Subgroups — #16015960

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718562	C₂	1	1	yes	maximal, characteristic
2	#13123433	C₂	1	2	yes	maximal
3	#13116625	C₂	1	2	yes	maximal, complemented
4	#13116623	C₂	1	4	yes	maximal, complemented
5	#13116617	C₂	1	2	yes	maximal, complemented
6	C₂×#11718096	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×#366	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88155	C₂²	1	1	yes	characteristic
10	#11719228	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#427	C₂²	1	2	yes	—
13	C₂×#88058	C₂²	1	2	yes	—
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	#11718656	—	2	2	no	—
16	#11718562	C₄	1	8	yes	complemented
17	C₂×#88038	C₂²	1	2	yes	—
18	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
19	#11718541	C₂²	1	4	yes	—
20	D₁₃×#107	C₂²	1	4	yes	—
21	#11718522	—	2	4	no	—
22	C₁₃×#367	C₂²	1	2	yes	—
23	C₁₃×#366	C₂²	1	1	yes	characteristic
24	C₁₃×#366	C₂²	1	1	yes	characteristic
25	#11718096	C₂²	1	2	yes	—
26	#11718096	C₂²	1	2	yes	—
27	#11718082	C₂²	1	4	yes	—
28	#11718081	C₂²	1	4	yes	—
29	C₂×#6127	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×#117	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×#6099	D₄	1	1	yes	characteristic
34	#88155	C₂×C₄	1	4	yes	—
35	#88155	Q₈	1	4	yes	—
36	C₂×#6094	D₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂³	1	2	yes	—
38	#88146	—	2	4	no	—
39	C₄⋊C₄×D₁₃	—	2	4	no	—
40	#88135	—	4	4	no	—
41	C₄×D₅₂	—	4	2	no	—
42	C₄×Dic₂₆	—	4	2	no	—
43	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₁₃×#107	C₂³	1	2	yes	—
47	C₁₃×#107	C₂³	1	2	yes	—
48	C₁₃×#106	—	2	2	no	—
49	#88058	—	2	2	no	—
50	#88058	C₂³	1	4	yes	—
51	#88052	C₂×C₄	1	4	yes	complemented
52	C₂×#6090	C₂³	1	2	yes	—
53	#88048	—	2	4	no	—
54	#88047	—	2	4	no	—
55	#88038	C₂×C₄	1	8	yes	complemented
56	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
57	#33410	—	13	1	no	maximal, Sylow
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₃	D₈	1	2	yes	—
60	C₂²×Dic₁₃	—	4	2	no	—
61	#6127	C₄⋊C₄	1	4	yes	—
62	#6127	Q₁₆	1	4	yes	—
63	C₂×D₅₂	D₈	1	1	yes	characteristic
64	C₂×C₄×D₁₃	—	4	4	no	—
65	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
66	C₂×Dic₂₆	D₈	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	2	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₄²×C₁₃	—	4	2	no	—
75	#6103	—	4	4	no	—
76	C₄⋊Dic₁₃	—	2	4	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	#6101	—	4	4	no	—
79	C₄×Dic₁₃	—	4	4	no	—
80	#6099	C₄⋊C₄	1	4	yes	—
81	C₂×#1055	C₂×D₄	1	2	yes	—
82	#6094	C₄⋊C₄	1	8	yes	—
83	C₈×D₁₃	C₂²×C₄	1	4	yes	—
84	C₈×D₁₃	C₂×Q₈	1	4	yes	—
85	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
86	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
87	#6090	C₂²×C₄	1	4	yes	—
88	C₂×#504	—	13	1	no	—
89	#2913	—	13	2	no	—
90	#2159	—	13	2	no	—
91	#2159	—	13	4	no	—
92	#2158	—	13	2	no	—
93	#2158	D₁₃	1	1	yes	characteristic, complemented
94	C₂×#366	—	13	1	no	—
95	C₂×#366	—	13	2	no	—
96	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂²×D₁₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₁₃	#107	1	8	yes	—
105	C₂×Dic₁₃	—	4	4	no	—
106	C₂×Dic₁₃	—	16	2	no	—
107	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
108	D₅₂	#107	1	4	yes	—
109	C₄×D₁₃	—	16	4	no	—
110	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₄×D₁₃	C₂×Q₁₆	1	4	yes	—
112	Dic₂₆	#107	1	4	yes	—
113	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	#1055	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×#131	—	13	1	no	—
118	#520	—	26	2	no	—
119	#504	—	13	8	no	—
120	C₂×M₃₂	—	13	1	no	—
121	C₂×M₃₂	—	13	2	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
124	#428	—	26	4	no	—
