Subgroups — #16807178

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131803	C₂	1	2	yes	maximal, complemented
2	#13131226	C₂	1	4	yes	maximal, complemented
3	#13131223	C₂	1	4	yes	maximal, complemented
4	#13131177	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718304	C₂	1	2	yes	maximal, complemented
6	#13131047	C₂	1	1	yes	maximal, characteristic, complemented
7	#13123739	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#575	C₄	1	1	yes	characteristic
9	C₂×#88202	C₂²	1	2	yes	—
10	#11719162	—	2	4	no	—
11	#11719162	C₂²	1	8	yes	complemented
12	C₂×#88106	—	2	4	no	—
13	C₂×#88106	C₂²	1	2	yes	—
14	C₂×#88106	C₂²	1	1	yes	characteristic
15	C₂²×#6092	C₂²	1	1	yes	characteristic
16	#11718897	C₄	1	2	yes	—
17	#11718858	C₄	1	4	yes	—
18	#11718857	C₂²	1	4	yes	—
19	#11718786	C₄	1	1	yes	characteristic
20	#11718783	C₂²	1	1	yes	characteristic
21	#11718304	C₂²	1	4	yes	complemented
22	#11718304	C₂²	1	4	yes	complemented
23	#11718261	C₂²	1	4	yes	complemented
24	#11718258	C₂²	1	2	yes	complemented
25	#11718257	C₂²	1	2	yes	complemented
26	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
27	C₁₃×#137	C₈	1	4	yes	—
28	C₁₃×AGL(1,ℤ/8)	C₈	1	4	yes	—
29	C₁₃×#135	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	#88202	—	2	2	no	—
32	#88202	—	2	2	no	—
33	#88202	C₂×C₄	1	4	yes	—
34	#88202	C₂³	1	4	yes	—
35	#88180	C₂×C₄	1	1	yes	characteristic
36	C₂×#6099	—	2	2	no	—
37	C₂×#6099	C₂×C₄	1	2	yes	—
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	C₂³	1	1	yes	characteristic
40	#88106	—	2	2	no	—
41	#88106	—	2	8	no	—
42	#88106	D₄	1	4	yes	—
43	#88106	C₂³	1	4	yes	—
44	C₂×#6092	—	2	2	no	—
45	C₂×#6092	—	2	2	no	—
46	C₂×#6092	—	2	8	no	—
47	C₂×#6092	C₂³	1	1	yes	characteristic
48	C₂×#6092	C₂³	1	1	yes	characteristic
49	C₂×#6092	C₂³	1	2	yes	—
50	#88077	C₂×C₄	1	8	yes	—
51	#88059	C₂×C₄	1	4	yes	—
52	#88055	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂³	1	2	yes	—
54	#33442	—	13	1	no	maximal, Sylow
55	C₁₃×#41	—	2	2	no	—
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	C₂×C₈	1	4	yes	—
58	C₁₃×#41	C₂²×C₄	1	4	yes	—
59	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	C₂×C₈	1	4	yes	—
65	C₁₃×SD₁₆	C₂×C₈	1	8	yes	—
66	C₁₃×D₈	C₂×C₈	1	4	yes	—
67	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²⋊Dic₁₃	—	2	1	no	—
70	C₄⋊Dic₁₃	—	2	1	no	—
71	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
72	#6099	—	2	2	no	—
73	#6099	—	2	4	no	—
74	#6099	—	2	4	no	—
75	#6099	C₂²×C₄	1	4	yes	—
76	C₂×#1055	—	2	2	no	—
77	C₂×#1055	—	2	4	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
80	C₂×#1055	C₂×D₄	1	2	yes	—
81	C₂×#1055	#41	1	2	yes	—
82	C₂×#1055	C₂⁴	1	1	yes	characteristic
83	#6092	—	2	4	no	—
84	#6092	—	2	1	no	—
85	#6092	—	2	1	no	—
86	#6092	C₂×D₄	1	4	yes	—
87	#3407	—	13	2	no	—
88	#2955	—	13	1	no	—
89	#2171	—	13	4	no	—
90	#2170	—	13	4	no	—
91	#2153	—	13	1	no	—
92	C₂×#350	—	13	2	no	—
93	#2108	—	13	1	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	#131	1	1	yes	characteristic
96	Q₈×C₁₃	—	2	2	no	—
97	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
98	D₄×C₁₃	—	2	4	no	—
99	D₄×C₁₃	—	2	4	no	—
100	D₄×C₁₃	—	2	2	no	—
101	D₄×C₁₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	2	no	—
110	C₈×C₁₃	C₂²×C₈	1	4	yes	—
111	#1055	—	2	4	no	—
112	#1055	C₄×D₄	1	2	yes	—
113	#1055	C₂²×D₄	1	1	yes	characteristic
114	#1055	C₂×#41	1	1	yes	characteristic
115	#575	#261	1	1	yes	characteristic
116	C₂×#131	—	13	2	no	—
117	#504	—	13	8	no	—
118	#504	—	26	4	no	—
119	C₂×M₃₂	—	13	1	no	—
120	C₂×M₃₂	—	13	2	no	—
121	C₂×M₃₂	—	26	4	no	—
122	C₂²×C₁₆	—	13	1	no	—
123	#444	—	13	4	no	—
124	#443	—	13	4	no	—
125	#430	—	13	1	no	—
