Subgroups — #20291510

Order 6784.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811675	C₂	1	2	yes	maximal, complemented
2	#16811674	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811673	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811673	C₂	1	2	yes	maximal, complemented
5	#16811672	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811629	C₂	1	2	yes	maximal
8	C₂×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
9	#16811587	C₂	1	2	yes	maximal
10	C₄×#11719940	C₂	1	1	yes	maximal, characteristic
11	#16811175	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719940	C₂²	1	1	yes	characteristic
13	C₂×#11719940	C₂²	1	1	yes	characteristic
14	C₂²×#88324	C₂²	1	2	yes	—
15	#13133284	C₂²	1	2	yes	—
16	#13133284	C₂²	1	2	yes	—
17	C₄⋊#88324	C₂²	1	1	yes	characteristic
18	C₄⋊#88324	C₂²	1	1	yes	characteristic
19	C₄⋊#88324	C₂²	1	2	yes	—
20	C₄⋊#88324	C₂²	1	2	yes	—
21	C₄⋊#88324	C₂²	1	2	yes	—
22	#13133272	C₂²	1	1	yes	characteristic
23	#13133272	C₂²	1	1	yes	characteristic
24	#13133272	C₂²	1	2	yes	—
25	#13133272	C₂²	1	2	yes	—
26	C₄×#88324	C₂²	1	1	yes	characteristic
27	C₄×#88324	C₂²	1	1	yes	characteristic
28	C₄×#88324	C₂²	1	2	yes	—
29	C₄×#88324	C₂²	1	2	yes	—
30	#13133216	C₄	1	1	yes	characteristic
31	#13133215	C₄	1	1	yes	characteristic
32	#13133207	C₄	1	2	yes	—
33	#13133196	C₄	1	1	yes	characteristic
34	#13133194	C₄	1	2	yes	—
35	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
36	#13133182	C₂²	1	1	yes	characteristic
37	#13133182	C₂²	1	2	yes	—
38	#13133173	C₂²	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
47	#11719940	—	2	2	no	—
48	#11719940	Q₈	1	4	yes	complemented
49	C₂×#88324	—	2	4	no	—
50	C₂×#88324	C₂³	1	2	yes	—
51	C₂×#88324	C₂³	1	1	yes	characteristic
52	C₂×#88324	C₂³	1	1	yes	characteristic
53	C₂×#88324	C₂³	1	2	yes	—
54	C₂×#88324	C₂³	1	1	yes	characteristic
55	C₂×#88324	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	#11719919	C₂×C₄	1	2	yes	—
63	#11719919	C₂×C₄	1	2	yes	—
64	#11719919	C₂×C₄	1	2	yes	—
65	#11719919	C₂×C₄	1	2	yes	—
66	#11719919	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₄×Dic₅₃	C₂³	1	2	yes	—
72	C₄×Dic₅₃	C₂³	1	2	yes	—
73	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₅₃	—	4	1	no	—
81	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂×Q₈	1	2	yes	—
87	C₂×Dic₅₃	#41	1	2	yes	—
88	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
89	Dic₁₀₆	—	2	2	no	—
90	Dic₁₀₆	—	4	2	no	—
91	#88324	—	2	2	no	—
92	#88324	—	2	2	no	—
93	#88324	—	2	1	no	—
94	#88324	—	2	1	no	—
95	#88324	C₂×Q₈	1	2	yes	—
96	#88324	C₂×Q₈	1	2	yes	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
100	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	#131	1	2	yes	—
105	Dic₅₃	—	2	2	no	—
106	Dic₅₃	—	2	1	no	—
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	C₄×Q₈	1	2	yes	—
110	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
111	Dic₅₃	C₂×#41	1	1	yes	characteristic
112	#2962	—	53	1	no	maximal, Sylow
113	C₂×C₅₃	—	2	2	no	—
114	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
116	C₂×C₅₃	#568	1	1	yes	characteristic
117	#533	—	53	1	no	—
