Subgroups — #28310731

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 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×#16811511	C₂	1	2	yes	maximal
3	#20289753	C₂	1	2	yes	maximal
4	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	C₄⋊#11719910	C₄	1	8	yes	complemented
7	#16811511	—	2	4	no	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₄³×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	—	2	4	no	—
17	C₂×#11719910	C₂×C₄	1	8	yes	—
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂³	1	1	yes	characteristic
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	C₈	1	4	yes	—
26	C₄×#88322	C₂×C₄	1	4	yes	—
27	C₄×#88322	D₄	1	2	yes	—
28	C₄×#88322	Q₈	1	2	yes	—
29	C₂×C₄²×C₅₃	C₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₅₃	—	2	4	no	—
40	C₄²×C₅₃	—	2	4	no	—
41	C₄²×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	C₄²	1	2	yes	—
44	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₅₃	C₂×C₈	1	2	yes	—
46	C₄²×C₅₃	M₁₆	1	2	yes	—
47	C₂×#88322	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	C₄²	1	2	yes	—
52	C₂×#88322	C₄⋊C₄	1	2	yes	—
53	C₂×#88322	C₂×C₈	1	2	yes	—
54	C₂×#88322	C₂×C₈	1	2	yes	—
55	C₂×#88322	M₁₆	1	2	yes	—
56	C₂×#88322	C₂²×C₄	1	2	yes	—
57	C₂×#88322	C₂×D₄	1	1	yes	characteristic
58	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
59	C₂×#88322	#41	1	2	yes	—
60	#11719910	—	2	8	no	—
61	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
71	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	#88322	—	2	2	no	—
79	#88322	—	2	2	no	—
80	#88322	C₄⋊C₈	1	4	yes	—
81	#88322	C₄×D₄	1	2	yes	—
82	#88322	C₄×Q₈	1	2	yes	—
83	#12653	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₅₃	#431	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
88	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	4	no	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	#346	1	4	yes	complemented
93	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₅₃	#431	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	—	53	2	no	—
98	C₂×#346	—	53	2	no	—
99	C₄²×C₈	—	53	1	no	—
100	#1374	—	53	2	no	—
101	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
103	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
104	C₂×C₅₃	#2150	1	2	yes	—
105	C₂×C₅₃	#2163	1	2	yes	—
106	C₂²×C₁₆	—	53	4	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₄³	#6157	1	1	yes	characteristic
114	C₄⋊C₁₆	—	53	8	no	—
115	#346	—	106	4	no	—
116	C₅₃	#12653	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	—	53	1	no	—
119	C₂²×C₈	—	53	1	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂×C₄²	#88324	1	2	yes	—
122	C₂×C₄²	#88324	1	2	yes	—
123	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
124	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
125	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
126	C₂×C₁₆	—	53	8	no	—
127	C₂×C₁₆	—	106	4	no	—
128	C₂×C₁₆	—	106	4	no	—
129	C₄×C₈	—	53	4	no	—
130	C₄×C₈	—	53	2	no	—
131	C₄×C₈	—	53	2	no	—
132	C₄×C₈	—	53	4	no	—
133	C₄×C₈	—	106	2	no	—
134	C₄×C₈	—	106	2	no	—
135	C₄×C₈	—	106	2	no	—
136	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
137	C₂²×C₄	C₂×#88324	1	2	yes	—
138	C₂²×C₄	C₂×#88324	1	2	yes	—
139	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	2	no	—
144	C₂×C₈	—	53	1	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	—	106	1	no	—
151	C₂×C₈	—	106	1	no	—
152	C₂×C₈	—	106	1	no	—
153	C₂×C₈	—	106	1	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	D₅₃⋊C₈	1	2	yes	—
159	C₄²	#11719936	1	2	yes	—
160	C₄²	C₄×#6157	1	2	yes	—
161	C₄²	C₄⋊#6157	1	2	yes	—
162	C₁₆	—	106	8	no	—
163	C₂³	C₂²×#88324	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₄×#88324	1	2	yes	—
170	C₂×C₄	C₄×#88324	1	2	yes	—
171	C₂×C₄	C₄⋊#88324	1	4	yes	—
172	C₂×C₄	#13133291	1	4	yes	—
173	C₂×C₄	#13133291	1	4	yes	—
174	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
176	C₂×C₄	#13133296	1	1	yes	characteristic
177	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
178	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
180	C₈	—	53	2	no	—
181	C₈	—	53	2	no	—
182	C₈	—	53	4	no	—
183	C₈	—	106	2	no	—
184	C₈	—	106	2	no	—
185	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
187	C₂²	#16811639	1	1	yes	characteristic
188	C₂²	C₂×#13133291	1	2	yes	—
189	C₂²	C₂×#13133291	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	#16811511	1	4	yes	complemented
194	C₄	C₄×C₄⋊#6157	1	1	yes	characteristic
195	C₄	C₄×#11719936	1	1	yes	characteristic
196	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
197	C₄	#16811642	1	1	yes	characteristic
198	C₂	C₄×C₄⋊#88324	1	1	yes	characteristic
199	C₂	C₂×#16811511	1	1	yes	characteristic
200	C₂	C₄×#13133291	1	1	yes	characteristic
201	C₂	#20291047	1	2	yes	—
202	C₂	#20291076	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄³×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₄³	#6157
hypocenter	C₅₃	#12653
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
6. C₄⋊#11719910	C₄	semidirect
92. C₄×C₅₃	#346	semidirect
116. C₅₃	#12653	semidirect
193. C₄	#16811511	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
C₄²×#88322	2
#12653	53