Subgroups — #108×#88324 #28302159

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290262	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#88324	C₂	1	2	yes	maximal
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
4	#20289548	C₂	1	2	yes	maximal
5	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	#16811486	C₄	1	2	yes	—
10	#16811484	C₄	1	2	yes	—
11	#16811464	—	2	2	no	—
12	C₈×#88324	—	2	2	no	—
13	C₈×#88324	C₄	1	2	yes	—
14	C₈×#88324	C₄	1	2	yes	—
15	C₂×#13133153	C₄	1	1	yes	characteristic
16	M₁₆×Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810387	C₄	1	2	yes	—
19	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	—	2	1	no	—
21	C₂²×#88324	Q₈	1	2	yes	—
22	C₄⋊#88324	C₂×C₄	1	4	yes	—
23	C₂²⋊#88324	—	2	2	no	—
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	C₂×C₄	1	2	yes	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	D₄	1	1	yes	characteristic
29	C₄×#88324	D₄	1	1	yes	characteristic
30	C₂×#11719917	C₂×C₄	1	2	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	#13133153	C₈	1	4	yes	complemented
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#108×C₅₃	C₈	1	4	yes	complemented
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₅₃	#41	1	1	yes	characteristic
46	C₂×#88324	—	2	4	no	—
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	2	4	no	—
49	C₂×#88324	—	2	4	no	—
50	C₂×#88324	C₄⋊C₄	1	2	yes	—
51	C₂×#88324	C₄⋊C₄	1	2	yes	—
52	M₁₆×C₅₃	—	2	2	no	—
53	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄×Dic₅₃	#41	1	1	yes	characteristic
63	#11719917	—	2	2	no	—
64	#11719917	C₂×C₈	1	4	yes	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	C₄²	1	1	yes	characteristic
69	C₂×#88322	C₄²	1	1	yes	characteristic
70	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×Dic₅₃	#117	1	1	yes	characteristic
81	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
83	#88324	—	2	4	no	—
84	#88324	—	2	2	no	—
85	#88324	—	4	4	no	—
86	#88324	#108	1	4	yes	direct product factor
87	C₈×C₅₃	—	2	4	no	—
88	C₈×C₅₃	C₄×C₈	1	2	yes	—
89	C₈×C₅₃	C₄×C₈	1	2	yes	—
90	#88322	—	2	1	no	—
91	#88322	—	2	1	no	—
92	#88322	—	2	4	no	—
93	C₈×#108	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
99	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
100	C₄×C₅₃	#405	1	1	yes	characteristic
101	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
102	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
103	Dic₅₃	—	2	1	no	—
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	C₂×#108	1	2	yes	—
106	Dic₅₃	#443	1	2	yes	—
107	C₄×#108	—	53	1	no	—
108	#1564	—	53	1	no	—
109	C₈×M₁₆	—	53	2	no	—
110	C₂×C₈²	—	53	1	no	—
111	#1278	—	53	2	no	—
112	C₂×C₅₃	—	2	1	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₅₃	#1772	1	1	yes	characteristic
116	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
117	#431	—	53	2	no	—
118	C₂×#108	—	53	1	no	—
119	C₂×#108	#6157	1	1	yes	characteristic
120	C₄×M₁₆	—	53	2	no	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	#341	—	53	2	no	—
125	#335	—	53	2	no	—
126	#334	—	53	2	no	—
127	#322	—	106	2	no	—
128	C₈²	—	53	2	no	—
129	C₈²	—	53	2	no	—
130	C₈²	—	106	2	no	—
131	C₅₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
132	C₂×M₁₆	—	53	2	no	—
133	C₂×M₁₆	C₂×#6157	1	2	yes	—
134	C₂²×C₈	—	53	2	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	106	1	no	—
137	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
138	C₂×C₄²	—	53	1	no	—
139	#108	—	53	4	no	—
140	#108	#88324	1	4	yes	direct product factor
141	C₄⋊C₈	—	53	4	no	—
142	C₂²⋊C₈	—	106	2	no	—
143	#97	—	106	2	no	—
144	C₄×C₈	—	53	2	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	—	53	1	no	—
147	C₄×C₈	—	53	1	no	—
148	C₄×C₈	—	53	1	no	—
149	C₄×C₈	—	53	1	no	—
150	C₄×C₈	—	53	1	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	106	2	no	—
153	C₄×C₈	—	106	2	no	—
154	C₄×C₈	—	106	1	no	—
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	53	4	no	—
160	M₁₆	—	106	2	no	—
161	M₁₆	C₂×#88324	1	4	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	106	4	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	4	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	4	no	—
174	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
175	C₂×C₈	C₄×#6157	1	1	yes	characteristic
176	C₂×C₈	C₄×#6157	1	1	yes	characteristic
177	C₂×C₈	C₂×#88324	1	2	yes	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₂³	Q₈×#6157	1	1	yes	characteristic
183	C₂×C₄	—	53	1	no	—
184	C₂×C₄	—	53	1	no	—
185	C₂×C₄	—	53	1	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	106	2	no	—
188	C₂×C₄	—	106	2	no	—
189	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
190	C₂×C₄	D₄×#6157	1	1	yes	characteristic
191	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
192	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
193	C₂×C₄	#13133304	1	1	yes	characteristic
194	C₂×C₄	#13133305	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₈	—	53	4	no	—
197	C₈	—	106	2	no	—
198	C₈	—	106	4	no	—
199	C₈	—	106	1	no	—
200	C₈	—	106	1	no	—
201	C₈	—	106	4	no	—
202	C₈	—	212	4	no	—
203	C₈	C₄×#88324	1	2	yes	—
204	C₈	C₄×#88324	1	2	yes	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
208	C₂²	Q₈×#88324	1	1	yes	characteristic
209	C₂²	#16811672	1	1	yes	characteristic
210	C₄	—	53	2	no	—
211	C₄	—	53	2	no	—
212	C₄	—	106	1	no	—
213	C₄	—	106	1	no	—
214	C₄	C₂×C₄×#88324	1	1	yes	characteristic
215	C₄	#16811588	1	1	yes	characteristic
216	C₄	D₄×#88324	1	1	yes	characteristic
217	C₄	#16811651	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	C₄⋊C₄×#88324	1	1	yes	characteristic
221	C₂	#20290520	1	1	yes	characteristic
222	C₂	#108×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	C₈×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
33. #13133153	C₈	semidirect
42. #108×C₅₃	C₈	semidirect
86. #88324	#108	direct (N × H)
131. C₅₃	C₈×#108	semidirect
140. #108	#88324	direct (N × H)

Maximal subgroups

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

subgroup	index
#20290262	2
M₁₆×#88324	2
C₂×C₈×#88324	2
#20289548	2
#108×Dic₅₃	2
C₈×#108	53