Subgroups — #16016168

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	#13127938	C₂	1	2	yes	maximal, complemented
2	#13123740	C₂	1	1	yes	maximal, characteristic
3	#13116679	C₂	1	4	yes	maximal, complemented
4	#13116674	C₂	1	4	yes	maximal, complemented
5	#13116602	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718100	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719453	C₄	1	1	yes	characteristic
9	C₂×#88155	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	#11718786	C₄	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	2	yes	—
15	#11718666	C₄	1	2	yes	—
16	#11718568	—	2	4	no	—
17	D₁₃×M₃₂	—	2	4	no	—
18	#11718562	C₂²	1	8	yes	complemented
19	#11718378	C₄	1	4	yes	—
20	#11718361	C₂²	1	4	yes	—
21	C₁₃×#346	C₂²	1	2	yes	complemented
22	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
23	#11718100	C₂²	1	4	yes	complemented
24	#11718100	C₂²	1	4	yes	complemented
25	#11718055	C₂²	1	4	yes	complemented
26	C₂×#6127	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
30	C₂×#6099	C₂×C₄	1	2	yes	—
31	#88163	C₈	1	4	yes	—
32	#88162	C₈	1	4	yes	—
33	#88158	C₂×C₄	1	4	yes	—
34	#88155	—	2	2	no	—
35	#88155	—	2	2	no	—
36	#88155	C₂×C₄	1	4	yes	—
37	#88155	C₂³	1	4	yes	—
38	C₂×#6094	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	C₁₃×M₃₂	D₄	1	4	yes	—
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₁₃×#97	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂³	1	2	yes	—
47	#88059	C₂×C₄	1	4	yes	—
48	#88052	—	2	2	no	—
49	#88052	C₂³	1	4	yes	—
50	C₂×#6090	—	2	2	no	—
51	#88045	C₂×C₄	1	8	yes	—
52	#88038	—	2	8	no	—
53	C₁₆×D₁₃	—	2	8	no	—
54	#33442	—	13	1	no	maximal, Sylow
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₁₃	—	2	2	no	—
57	#6127	—	2	2	no	—
58	#6127	—	2	2	no	—
59	#6127	C₂×C₈	1	4	yes	—
60	#6127	C₂²×C₄	1	4	yes	—
61	C₂×D₅₂	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₁₃	—	2	2	no	—
63	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
64	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	#41	1	2	yes	—
69	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	#6099	—	2	2	no	—
74	#6099	C₂²×C₄	1	4	yes	—
75	C₂×#1055	—	2	2	no	—
76	Dic₅₂	C₂×C₈	1	4	yes	—
77	D₁₀₄	C₂×C₈	1	4	yes	—
78	#6095	C₂×C₈	1	8	yes	—
79	#6094	—	2	4	no	—
80	#6094	—	2	4	no	—
81	C₈×D₁₃	—	2	4	no	—
82	C₈×D₁₃	—	2	4	no	—
83	C₁₃×C₁₆	—	2	1	no	—
84	C₁₃×C₁₆	—	2	1	no	—
85	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
86	#6090	—	2	4	no	—
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	D₁₃	1	1	yes	characteristic, complemented
94	C₂³×C₁₃	#131	1	1	yes	characteristic
95	C₂²×D₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₂²⋊D₁₃	—	2	4	no	—
102	C₂²⋊D₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	D₅₂	—	2	2	no	—
105	D₅₂	C₂²×C₈	1	4	yes	—
106	C₄×D₁₃	—	2	4	no	—
107	C₄×D₁₃	—	2	4	no	—
108	Dic₂₆	—	2	2	no	—
109	Dic₂₆	C₂²×C₈	1	4	yes	—
110	C₈×C₁₃	C₄×D₄	1	2	yes	—
111	C₈×C₁₃	C₂²×C₈	1	4	yes	—
112	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
114	#1055	—	2	4	no	—
115	#575	—	13	1	no	—
116	C₂×#131	—	13	2	no	—
117	#504	—	13	8	no	—
118	#504	—	26	4	no	—
119	C₂×M₃₂	—	13	2	no	—
120	C₂×M₃₂	—	26	4	no	—
121	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
123	#444	—	13	4	no	—
124	#443	—	13	4	no	—
125	#430	—	13	1	no	—
126	#421	—	13	2	no	—
127	#405	C₂×D₁₃	1	1	yes	characteristic
128	#364	—	13	4	no	—
129	#350	—	13	4	no	—
130	#350	—	13	4	no	—
131	#346	C₂×D₁₃	1	2	yes	complemented
132	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	C₈×D₄	1	2	yes	—