118	#446	—	53	1	no	—
119	#446	—	53	2	no	—
120	#446	—	53	1	no	—
121	#446	—	53	2	no	—
122	C₈×Q₈	—	53	1	no	—
123	C₈×Q₈	—	53	1	no	—
124	#424	—	53	2	no	—
125	C₂×C₄⋊C₈	—	53	1	no	—
126	#405	—	53	2	no	—
127	C₄×M₁₆	—	53	1	no	—
128	C₅₃	#2962	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	53	1	no	—
130	C₂×M₁₆	—	53	1	no	—
131	C₂²×C₈	—	53	2	no	—
132	C₄⋊Q₈	—	53	1	no	—
133	C₄⋊Q₈	—	53	1	no	—
134	#125	—	53	2	no	—
135	C₂²⋊Q₈	—	53	1	no	—
136	C₂²⋊Q₈	—	53	2	no	—
137	C₂²⋊Q₈	#6157	1	1	yes	characteristic
138	C₄×Q₈	—	53	1	no	—
139	C₄×Q₈	—	53	1	no	—
140	C₄×Q₈	—	53	2	no	—
141	#117	—	53	2	no	—
142	C₂×C₄²	—	53	1	no	—
143	C₄⋊C₈	—	53	1	no	—
144	C₄⋊C₈	—	53	1	no	—
145	C₄⋊C₈	—	53	2	no	—
146	C₄⋊C₈	—	53	2	no	—
147	C₄⋊C₈	—	53	2	no	—
148	C₄⋊C₈	—	53	2	no	—
149	C₄⋊C₈	—	53	2	no	—
150	#97	—	53	1	no	—
151	#97	—	53	1	no	—
152	#97	—	53	2	no	—
153	#97	—	53	2	no	—
154	C₄×C₈	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	2	no	—
157	C₄×C₈	—	53	2	no	—
158	C₂×Q₈	—	53	1	no	—
159	C₂×Q₈	—	53	2	no	—
160	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
164	M₁₆	—	53	4	no	—
165	M₁₆	—	106	2	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	1	no	—
170	C₂×C₈	—	53	1	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	106	4	no	—
173	C₄⋊C₄	—	53	2	no	—
174	C₄⋊C₄	—	53	2	no	—
175	C₄⋊C₄	—	53	2	no	—
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	53	2	no	—
178	C₄⋊C₄	—	53	2	no	—
179	C₄⋊C₄	—	53	1	no	—
180	C₄⋊C₄	C₂×#6157	1	2	yes	—
181	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
182	C₂²⋊C₄	—	53	2	no	—
183	C₂²⋊C₄	C₂×#6157	1	2	yes	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	2	no	—
186	C₄²	—	53	2	no	—
187	C₄²	—	53	1	no	—
188	C₄²	—	53	1	no	—
189	C₄²	—	53	1	no	—
190	C₂³	C₂²×#6157	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	106	2	no	—
193	Q₈	—	212	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	53	1	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	53	2	no	—
202	C₂×C₄	—	53	2	no	—
203	C₂×C₄	—	212	1	no	—
204	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
205	C₂×C₄	C₂²×#6157	1	2	yes	—
206	C₂×C₄	C₂²×#6157	1	2	yes	—
207	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
208	C₈	—	53	2	no	—
209	C₈	—	53	2	no	—
210	C₈	—	106	2	no	—
211	C₈	—	106	2	no	—
212	C₈	—	106	1	no	—
213	C₈	—	106	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈×#6157	1	2	yes	—
217	C₂²	C₂³×#6157	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	53	1	no	—
222	C₄	—	53	1	no	—
223	C₄	—	53	2	no	—
224	C₄	—	106	2	no	—
225	C₄	—	106	1	no	—
226	C₄	—	106	1	no	—
227	C₄	—	106	2	no	—
228	C₄	#13133304	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈×#6157	1	1	yes	characteristic
231	C₂	C₂×#13133304	1	1	yes	characteristic
232	C₂	#16811702	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂²⋊Q₈×C₅₃	C₄
hypercenter	C₂²⋊Q₈	#6157
hypocenter	C₅₃	#2962
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊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. #16811673	C₂	semidirect
4. #16811673	C₂	semidirect
5. #16811672	C₂	semidirect
6. Q₈×#88324	C₂	semidirect
48. #11719940	Q₈	semidirect
128. C₅₃	#2962	semidirect