Subgroups — #28311972

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 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×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	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	C₄×#11719910	—	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	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#95×C₅₃	C₈	1	2	yes	—
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	2	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₄⋊C₄×C₅₃	C₁₆	1	4	yes	complemented
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₄²×C₅₃	M₁₆	1	2	yes	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	C₄⋊C₄	1	2	yes	—
56	C₂×#88322	C₂×D₄	1	1	yes	characteristic
57	C₂×#88322	C₂×D₄	1	1	yes	characteristic
58	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
59	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
60	C₂×#88322	#41	1	1	yes	characteristic
61	C₂×#88322	#41	1	1	yes	characteristic
62	#11719910	—	2	2	no	—
63	#11719910	—	2	2	no	—
64	#11719910	—	4	2	no	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
73	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
84	#88322	—	2	2	no	—
85	#88322	—	4	2	no	—
86	#88322	#121	1	1	yes	characteristic
87	#88322	C₂²⋊Q₈	1	1	yes	characteristic
88	#88322	#125	1	1	yes	characteristic
89	#88322	C₄⋊Q₈	1	1	yes	characteristic
90	#13159	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
94	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₅₃	#504	1	1	yes	characteristic
97	C₂²×C₅₃	#504	1	1	yes	characteristic
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	4	no	—
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	complemented
102	C₄×C₅₃	#389	1	1	yes	characteristic
103	C₄×C₅₃	#424	1	1	yes	characteristic
104	C₄×C₅₃	#436	1	1	yes	characteristic
105	C₄×C₅₃	#446	1	1	yes	characteristic
106	C₂×C₄⋊C₁₆	—	53	1	no	—
107	C₂×C₄⋊C₁₆	—	53	2	no	—
108	C₂×C₄×C₁₆	—	53	1	no	—
109	C₈×C₄⋊C₄	—	53	1	no	—
110	#1374	—	53	2	no	—
111	C₂×C₅₃	#1939	1	1	yes	characteristic
112	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
113	C₂×C₅₃	#2151	1	1	yes	characteristic
114	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
115	C₂×C₅₃	#2169	1	1	yes	characteristic
116	C₂×C₅₃	Q₈×C₁₆	1	1	yes	characteristic
117	C₂×C₅₃	#2183	1	1	yes	characteristic
118	C₂²×C₁₆	—	53	1	no	—
119	C₂²×C₁₆	—	53	1	no	—
120	C₂²×C₁₆	—	53	2	no	—
121	C₂×C₄⋊C₈	—	53	1	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	2	no	—
124	C₄×C₄⋊C₄	#6157	1	1	yes	characteristic
125	C₄⋊C₁₆	—	106	2	no	—
126	C₄⋊C₁₆	—	106	4	no	—
127	C₄×C₁₆	—	106	2	no	—
128	#336	—	53	2	no	—
129	C₅₃	#13159	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	2	no	—
133	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
134	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
135	C₂×C₄²	C₂×#6157	1	2	yes	—
136	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
137	C₂×C₁₆	—	53	2	no	—
138	C₂×C₁₆	—	53	2	no	—
139	C₂×C₁₆	—	106	1	no	—
140	C₂×C₁₆	—	106	1	no	—
141	C₂×C₁₆	—	106	1	no	—
142	C₂×C₁₆	—	106	1	no	—
143	C₂×C₁₆	—	106	2	no	—
144	C₂×C₁₆	—	106	2	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₄⋊C₈	—	106	2	no	—
147	C₄×C₈	—	53	2	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	106	2	no	—
150	C₄×C₈	—	106	2	no	—
151	#95	#88324	1	2	yes	—
152	C₂²×C₄	C₂×#88324	1	2	yes	—
153	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
154	C₂²×C₄	C₂×#88324	1	2	yes	—
155	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
156	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
157	C₂×C₈	—	53	1	no	—
158	C₂×C₈	—	53	1	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	1	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	106	1	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	#11719910	1	4	yes	complemented
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	#11719936	1	2	yes	—
174	C₄²	C₄⋊#6157	1	2	yes	—
175	C₁₆	—	106	2	no	—
176	C₁₆	—	106	2	no	—
177	C₁₆	—	212	2	no	—
178	C₂³	C₂²×#88324	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₄⋊#88324	1	2	yes	—
187	C₂×C₄	C₄⋊#88324	1	2	yes	—
188	C₂×C₄	C₂×#11719910	1	4	yes	—
189	C₂×C₄	C₂×#11719910	1	2	yes	—
190	C₂×C₄	#13133291	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
192	C₂×C₄	D₄×#6157	1	1	yes	characteristic
193	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
194	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
195	C₂×C₄	#13133304	1	1	yes	characteristic
196	C₂×C₄	#13133305	1	1	yes	characteristic
197	C₈	—	53	1	no	—
198	C₈	—	53	1	no	—
199	C₈	—	53	1	no	—
200	C₈	—	53	1	no	—
201	C₈	—	106	2	no	—
202	C₈	—	212	2	no	—
203	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
204	C₂²	D₄×#88324	1	1	yes	characteristic
205	C₂²	Q₈×#88324	1	1	yes	characteristic
206	C₂²	C₂²×#11719910	1	1	yes	characteristic
207	C₂²	C₂×#13133291	1	1	yes	characteristic
208	C₂²	#16811681	1	1	yes	characteristic
209	C₂²	#16811681	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	C₄⋊#11719910	1	4	yes	complemented
214	C₄	#16811574	1	1	yes	characteristic
215	C₄	#16811630	1	1	yes	characteristic
216	C₄	#16811657	1	1	yes	characteristic
217	C₄	#16811673	1	1	yes	characteristic
218	C₂	#20290812	1	1	yes	characteristic
219	C₂	C₂×C₄⋊#11719910	1	1	yes	characteristic
220	C₂	#20291050	1	1	yes	characteristic
221	C₂	D₄×#11719910	1	1	yes	characteristic
222	C₂	#20291099	1	1	yes	characteristic
223	C₂	Q₈×#11719910	1	1	yes	characteristic
224	C₂	#20291130	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₄×C₄⋊C₄×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×C₄⋊C₄	#6157
hypocenter	C₅₃	#13159
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
45. C₄⋊C₄×C₅₃	C₁₆	semidirect
101. C₄×C₅₃	C₄⋊C₁₆	semidirect
129. C₅₃	#13159	semidirect
170. C₄⋊C₄	#11719910	semidirect
213. C₄	C₄⋊#11719910	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×C₄×#11719910	2
#20289753	2
C₄⋊C₄×#88322	2
#13159	53