Subgroups — #28247725

Order 13568.

2738 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811390	C₂	1	2	yes	maximal
2	#20283283	C₂	1	16	yes	maximal, complemented
3	#20283279	C₂	1	6	yes	maximal, complemented
4	C₂×#16810647	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811390	—	2	16	no	—
7	C₂×#13133224	C₂²	1	16	yes	—
8	C₂²×#11719917	C₂²	1	12	yes	—
9	#16810650	C₂²	1	48	yes	complemented
10	#16810647	C₂²	1	4	yes	—
11	#16810647	C₂²	1	12	yes	—
12	#16810647	C₂²	1	24	yes	complemented
13	C₂×C₄⋊#88322	C₂²	1	24	yes	complemented
14	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₄	1	4	yes	—
16	C₄×#41×C₅₃	C₄	1	4	yes	—
17	C₄×#41×C₅₃	C₂²	1	8	yes	—
18	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×D₄×C₅₃	C₄	1	6	yes	—
20	C₂×#117×C₅₃	C₂²	1	3	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
22	#13133224	—	2	64	no	—
23	C₂×#11719917	—	2	48	no	—
24	C₂×#11719917	C₂³	1	8	yes	—
25	C₂×#11719917	C₂³	1	24	yes	—
26	C₂²×#88322	C₂³	1	24	yes	—
27	C₄⋊#88322	C₂³	1	16	yes	—
28	C₄⋊#88322	C₂³	1	48	yes	—
29	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×#41×C₅₃	D₄	1	8	yes	—
31	C₂×#41×C₅₃	Q₈	1	8	yes	—
32	C₂×#41×C₅₃	C₂³	1	4	yes	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₅₃	C₂×C₄	1	3	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
37	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
38	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
39	C₄×D₄×C₅₃	C₂×C₄	1	24	yes	—
40	C₄×D₄×C₅₃	C₂×C₄	1	24	yes	—
41	#117×C₅₃	C₂×C₄	1	12	yes	—
42	#117×C₅₃	C₂³	1	12	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
44	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
51	C₂⁴×C₅₃	C₂²×C₄	1	3	yes	—
52	#41×C₅₃	C₄⋊C₄	1	32	yes	—
53	#41×C₅₃	C₂×D₄	1	16	yes	—
54	#41×C₅₃	C₂×Q₈	1	16	yes	—
55	C₂×Q₈×C₅₃	C₄⋊C₄	1	8	yes	—
56	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₅₃	C₄⋊C₄	1	24	yes	—
58	C₂×D₄×C₅₃	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₅₃	—	4	12	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
70	C₄⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
71	C₂²⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
76	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
77	C₄²×C₅₃	C₂⁴	1	2	yes	—
78	C₄²×C₅₃	C₂⁴	1	6	yes	—
79	#11719917	—	2	16	no	—
80	#11719917	—	2	48	no	—
81	C₂×#88322	—	2	48	no	—
82	C₂×#88322	C₂⁴	1	16	yes	—
83	C₂³×C₅₃	—	4	6	no	—
84	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
85	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
87	Q₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
88	D₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	24	no	—
91	C₂×C₄×C₅₃	—	4	12	no	—
92	C₂×C₄×C₅₃	—	4	6	no	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
102	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
104	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
105	#88322	—	2	16	no	—
106	#60099	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	12	no	—
109	C₂²×C₅₃	—	4	12	no	—
110	C₂²×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
111	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
112	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	12	no	—
115	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
116	C₄×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
117	C₄×C₅₃	#568	1	4	yes	complemented
118	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
120	C₂×#568	—	53	2	no	—
121	C₂×C₄×#41	D₅₃	1	1	yes	characteristic
122	#2906	—	53	16	no	—
123	#2905	—	53	6	no	—
124	C₂×#423	—	53	6	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	2	12	no	—
127	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#568	1	2	yes	—
129	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
130	#568	—	106	16	no	—
131	C₂×#131	—	53	16	no	—
132	C₂²×M₁₆	—	53	12	no	—
133	C₄×#41	Dic₅₃	1	4	yes	—
134	C₄×#41	Dic₅₃	1	4	yes	—
135	C₄×#41	C₂×D₅₃	1	8	yes	—
136	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
137	C₂×C₄×D₄	Dic₅₃	1	6	yes	—
138	C₂×#117	C₂×D₅₃	1	3	yes	—
139	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
140	#424	—	53	48	no	—
141	#423	—	53	4	no	—
142	#423	—	53	12	no	—
143	#423	—	53	24	no	—
144	C₂×C₄⋊C₈	—	53	24	no	—
145	C₅₃	#60099	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	Dic₁₀₆	1	8	yes	—
