Subgroups — #28311975

Order 13568.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811511	C₂	1	1	yes	maximal, characteristic
4	#20289753	C₂	1	2	yes	maximal
5	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719910	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	1	yes	characteristic
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	C₄⋊#11719910	—	2	4	no	—
10	C₄⋊#11719910	—	2	2	no	—
11	#16811511	—	2	2	no	—
12	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	#16810378	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719910	—	2	1	no	—
18	C₂×#11719910	—	2	1	no	—
19	C₂×#11719910	—	2	1	no	—
20	C₂×#11719910	—	2	1	no	—
21	C₂×#11719910	—	2	2	no	—
22	C₂×#11719910	—	2	2	no	—
23	C₂×#11719910	—	2	2	no	—
24	C₂×#11719910	D₄	1	2	yes	—
25	C₂×#11719910	Q₈	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	C₄⋊#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	D₄	1	2	yes	—
33	C₄×#88322	Q₈	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#95×C₅₃	C₈	1	1	yes	characteristic
38	#95×C₅₃	C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₄²×C₅₃	M₁₆	1	2	yes	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	C₄⋊C₄	1	2	yes	—
55	C₂×#88322	C₂×D₄	1	1	yes	characteristic
56	C₂×#88322	C₂×D₄	1	1	yes	characteristic
57	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
58	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
59	C₂×#88322	#41	1	1	yes	characteristic
60	C₂×#88322	#41	1	1	yes	characteristic
61	#11719910	—	2	2	no	—
62	#11719910	—	2	2	no	—
63	#11719910	—	4	2	no	—
64	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
81	#88322	—	2	2	no	—
82	#88322	—	4	2	no	—
83	#88322	#121	1	1	yes	characteristic
84	#88322	C₂²⋊Q₈	1	1	yes	characteristic
85	#88322	#125	1	1	yes	characteristic
86	#88322	C₄⋊Q₈	1	1	yes	characteristic
87	#13160	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
92	C₂²×C₅₃	#504	1	2	yes	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	#389	1	1	yes	characteristic
98	C₄×C₅₃	#424	1	1	yes	characteristic
99	C₄×C₅₃	#436	1	1	yes	characteristic
100	C₄×C₅₃	#446	1	1	yes	characteristic
101	C₂×C₄⋊C₁₆	—	53	1	no	—
102	C₂×C₄⋊C₁₆	—	53	2	no	—
103	C₂×#346	—	53	1	no	—
104	C₈×C₄⋊C₄	—	53	1	no	—
105	#1374	—	53	2	no	—
106	C₂×C₅₃	#1939	1	1	yes	characteristic
107	C₂×C₅₃	#2150	1	1	yes	characteristic
108	C₂×C₅₃	#2151	1	1	yes	characteristic
109	C₂×C₅₃	#2168	1	2	yes	—
110	C₂×C₅₃	#2183	1	2	yes	—
111	C₂²×C₁₆	—	53	1	no	—
112	C₂²×C₁₆	—	53	1	no	—
113	C₂²×C₁₆	—	53	2	no	—
114	C₂×C₄⋊C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₄×C₄⋊C₄	#6157	1	1	yes	characteristic
118	C₄⋊C₁₆	—	106	2	no	—
119	C₄⋊C₁₆	—	106	4	no	—
120	#346	—	106	2	no	—
121	#336	—	53	2	no	—
122	C₅₃	#13160	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	2	no	—
126	C₂×C₄⋊C₄	#88324	1	2	yes	—
127	C₂×C₄²	C₂×#6157	1	2	yes	—
128	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
129	C₂×C₁₆	—	53	2	no	—
130	C₂×C₁₆	—	53	2	no	—
131	C₂×C₁₆	—	106	1	no	—
132	C₂×C₁₆	—	106	1	no	—
133	C₂×C₁₆	—	106	1	no	—
134	C₂×C₁₆	—	106	1	no	—
135	C₂×C₁₆	—	106	2	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₄⋊C₈	—	106	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	—	106	2	no	—
143	#95	#88324	1	1	yes	characteristic
144	#95	#88324	1	1	yes	characteristic
145	C₂²×C₄	C₂×#88324	1	2	yes	—
146	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
147	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
148	C₂²×C₄	C₂×#88324	1	2	yes	—
149	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
150	C₂×C₈	—	53	1	no	—
151	C₂×C₈	—	53	1	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	106	1	no	—
158	C₂×C₈	—	106	1	no	—
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#11719936	1	2	yes	—
166	C₄²	C₄⋊#6157	1	2	yes	—
167	C₁₆	—	106	2	no	—
168	C₁₆	—	106	2	no	—
169	C₁₆	—	212	2	no	—
170	C₂³	C₂²×#88324	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₄⋊#88324	1	2	yes	—
179	C₂×C₄	C₄⋊#88324	1	2	yes	—
180	C₂×C₄	#13133291	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
182	C₂×C₄	D₄×#6157	1	1	yes	characteristic
183	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
185	C₂×C₄	#13133304	1	1	yes	characteristic
186	C₂×C₄	#13133305	1	1	yes	characteristic
187	C₈	—	53	1	no	—
188	C₈	—	53	1	no	—
189	C₈	—	53	1	no	—
190	C₈	—	53	1	no	—
191	C₈	—	106	2	no	—
192	C₈	—	212	2	no	—
193	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
194	C₂²	D₄×#88324	1	1	yes	characteristic
195	C₂²	Q₈×#88324	1	1	yes	characteristic
196	C₂²	C₂×#13133291	1	2	yes	—
197	C₂²	#16811681	1	2	yes	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	#16811574	1	1	yes	characteristic
203	C₄	#16811630	1	1	yes	characteristic
204	C₄	#16811657	1	1	yes	characteristic
205	C₄	#16811673	1	1	yes	characteristic
206	C₂	#20290812	1	1	yes	characteristic
207	C₂	#20291047	1	1	yes	characteristic
208	C₂	#20291050	1	1	yes	characteristic
209	C₂	#20291095	1	2	yes	—
210	C₂	#20291130	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₅₃	C₄
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×C₄⋊C₄	#6157
hypocenter	C₅₃	#13160
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
122. C₅₃	#13160	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719910	2
C₂×#16811511	2
#20289753	2
C₄⋊C₄×#88322	2
#13160	53