×C₄×#11719917	1	1	yes	characteristic
272	C₂	C₄×#13133224	1	1	yes	characteristic
273	C₂	C₂×#16810754	1	2	yes	—
274	C₂	#20283682	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	C₄×#446
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
9. #16810754	C₄	direct (N × H)
40. C₄×#88322	Q₈	semidirect
117. #88322	C₄×Q₈	semidirect
130. C₄×C₅₃	#446	semidirect
166. C₅₃	C₄×#446	semidirect
267. C₄	#16810754	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×#16810754	2
#20274590	2
#20274584	2
C₄×C₄⋊#88322	2
C₄×#13133128	2
C₄²×#88322	2
C₄²×Q₈×C₅₃	2
C₄×#446	53

Subgroups — C4×#16810754 #27382719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810754 #27382719