147	C₂×#41	D₂₁₂	1	8	yes	—
148	C₂×#41	C₂×Dic₅₃	1	4	yes	—
149	C₂×#41	C₂²×D₅₃	1	4	yes	—
150	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂²×D₄	C₂×Dic₅₃	1	3	yes	—
152	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
153	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
154	#131	—	106	64	no	—
155	C₂×M₁₆	—	53	8	no	—
156	C₂×M₁₆	—	53	24	no	—
157	C₂×M₁₆	—	106	48	no	—
158	C₂²×C₈	—	53	24	no	—
159	C₄×Q₈	C₂×Dic₅₃	1	8	yes	—
160	C₄×Q₈	C₂×Dic₅₃	1	8	yes	—
161	C₄×D₄	C₂×Dic₅₃	1	24	yes	—
162	C₄×D₄	C₂×Dic₅₃	1	24	yes	—
163	#117	C₂×Dic₅₃	1	12	yes	—
164	#117	C₂²×D₅₃	1	12	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
166	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	6	yes	—
167	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
168	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
169	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
170	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
171	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
172	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
173	C₄⋊C₈	—	53	16	no	—
174	C₄⋊C₈	—	53	48	no	—
175	C₂⁴	C₂²×Dic₅₃	1	3	yes	—
176	#41	C₄⋊Dic₅₃	1	32	yes	—
177	#41	C₂×Dic₁₀₆	1	16	yes	—
178	#41	C₂×D₂₁₂	1	16	yes	—
179	C₂×Q₈	C₄⋊Dic₅₃	1	8	yes	—
180	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
181	C₂×D₄	C₄⋊Dic₅₃	1	24	yes	—
182	C₂×D₄	C₂²×Dic₅₃	1	12	yes	—
183	C₂²×C₄	—	4	12	no	—
184	C₂²×C₄	C₂×Dic₁₀₆	1	12	yes	—
185	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
186	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
188	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
189	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
190	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
191	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
192	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
194	M₁₆	—	106	16	no	—
195	M₁₆	—	106	48	no	—
196	C₂×C₈	—	53	16	no	—
197	C₂×C₈	—	106	48	no	—
198	C₄⋊C₄	C₂²×Dic₅₃	1	24	yes	—
199	C₂²⋊C₄	C₂²×Dic₅₃	1	24	yes	—
200	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₄²	C₂²×Dic₅₃	1	6	yes	—
203	C₄²	C₂²×Dic₅₃	1	12	yes	—
204	C₄²	C₂²×Dic₅₃	1	12	yes	—
205	C₄²	C₂³×D₅₃	1	2	yes	—
206	C₄²	C₂³×D₅₃	1	6	yes	—
207	C₂³	—	4	6	no	—
208	C₂³	C₂×C₄⋊Dic₅₃	1	12	yes	—
209	C₂³	C₂³×Dic₅₃	1	6	yes	—
210	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
211	Q₈	C₂×C₄⋊Dic₅₃	1	16	yes	—
212	D₄	C₂×C₄⋊Dic₅₃	1	48	yes	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	24	no	—
215	C₂×C₄	—	4	12	no	—
216	C₂×C₄	—	4	6	no	—
217	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₅₃	1	24	yes	—
219	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
220	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
221	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
223	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
224	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
225	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
226	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
227	C₂×C₄	C₂²×Dic₁₀₆	1	12	yes	—
228	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
229	C₈	—	106	16	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	12	no	—
232	C₂²	—	4	12	no	—
233	C₂²	C₂²×C₄⋊Dic₅₃	1	2	yes	—
234	C₂²	C₂²×C₄⋊Dic₅₃	1	12	yes	—
235	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	12	no	—
238	C₄	C₂²×C₄⋊Dic₅₃	1	2	yes	—
239	C₄	C₂²×C₄⋊Dic₅₃	1	12	yes	—
240	C₄	#16811390	1	4	yes	complemented
241	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
242	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	2	12	no	—
245	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
246	C₂	C₂×#16811390	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₂×C₄×#41×C₅₃	C₂
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×C₄×#41	D₅₃
hypocenter	C₅₃	#60099
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #20283283	C₂	semidirect
3. #20283279	C₂	semidirect
4. C₂×#16810647	C₂	semidirect
9. #16810650	C₂²	semidirect
12. #16810647	C₂²	semidirect
13. C₂×C₄⋊#88322	C₂²	semidirect
117. C₄×C₅₃	#568	semidirect
145. C₅₃	#60099	semidirect
240. C₄	#16811390	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811390	2
#20283283	2
#20283279	2
C₂×#16810647	2
C₂×C₄×#41×C₅₃	2
#60099	53