126	#421	—	13	2	no	—
127	#405	—	13	1	no	—
128	#364	—	13	4	no	—
129	#350	—	13	4	no	—
130	#350	—	13	4	no	—
131	#346	—	13	2	no	—
132	C₄×C₁₆	—	13	2	no	—
133	C₂²×C₁₃	—	2	4	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	8	2	no	—
136	C₂²×C₁₃	C₈×D₄	1	2	yes	—
137	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
138	C₄×C₁₃	—	2	4	no	—
139	C₄×C₁₃	C₈×D₄	1	2	yes	—
140	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
142	Dic₁₃	—	4	2	no	—
143	C₂×#41	C₂×#261	1	2	yes	—
144	#137	#1057	1	4	yes	—
145	AGL(1,ℤ/8)	#1057	1	4	yes	—
146	#135	C₂×#261	1	4	yes	—
147	#131	—	13	4	no	—
148	#131	—	13	4	no	—
149	#131	—	26	2	no	—
150	#131	—	26	2	no	—
151	C₂×M₁₆	—	13	2	no	—
152	C₂×M₁₆	—	26	2	no	—
153	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	26	2	no	—
156	#117	—	13	1	no	—
157	M₃₂	—	13	4	no	—
158	M₃₂	—	13	4	no	—
159	M₃₂	—	26	8	no	—
160	M₃₂	—	26	2	no	—
161	C₂×C₁₆	—	13	2	no	—
162	C₂×C₁₆	—	13	1	no	—
163	C₂×C₁₆	—	13	1	no	—
164	C₂×C₁₆	—	26	8	no	—
165	C₂×C₁₆	—	26	2	no	—
166	C₂×C₁₆	—	26	2	no	—
167	#108	—	13	4	no	—
168	#104	—	13	8	no	—
169	#97	—	13	2	no	—
170	C₄×C₈	—	13	2	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	2	no	—
173	C₂×C₁₃	—	4	4	no	—
174	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	C₂×#1057	1	4	yes	—
178	#41	C₂²×#261	1	4	yes	—
179	C₂×Q₈	C₂×#1057	1	2	yes	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	C₂×#1057	1	2	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
184	Q₁₆	C₂×#1057	1	4	yes	—
185	SD₁₆	C₂×#1057	1	8	yes	—
186	D₈	C₂×#1057	1	4	yes	—
187	M₁₆	—	13	4	no	—
188	M₁₆	—	26	4	no	—
189	M₁₆	—	26	4	no	—
190	M₁₆	—	26	2	no	—
191	M₁₆	C₂×#1057	1	4	yes	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	26	4	no	—
197	C₂×C₈	—	26	4	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	C₂²×#261	1	2	yes	—
200	C₄⋊C₄	—	26	1	no	—
201	C₂²⋊C₄	—	26	1	no	—
202	C₄²	—	13	2	no	—
203	C₁₆	—	13	4	no	—
204	C₁₆	—	26	1	no	—
205	C₁₆	—	26	1	no	—
206	C₁₆	—	26	4	no	—
207	C₁₃	#33442	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	#88238	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×#1057	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×#1057	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	#88238	1	1	yes	characteristic
221	C₂×C₄	D₄×#261	1	2	yes	—
222	C₂×C₄	C₂²×#1057	1	2	yes	—
223	C₂×C₄	C₂³×#261	1	1	yes	characteristic
224	C₈	—	13	1	no	—
225	C₈	—	13	1	no	—
226	C₈	—	13	2	no	—
227	C₈	—	26	4	no	—
228	C₈	C₂²×#1057	1	4	yes	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	D₄×#1057	1	2	yes	—
233	C₂²	C₂×#88238	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	52	2	no	—
236	C₄	D₄×#1057	1	2	yes	—
237	C₄	C₂×D₄×#261	1	1	yes	characteristic
238	C₄	C₂³×#1057	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	C₂	C₂×D₄×#1057	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting, prime radical	C₁₃×#575	C₄
socle	C₂×C₁₃	C₂×C₈×D₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#575	#261
hypocenter	C₁₃	#33442
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

Complements

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

N	H ≅ G/N	type
1. #13131803	C₂	semidirect
2. #13131226	C₂	semidirect
3. #13131223	C₂	semidirect
4. #13131177	C₂	semidirect
5. C₂×#11718304	C₂	semidirect
6. #13131047	C₂	semidirect
11. #11719162	C₂²	semidirect
21. #11718304	C₂²	semidirect
22. #11718304	C₂²	semidirect
23. #11718261	C₂²	semidirect
24. #11718258	C₂²	semidirect
25. #11718257	C₂²	semidirect
207. C₁₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#13131803	2
#13131226	2
#13131223	2
#13131177	2
C₂×#11718304	2
#13131047	2
#13123739	2
#33442	13