Subgroups — #20283381

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810684	C₂	1	4	yes	maximal, complemented
2	#16810680	C₂	1	2	yes	maximal, complemented
3	#16810676	C₂	1	2	yes	maximal, complemented
4	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810590	C₂	1	2	yes	maximal
7	#16810579	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719917	C₂²	1	2	yes	—
10	C₂²×#88322	C₂²	1	2	yes	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	2	yes	—
14	C₄⋊#88322	C₂²	1	2	yes	—
15	C₄⋊#88322	C₂²	1	2	yes	—
16	C₂²⋊#88322	C₂²	1	4	yes	—
17	C₂²⋊#88322	C₂²	1	4	yes	—
18	#13133128	C₂²	1	4	yes	—
19	C₄×#88322	C₂²	1	4	yes	—
20	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
24	C₄×D₄×C₅₃	C₄	1	2	yes	—
25	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
26	C₄×D₄×C₅₃	C₄	1	2	yes	—
27	#117×C₅₃	C₂²	1	1	yes	characteristic
28	#117×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
31	#41×C₅₃	—	2	2	no	—
32	#41×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄²×C₅₃	C₂³	1	2	yes	—
55	C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	#11719917	—	2	4	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	C₂³	1	2	yes	—
59	C₂×#88322	C₂³	1	4	yes	—
60	C₂×#88322	C₂³	1	2	yes	—
61	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	Q₈×C₅₃	—	2	2	no	—
64	D₄×C₅₃	—	2	4	no	—
65	D₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	4	1	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#41	1	2	yes	—
81	C₂×C₄×C₅₃	#41	1	2	yes	—
82	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	#88322	—	2	2	no	—
84	#88322	—	2	2	no	—
85	#88322	—	2	4	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	#117	1	2	yes	—
92	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	#117	1	2	yes	—
100	C₄×C₅₃	#131	1	2	yes	—
101	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
103	#2925	—	53	1	no	maximal, Sylow
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	—	4	2	no	—
106	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
108	C₂×C₅₃	#568	1	1	yes	characteristic
109	C₄×#41	D₅₃	1	1	yes	characteristic
110	#433	—	53	4	no	—
111	#432	—	53	2	no	—
112	#431	—	53	2	no	—
113	#423	—	53	1	no	—
114	C₂×C₄⋊C₈	—	53	1	no	—
115	#408	—	53	2	no	—
116	#405	—	53	2	no	—
117	C₅₃	#2925	1	1	yes	characteristic, complemented, Sylow
118	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
119	C₂×M₁₆	—	53	2	no	—
120	C₂²×C₈	—	53	2	no	—
121	C₄×Q₈	Dic₅₃	1	1	yes	characteristic
122	C₄×Q₈	Dic₅₃	1	1	yes	characteristic
123	C₄×D₄	Dic₅₃	1	2	yes	—
124	C₄×D₄	Dic₅₃	1	1	yes	characteristic
125	C₄×D₄	Dic₅₃	1	2	yes	—
126	C₄×D₄	Dic₅₃	1	1	yes	characteristic
127	#117	C₂×D₅₃	1	1	yes	characteristic
128	#117	C₂×D₅₃	1	2	yes	—
129	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₅₃	1	2	yes	—
131	C₄⋊C₈	—	53	1	no	—
132	C₄⋊C₈	—	53	1	no	—
133	C₄⋊C₈	—	53	2	no	—
134	C₄⋊C₈	—	53	2	no	—
135	C₄⋊C₈	—	53	2	no	—
136	C₂²⋊C₈	—	53	4	no	—
137	C₂²⋊C₈	—	53	4	no	—
138	#97	—	53	4	no	—
139	C₄×C₈	—	53	4	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
144	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
146	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
148	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
150	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
151	M₁₆	—	106	4	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	106	4	no	—
156	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
163	C₄²	C₂×Dic₅₃	1	2	yes	—
164	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
165	C₄²	C₂×Dic₅₃	1	2	yes	—
166	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	C₄²	C₂²×D₅₃	1	2	yes	—
168	C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	C₂³	C₂²×Dic₅₃	1	2	yes	—
171	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#11719945	1	2	yes	—
182	C₂×C₄	#11719945	1	2	yes	—
183	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
185	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
186	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
188	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
192	C₈	—	106	2	no	—
193	C₈	—	106	2	no	—
194	C₈	—	106	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#13133176	1	2	yes	—
201	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#13133176	1	2	yes	—
209	C₄	#13133224	1	2	yes	—
210	C₄	C₂×#11719945	1	1	yes	characteristic
211	C₄	C₂×#11719945	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#13133176	1	1	yes	characteristic
215	C₂	C₂×#13133224	1	1	yes	characteristic
216	C₂	#16811390	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×#41×C₅₃	C₂
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#2925
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #16810684	C₂	semidirect
2. #16810680	C₂	semidirect
3. #16810676	C₂	semidirect
4. #16810647	C₂	semidirect
5. C₂×C₄⋊#88322	C₂	semidirect
7. #16810579	C₂	semidirect
117. C₅₃	#2925	semidirect