Subgroups — #28310778

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 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	1	yes	maximal, characteristic
2	C₂×#16811511	C₂	1	2	yes	maximal
3	C₂×C₄×#11719910	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	1	yes	characteristic
7	C₂²×#11719910	C₂²	1	2	yes	—
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	C₄⋊#11719910	C₄	1	4	yes	complemented
10	#16811511	—	2	4	no	—
11	C₄×#11719910	C₄	1	4	yes	complemented
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	2	no	—
18	C₂×#11719910	—	2	1	no	—
19	C₂×#11719910	—	2	1	no	—
20	C₂×#11719910	—	2	2	no	—
21	C₂×#11719910	C₂×C₄	1	4	yes	—
22	C₂×#11719910	C₂×C₄	1	2	yes	—
23	C₂×#11719910	C₂×C₄	1	2	yes	—
24	C₂×#11719910	D₄	1	2	yes	—
25	C₂×#11719910	Q₈	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	C₄⋊#88322	C₈	1	4	yes	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	C₂×C₄	1	2	yes	—
33	C₄×#88322	C₂×C₄	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	2	yes	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	C₄²	1	2	yes	—
46	C₄²×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	C₄²	1	2	yes	—
52	C₂×#88322	C₂×C₈	1	4	yes	—
53	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
54	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
55	C₂×#88322	C₂×D₄	1	1	yes	characteristic
56	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
57	C₂×#88322	#41	1	1	yes	characteristic
58	C₂×#88322	#41	1	1	yes	characteristic
59	#11719910	—	2	2	no	—
60	#11719910	—	2	4	no	—
61	#11719910	C₄⋊C₄	1	4	yes	complemented
62	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
76	#88322	—	2	2	no	—
77	#88322	—	2	4	no	—
78	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
79	#88322	#117	1	1	yes	characteristic
80	#88322	C₄×D₄	1	1	yes	characteristic
81	#88322	C₄×Q₈	1	1	yes	characteristic
82	#12662	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
85	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
87	C₂²×C₅₃	#504	1	2	yes	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	4	4	no	—
90	C₄×C₅₃	#346	1	4	yes	complemented
91	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₅₃	#405	1	1	yes	characteristic
93	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
94	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
95	C₂×C₄⋊C₁₆	—	53	1	no	—
96	C₂×#346	—	53	2	no	—
97	C₂×C₄×C₁₆	—	53	1	no	—
98	C₈×C₄⋊C₄	—	53	1	no	—
99	#1374	—	53	2	no	—
100	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
102	C₂×C₅₃	#2108	1	1	yes	characteristic
103	C₂×C₅₃	#2169	1	2	yes	—
104	C₂×C₅₃	#2183	1	2	yes	—
105	C₂²×C₁₆	—	53	1	no	—
106	C₂²×C₁₆	—	53	1	no	—
107	C₂²×C₁₆	—	53	2	no	—
108	C₂×C₄⋊C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	2	no	—
111	C₄×C₄⋊C₄	#6157	1	1	yes	characteristic
112	C₄⋊C₁₆	—	53	4	no	—
113	#346	—	106	4	no	—
114	C₄×C₁₆	—	53	4	no	—
115	#336	—	53	2	no	—
116	C₅₃	#12662	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	—	53	1	no	—
119	C₂²×C₈	—	53	2	no	—
120	C₂×C₄⋊C₄	#88324	1	2	yes	—
121	C₂×C₄²	C₂×#6157	1	2	yes	—
122	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
123	C₂×C₁₆	—	53	2	no	—
124	C₂×C₁₆	—	53	2	no	—
125	C₂×C₁₆	—	53	2	no	—
126	C₂×C₁₆	—	53	2	no	—
127	C₂×C₁₆	—	53	4	no	—
128	C₂×C₁₆	—	106	2	no	—
129	C₂×C₁₆	—	106	1	no	—
130	C₂×C₁₆	—	106	1	no	—
131	C₂×C₁₆	—	106	2	no	—
132	C₄⋊C₈	—	53	4	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	106	2	no	—
136	C₄×C₈	—	106	2	no	—
137	#95	#88324	1	2	yes	—
138	C₂²×C₄	C₂×#88324	1	4	yes	—
139	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
140	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
141	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	1	no	—
145	C₂×C₈	—	53	1	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	1	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	4	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	106	1	no	—
152	C₂×C₈	—	106	1	no	—
153	C₂×C₈	—	106	2	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	D₅₃⋊C₈	1	2	yes	—
158	C₄²	C₄×#6157	1	2	yes	—
159	C₁₆	—	53	4	no	—
160	C₁₆	—	106	2	no	—
161	C₁₆	—	106	4	no	—
162	C₂³	C₂²×#88324	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×#88324	1	2	yes	—
168	C₂×C₄	C₄×#88324	1	2	yes	—
169	C₂×C₄	#13133291	1	4	yes	—
170	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
171	C₂×C₄	D₄×#6157	1	1	yes	characteristic
172	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
173	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
174	C₂×C₄	#13133304	1	1	yes	characteristic
175	C₂×C₄	#13133305	1	1	yes	characteristic
176	C₈	—	53	1	no	—
177	C₈	—	53	1	no	—
178	C₈	—	53	1	no	—
179	C₈	—	53	1	no	—
180	C₈	—	106	2	no	—
181	C₈	—	106	4	no	—
182	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
183	C₂²	D₄×#88324	1	1	yes	characteristic
184	C₂²	Q₈×#88324	1	1	yes	characteristic
185	C₂²	C₂×#13133291	1	2	yes	—
186	C₂²	#16811681	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	#16811511	1	4	yes	complemented
190	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
191	C₄	#16811588	1	1	yes	characteristic
192	C₄	#16811651	1	1	yes	characteristic
193	C₄	#16811672	1	1	yes	characteristic
194	C₂	C₄⋊C₄×#88324	1	1	yes	characteristic
195	C₂	C₂×#16811511	1	1	yes	characteristic
196	C₂	#20290924	1	1	yes	characteristic
197	C₂	#20291099	1	2	yes	—
198	C₂	#20291130	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×C₄⋊C₄×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×C₄⋊C₄	#6157
hypocenter	C₅₃	#12662
LCS	G ▷ C₂²×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
9. C₄⋊#11719910	C₄	semidirect
11. C₄×#11719910	C₄	semidirect
61. #11719910	C₄⋊C₄	semidirect
90. C₄×C₅₃	#346	semidirect
116. C₅₃	#12662	semidirect
189. 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
C₂×C₄×#11719910	2
#20289753	2
C₄⋊C₄×#88322	2
#12662	53