Subgroups — #20291517

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811675	C₂	1	4	yes	maximal, complemented
2	#16811674	C₂	1	2	yes	maximal, complemented
3	#16811672	C₂	1	2	yes	maximal, complemented
4	#16811628	C₂	1	2	yes	maximal
5	#16811627	C₂	1	1	yes	maximal, characteristic
6	C₄×#11719936	C₂	1	1	yes	maximal, characteristic
7	C₄×#11719936	C₂	1	2	yes	maximal
8	C₄×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719936	C₂²	1	2	yes	—
10	C₂×#11719936	C₂²	1	1	yes	characteristic
11	C₂×#11719936	C₂²	1	1	yes	characteristic
12	#13133284	C₂²	1	4	yes	—
13	#13133284	C₂²	1	4	yes	—
14	C₄⋊#88324	C₂²	1	2	yes	—
15	C₄⋊#88324	C₂²	1	2	yes	—
16	#13133272	C₂²	1	4	yes	—
17	#13133272	C₂²	1	2	yes	—
18	C₄×#88324	C₂²	1	2	yes	—
19	C₄×#88324	C₂²	1	4	yes	—
20	C₂×Q₈×D₅₃	C₄	1	1	yes	characteristic
21	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₀₆	C₄	1	2	yes	—
23	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
24	#13133182	C₂²	1	1	yes	characteristic
25	#13133182	C₂²	1	2	yes	—
26	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
27	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
28	C₄²×D₅₃	C₂²	1	1	yes	characteristic
29	C₄²×D₅₃	C₂²	1	2	yes	—
30	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
31	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	Q₈×D₅₃	—	2	4	no	—
33	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
38	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
40	C₂×#88324	C₂³	1	4	yes	—
41	C₂×#88324	C₂³	1	2	yes	—
42	C₂×#88324	C₂³	1	2	yes	—
43	#11719936	—	2	4	no	—
44	#11719936	—	2	2	no	—
45	#11719936	Q₈	1	4	yes	complemented
46	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×C₄	1	2	yes	—
49	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	#11719919	C₂×C₄	1	2	yes	—
53	#11719919	C₂×C₄	1	2	yes	—
54	#11719919	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₄×Dic₅₃	C₂³	1	2	yes	—
58	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂³	1	2	yes	—
61	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
71	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
73	C₄×D₅₃	—	2	2	no	—
74	C₄×D₅₃	—	2	4	no	—
75	C₄×D₅₃	—	4	1	no	—
76	C₄×D₅₃	—	4	2	no	—
77	C₄×D₅₃	C₂×Q₈	1	2	yes	—
78	C₄×D₅₃	#41	1	2	yes	—
79	Dic₁₀₆	—	2	4	no	—
80	Dic₁₀₆	M₁₆	1	4	yes	complemented
81	#88324	—	2	4	no	—
82	#88324	—	2	2	no	—
83	#88324	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×D₅₃	—	2	1	no	—
86	C₂×D₅₃	—	2	1	no	—
87	C₂×D₅₃	C₄×Q₈	1	2	yes	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	C₄×Q₈	1	2	yes	—
92	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
93	Dic₅₃	—	2	1	no	—
94	Dic₅₃	—	2	1	no	—
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	C₂×M₁₆	1	4	yes	—
97	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
98	Dic₅₃	C₂×#41	1	1	yes	characteristic
99	Q₈×M₁₆	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₅₃	#568	1	1	yes	characteristic
103	D₅₃	—	2	2	no	—
104	C₂×C₄×Q₈	—	53	1	no	—
