Subgroups — #20291619

Order 6784.

322 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#16811647	C₂	1	2	yes	maximal
8	#16811647	C₂	1	2	yes	maximal
9	#16811644	C₂	1	1	yes	maximal, characteristic
10	#16811639	C₂	1	1	yes	maximal, characteristic
11	#16811175	C₂	1	1	yes	maximal, characteristic
12	#13133284	C₂²	1	2	yes	—
13	#13133284	C₂²	1	2	yes	—
14	C₄⋊#88324	C₂²	1	1	yes	characteristic
15	C₄⋊#88324	C₂²	1	1	yes	characteristic
16	C₄⋊#88324	C₂²	1	1	yes	characteristic
17	C₄⋊#88324	C₂²	1	1	yes	characteristic
18	C₄⋊#88324	C₂²	1	2	yes	—
19	C₄⋊#88324	C₂²	1	2	yes	—
20	C₂²⋊#88324	C₂²	1	1	yes	characteristic
21	C₂²⋊#88324	C₂²	1	1	yes	characteristic
22	C₂²⋊#88324	C₂²	1	2	yes	—
23	#13133272	C₂²	1	2	yes	—
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	C₂×#88324	C₂³	1	1	yes	characteristic
48	C₂×#88324	C₂³	1	1	yes	characteristic
49	C₂×#88324	C₂³	1	1	yes	characteristic
50	C₂×#88324	C₂³	1	2	yes	—
51	C₂×#88324	C₂³	1	1	yes	characteristic
52	C₂×#88324	C₂³	1	2	yes	—
53	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	#11719919	C₂×C₄	1	2	yes	—
60	#11719919	C₂×C₄	1	2	yes	—
61	#11719919	C₂×C₄	1	2	yes	—
62	#11719919	C₂×C₄	1	2	yes	—
63	#11719919	C₂×C₄	1	2	yes	—
64	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂³	1	2	yes	—
69	C₄×Dic₅₃	C₂³	1	2	yes	—
70	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
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×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	4	1	no	—
80	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
86	Dic₁₀₆	—	2	2	no	—
87	Dic₁₀₆	—	4	2	no	—
88	#88324	—	2	1	no	—
89	#88324	—	2	1	no	—
90	#88324	—	2	1	no	—
91	#88324	—	2	2	no	—
92	#88324	—	2	1	no	—
93	#88324	—	2	2	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	#131	1	2	yes	—
102	Dic₅₃	—	2	1	no	—
103	Dic₅₃	—	2	2	no	—
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	2	2	no	—
106	Dic₅₃	—	2	1	no	—
107	Dic₅₃	2₋¹⁺⁴	1	1	yes	characteristic
108	Dic₅₃	2₋¹⁺⁴	1	1	yes	characteristic
109	#2985	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	4	1	no	—
111	C₂×C₅₃	#519	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
113	C₂×C₅₃	#568	1	1	yes	characteristic
114	#533	—	53	1	no	—
115	#446	—	53	1	no	—
116	#446	—	53	2	no	—
117	#446	—	53	1	no	—
118	#446	—	53	2	no	—
119	C₈×Q₈	—	53	1	no	—
120	C₈×Q₈	—	53	1	no	—
121	#433	—	53	2	no	—
122	#433	—	53	2	no	—
123	#432	—	53	1	no	—
124	#431	—	53	1	no	—
125	C₅₃	#2985	1	1	yes	characteristic, complemented, Sylow
126	C₄⋊Q₈	—	53	1	no	—
127	C₄⋊Q₈	—	53	1	no	—
128	#125	—	53	2	no	—
129	C₂²⋊Q₈	—	53	1	no	—
130	C₂²⋊Q₈	—	53	2	no	—
131	C₂²⋊Q₈	#6157	1	1	yes	characteristic
132	C₄×Q₈	—	53	1	no	—
133	C₄×Q₈	—	53	1	no	—
134	C₄×Q₈	—	53	2	no	—
135	#117	—	53	2	no	—
136	C₂×C₄²	—	53	1	no	—
137	C₄⋊C₈	—	53	1	no	—
138	C₄⋊C₈	—	53	1	no	—
139	C₄⋊C₈	—	53	1	no	—
140	C₄⋊C₈	—	53	1	no	—
141	C₄⋊C₈	—	53	2	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	—	53	2	no	—
144	C₄⋊C₈	—	53	2	no	—
145	C₂²⋊C₈	—	53	1	no	—
146	C₂²⋊C₈	—	53	1	no	—
147	C₂²⋊C₈	—	53	2	no	—
148	#97	—	53	2	no	—
149	#97	—	53	2	no	—
150	#97	—	53	2	no	—
151	C₄×C₈	—	53	1	no	—
152	C₄×C₈	—	53	1	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	53	2	no	—
155	C₂×Q₈	—	53	1	no	—
156	C₂×Q₈	—	53	2	no	—
157	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	2	no	—
167	C₄⋊C₄	—	53	2	no	—
168	C₄⋊C₄	—	53	2	no	—
169	C₄⋊C₄	—	53	2	no	—
170	C₄⋊C₄	—	53	2	no	—
171	C₄⋊C₄	—	53	2	no	—
172	C₄⋊C₄	—	53	2	no	—
173	C₄⋊C₄	—	53	1	no	—
174	C₄⋊C₄	C₂×#6157	1	2	yes	—
175	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
176	C₂²⋊C₄	—	53	2	no	—
177	C₂²⋊C₄	C₂×#6157	1	2	yes	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	2	no	—
180	C₄²	—	53	2	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₂³	C₂²×#6157	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	Q₈	—	106	2	no	—
187	Q₈	—	212	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	53	1	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	106	1	no	—
196	C₂×C₄	—	106	1	no	—
197	C₂×C₄	—	212	1	no	—
198	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
199	C₂×C₄	C₂²×#6157	1	2	yes	—
200	C₂×C₄	C₂²×#6157	1	2	yes	—
201	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
202	C₈	—	106	1	no	—
203	C₈	—	106	1	no	—
204	C₈	—	106	1	no	—
205	C₈	—	106	2	no	—
206	C₈	—	106	1	no	—
207	C₈	—	106	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂³×#6157	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	53	1	no	—
216	C₄	—	53	1	no	—
217	C₄	—	106	1	no	—
218	C₄	—	106	2	no	—
219	C₄	—	106	1	no	—
220	C₄	—	106	1	no	—
221	C₄	—	106	2	no	—
222	C₄	#13133304	1	2	yes	—
223	C₂	—	4	1	no	—
224	C₂	#16811691	1	1	yes	characteristic
225	C₂	C₂×#13133304	1	1	yes	characteristic
226	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	#2985
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
125. C₅₃	#2985	semidirect