Subgroups — #107×#88322 #27275323

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#88322	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810376	C₄	1	4	yes	complemented
12	C₈×#88322	C₄	1	4	yes	complemented
13	C₂×#107×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#340	C₄	1	2	yes	—
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	D₄	1	1	yes	characteristic
20	C₂²×#88322	D₄	1	1	yes	characteristic
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	C₄×#88322	—	2	1	no	—
24	C₄×#88322	—	4	4	no	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₄×#88322	Q₈	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#107×C₅₃	C₈	1	8	yes	complemented
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	#95×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	C₄²	1	2	yes	—
43	C₂×C₈×C₅₃	C₄²	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
47	C₄²×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₅₃	#41	1	1	yes	characteristic
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	1	no	—
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	C₄⋊C₄	1	2	yes	—
61	C₂×#88322	C₄⋊C₄	1	2	yes	—
62	C₂×#88322	D₈	1	2	yes	—
63	C₂×#88322	Q₁₆	1	2	yes	—
64	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	8	no	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₅₃	#131	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
83	#88322	—	2	4	no	—
84	#88322	—	2	2	no	—
85	#88322	—	4	4	no	—
86	#88322	#107	1	4	yes	direct product factor
87	C₈×#107	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₃	C₈×D₄	1	2	yes	—
90	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
92	C₂²×C₅₃	#443	1	2	yes	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	4	8	no	—
96	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₅₃	#405	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
99	C₄×C₅₃	#427	1	1	yes	characteristic
100	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
101	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
102	C₄×C₅₃	C₈×Q₈	1	2	yes	—
103	C₄×#107	D₅₃	1	1	yes	characteristic
104	C₈×C₄⋊C₄	—	53	2	no	—
105	C₂×#335	—	53	1	no	—
106	C₂×C₈²	—	53	1	no	—
107	#1279	—	53	2	no	—
108	C₂×C₅₃	C₈×D₈	1	2	yes	—
109	C₂×C₅₃	C₈×Q₁₆	1	2	yes	—
110	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₅₃	#1772	1	1	yes	characteristic
112	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
113	C₂×#107	Dic₅₃	1	2	yes	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	106	2	no	—
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
120	#340	Dic₅₃	1	2	yes	—
121	#336	—	106	2	no	—
122	#335	—	53	4	no	—
123	C₈²	—	53	4	no	—
124	C₅₃	C₈×#107	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	106	1	no	—
128	C₂²×C₈	—	106	2	no	—
129	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	#107	#88322	1	8	yes	direct product factor
135	C₄⋊C₈	—	53	4	no	—
136	C₄⋊C₈	—	106	2	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	106	1	no	—
141	C₄×C₈	—	212	4	no	—
142	C₄×C₈	C₄×D₅₃	1	2	yes	—
143	C₄×C₈	C₄×D₅₃	1	2	yes	—
144	#95	—	2	2	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	#11719947	1	1	yes	characteristic
148	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	53	2	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	106	4	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	1	no	—
157	C₂×C₈	—	106	1	no	—
158	C₂×C₈	—	106	4	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	C₂×#88322	1	4	yes	—
162	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
163	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	C₂×#88322	1	8	yes	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₄²	Q₈×D₅₃	1	1	yes	characteristic
171	C₄²	#11719949	1	1	yes	characteristic
172	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	#13133175	1	1	yes	characteristic
180	C₂×C₄	#13133179	1	1	yes	characteristic
181	C₂×C₄	#13133182	1	1	yes	characteristic
182	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×#88322	1	2	yes	—
184	C₂×C₄	#13133224	1	2	yes	—
185	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
186	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
187	C₂×C₄	#13133236	1	1	yes	characteristic
188	C₂×C₄	#13133240	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	53	4	no	—
191	C₈	—	106	4	no	—
192	C₈	—	106	2	no	—
193	C₈	—	212	4	no	—
194	C₈	C₄×#88322	1	4	yes	complemented
195	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
196	C₂²	D₄×#88322	1	2	yes	—
197	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
198	C₂²	Q₁₆×Dic₅₃	1	1	yes	characteristic
199	C₂²	#16810740	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	8	no	—
203	C₄	C₂×C₄×#88322	1	1	yes	characteristic
204	C₄	#16810579	1	1	yes	characteristic
205	C₄	#107×D₅₃	1	1	yes	characteristic
206	C₄	#16810660	1	1	yes	characteristic
207	C₄	#16810705	1	1	yes	characteristic
208	C₄	#16810717	1	1	yes	characteristic
209	C₄	Q₈×#88322	1	2	yes	—
210	C₂	D₈×#88322	1	2	yes	—
211	C₂	Q₁₆×#88322	1	2	yes	—
212	C₂	C₄⋊C₄×#88322	1	1	yes	characteristic
213	C₂	#20274596	1	1	yes	characteristic
214	C₂	#107×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	C₈×#107
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #16810376	C₄	semidirect
12. C₈×#88322	C₄	semidirect
33. #107×C₅₃	C₈	semidirect
82. C₈×C₅₃	C₄×C₈	semidirect
86. #88322	#107	direct (N × H)
124. C₅₃	C₈×#107	semidirect
134. #107	#88322	direct (N × H)
194. C₈	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#88322	2
C₂×#16810376	2
C₂×C₈×#88322	2
#20272556	2
C₄×#107×C₅₃	2
C₈×#107	53