135	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
136	C₂×D₁₃	—	2	4	no	—
137	C₂×D₁₃	—	8	2	no	—
138	C₄×C₁₃	—	4	2	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	—	2	4	no	—
143	C₂×#41	—	13	2	no	—
144	#137	—	13	4	no	—
145	AGL(1,ℤ/8)	—	13	4	no	—
146	#135	—	13	4	no	—
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	C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₈	—	26	2	no	—
155	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	#117	C₄×D₁₃	1	1	yes	characteristic
157	M₃₂	—	13	4	no	—
158	M₃₂	—	26	2	no	—
159	M₃₂	—	26	8	no	—
160	M₃₂	D₅₂	1	4	yes	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	26	2	no	—
163	C₂×C₁₆	—	26	8	no	—
164	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
167	#108	—	13	4	no	—
168	#104	—	13	8	no	—
169	#97	C₄×D₁₃	1	2	yes	—
170	C₄×C₈	C₂²×D₁₃	1	2	yes	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	2	no	—
173	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
174	D₁₃	—	4	4	no	—
175	#41	—	13	4	no	—
176	#41	—	13	4	no	—
177	#41	—	26	2	no	—
178	#41	—	26	2	no	—
179	C₂×Q₈	—	13	2	no	—
180	C₂×D₄	—	13	2	no	—
181	C₂×D₄	—	26	2	no	—
182	C₂²×C₄	—	26	2	no	—
183	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
184	Q₁₆	—	13	4	no	—
185	SD₁₆	—	13	8	no	—
186	D₈	—	13	4	no	—
187	M₁₆	—	13	4	no	—
188	M₁₆	—	26	2	no	—
189	M₁₆	—	26	4	no	—
190	M₁₆	—	26	4	no	—
191	M₁₆	C₈×D₁₃	1	4	yes	—
192	C₂×C₈	—	26	2	no	—
193	C₂×C₈	—	26	4	no	—
194	C₂×C₈	—	26	4	no	—
195	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
197	C₂×C₈	C₂×D₅₂	1	2	yes	—
198	C₂×C₈	#6127	1	2	yes	—
199	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×C₄×D₁₃	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	—	26	4	no	—
206	C₁₆	C₂×D₅₂	1	4	yes	—
207	C₁₃	#33442	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	26	2	no	—
209	C₂³	#88155	1	1	yes	characteristic
210	Q₈	—	13	4	no	—
211	Q₈	—	26	2	no	—
212	D₄	—	13	4	no	—
213	D₄	—	26	4	no	—
214	D₄	—	26	4	no	—
215	D₄	—	26	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	—	26	4	no	—
219	C₂×C₄	—	26	4	no	—
220	C₂×C₄	C₄×D₅₂	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
222	C₂×C₄	#88155	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
224	C₈	—	26	4	no	—
225	C₈	C₄×D₅₂	1	2	yes	—
226	C₈	C₂×C₈×D₁₃	1	4	yes	—
227	C₈	C₂²×D₅₂	1	1	yes	characteristic
228	C₈	C₂×#6127	1	1	yes	characteristic
229	C₂²	—	4	1	no	—
230	C₂²	—	26	4	no	—
231	C₂²	—	104	2	no	—
232	C₂²	C₈×D₅₂	1	2	yes	—
233	C₂²	C₂×#88155	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	26	4	no	—
236	C₄	C₈×D₅₂	1	2	yes	—
237	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
238	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	52	4	no	—
242	C₂	C₂×C₈×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₅₂
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	#11719453	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33442
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #13127938	C₂	semidirect
3. #13116679	C₂	semidirect
4. #13116674	C₂	semidirect
5. #13116602	C₂	semidirect
6. C₂×#11718100	C₂	semidirect
7. C₁₃×#2108	C₂	semidirect
18. #11718562	C₂²	semidirect
21. C₁₃×#346	C₂²	semidirect
22. C₄×C₁₃×C₁₆	C₂²	semidirect
23. #11718100	C₂²	semidirect
24. #11718100	C₂²	semidirect
25. #11718055	C₂²	semidirect
93. #2108	D₁₃	semidirect
131. #346	C₂×D₁₃	semidirect
132. C₄×C₁₆	C₂×D₁₃	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
#13127938	2
#13123740	2
#13116679	2
#13116674	2
#13116602	2
C₂×#11718100	2
C₁₃×#2108	2
#33442	13