105	#446	—	53	2	no	—
106	#446	—	53	4	no	—
107	C₈×Q₈	—	53	2	no	—
108	#423	—	53	1	no	—
109	#423	—	53	2	no	—
110	C₄×M₁₆	—	53	1	no	—
111	C₄×M₁₆	—	53	2	no	—
112	C₅₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	53	1	no	—
114	C₂×M₁₆	—	53	1	no	—
115	C₂×M₁₆	—	53	2	no	—
116	C₂×M₁₆	—	53	1	no	—
117	C₄×Q₈	—	53	1	no	—
118	C₄×Q₈	—	53	1	no	—
119	C₄×Q₈	—	53	2	no	—
120	C₄×Q₈	—	53	1	no	—
121	C₄×Q₈	—	53	2	no	—
122	C₄×Q₈	#6157	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	53	2	no	—
124	C₂×C₄⋊C₄	—	53	1	no	—
125	C₂×C₄²	—	53	1	no	—
126	C₂×C₄²	—	53	2	no	—
127	C₄⋊C₈	—	53	2	no	—
128	C₄⋊C₈	—	53	4	no	—
129	C₄⋊C₈	—	53	2	no	—
130	C₄⋊C₈	—	53	4	no	—
131	#97	—	53	4	no	—
132	#97	—	53	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	4	no	—
135	C₂×Q₈	—	53	1	no	—
136	C₂×Q₈	—	53	2	no	—
137	C₂×Q₈	—	106	4	no	—
138	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
139	C₂²×C₄	—	53	1	no	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	53	1	no	—
143	C₂²×C₄	—	53	2	no	—
144	M₁₆	—	53	4	no	—
145	M₁₆	—	106	4	no	—
146	M₁₆	—	106	2	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	—	53	2	no	—
150	C₄⋊C₄	—	53	2	no	—
151	C₄⋊C₄	—	53	2	no	—
152	C₄⋊C₄	—	53	2	no	—
153	C₄⋊C₄	—	53	2	no	—
154	C₄⋊C₄	—	53	1	no	—
155	C₄⋊C₄	C₂×#6157	1	2	yes	—
156	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
157	C₄²	—	53	2	no	—
158	C₄²	—	53	1	no	—
159	C₄²	—	53	2	no	—
160	C₄²	—	53	1	no	—
161	C₄²	—	53	2	no	—
162	C₄²	—	53	1	no	—
163	C₄²	C₂×#6157	1	2	yes	—
164	C₄²	C₂×#6157	1	1	yes	characteristic
165	C₂³	—	53	1	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	53	4	no	—
168	Q₈	—	106	4	no	—
169	C₂×C₄	—	53	2	no	—
170	C₂×C₄	—	53	1	no	—
171	C₂×C₄	—	53	2	no	—
172	C₂×C₄	—	53	2	no	—
173	C₂×C₄	—	53	1	no	—
174	C₂×C₄	—	53	2	no	—
175	C₂×C₄	—	53	1	no	—
176	C₂×C₄	—	106	4	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	—	212	1	no	—
180	C₂×C₄	C₂²×#6157	1	2	yes	—
181	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
182	C₂×C₄	C₂²×#6157	1	2	yes	—
183	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
184	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
185	C₈	—	53	4	no	—
186	C₈	—	106	4	no	—
187	C₈	—	106	2	no	—
188	C₂²	—	53	2	no	—
189	C₂²	—	106	1	no	—
190	C₂²	—	106	1	no	—
191	C₂²	C₂³×#6157	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	53	1	no	—
196	C₄	—	53	1	no	—
197	C₄	—	53	4	no	—
198	C₄	—	106	1	no	—
199	C₄	—	106	1	no	—
200	C₄	—	106	2	no	—
201	C₄	Q₈×#6157	1	2	yes	—
202	C₄	D₅₃⋊M₁₆	1	2	yes	—
203	C₂	—	106	2	no	—
204	C₂	C₂×Q₈×#6157	1	1	yes	characteristic
205	C₂	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
206	C₂	#16811702	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×Q₈×C₅₃	C₄
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #16811675	C₂	semidirect
2. #16811674	C₂	semidirect
3. #16811672	C₂	semidirect
45. #11719936	Q₈	semidirect
80. Dic₁₀₆	M₁₆	semidirect
83. #88324	C₂×Q₈	semidirect
112. C₅₃	Q₈×M₁₆	semidirect