125	#428	—	26	2	no	—
126	#427	—	13	4	no	—
127	#427	C₂×D₁₃	1	2	yes	—
128	C₂×#107	—	13	2	no	—
129	C₂×#107	—	13	4	no	—
130	#367	—	13	4	no	—
131	#367	C₂×D₁₃	1	2	yes	—
132	#366	—	13	2	no	—
133	#366	—	13	2	no	—
134	#366	—	13	4	no	—
135	#366	C₂×D₁₃	1	1	yes	characteristic
136	#366	C₂×D₁₃	1	1	yes	characteristic
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	C₂×#107	1	2	yes	—
140	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₁₃	—	2	2	no	—
142	C₂×D₁₃	—	2	2	no	—
143	C₂×D₁₃	C₂×#107	1	4	yes	—
144	C₄×C₁₃	—	8	2	no	—
145	C₄×C₁₃	—	8	2	no	—
146	C₄×C₁₃	C₂×#107	1	2	yes	—
147	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
149	Dic₁₃	—	8	4	no	—
150	Dic₁₃	C₂×#107	1	4	yes	—
151	C₂×#41	—	13	1	no	—
152	#131	—	13	4	no	—
153	#131	—	13	4	no	—
154	C₂×M₁₆	—	13	1	no	—
155	C₂×M₁₆	—	13	2	no	—
156	C₂²×C₈	—	13	2	no	—
157	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
158	C₄×Q₈	—	52	2	no	—
159	C₄×D₄	—	52	2	no	—
160	C₄×D₄	—	52	4	no	—
161	#117	—	2	2	no	—
162	#117	—	26	4	no	—
163	C₂×C₄⋊C₄	—	26	4	no	—
164	C₂×C₄⋊C₄	—	26	2	no	—
165	M₃₂	—	13	4	no	—
166	M₃₂	—	13	8	no	—
167	C₂×C₁₆	—	13	2	no	—
168	C₂×C₁₆	—	13	8	no	—
169	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
170	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
172	#107	—	13	4	no	—
173	#107	—	26	4	no	—
174	#107	—	26	2	no	—
175	#107	C₂²×D₁₃	1	2	yes	—
176	#107	C₂²×D₁₃	1	2	yes	—
177	#106	—	2	2	no	—
178	#106	—	26	4	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
181	C₂×C₁₃	#3415	1	1	yes	characteristic
182	C₂×C₁₃	#3417	1	1	yes	characteristic
183	D₁₃	—	2	4	no	—
184	#41	—	13	4	no	—
185	#41	—	13	4	no	—
186	C₂×Q₈	—	13	1	no	—
187	C₂×D₄	—	13	2	no	—
188	C₂×D₄	—	13	1	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	52	4	no	—
191	C₂²×C₄	—	52	2	no	—
192	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
193	M₁₆	—	13	4	no	—
194	M₁₆	—	13	8	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	4	no	—
197	C₂×C₈	—	13	4	no	—
198	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
199	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
200	C₂×C₈	Q₈×D₁₃	1	2	yes	—
201	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	26	4	no	—
205	C₄⋊C₄	—	52	4	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	52	4	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	52	4	no	—
211	C₁₆	—	13	4	no	—
212	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
213	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
214	C₁₃	#33410	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	13	2	no	—
216	C₂³	D₈×D₁₃	1	1	yes	characteristic
217	Q₈	—	13	4	no	—
218	D₄	—	13	8	no	—
219	D₄	—	13	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	13	2	no	—
223	C₂×C₄	—	13	4	no	—
224	C₂×C₄	—	13	4	no	—
225	C₂×C₄	—	52	4	no	—
226	C₂×C₄	—	208	4	no	—
227	C₂×C₄	—	208	2	no	—
228	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
229	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
230	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
231	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
232	C₈	—	13	4	no	—
233	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
234	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
235	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	13	4	no	—
239	C₂²	—	26	2	no	—
240	C₂²	—	26	2	no	—
241	C₂²	D₁₃×#107	1	2	yes	—
242	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	—	13	4	no	—
246	C₄	—	104	4	no	—
247	C₄	D₁₃×#107	1	2	yes	—
248	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
249	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	26	4	no	—
252	C₂	C₂×D₁₃×#107	1	1	yes	characteristic
253	C₂	#13127974	1	1	yes	characteristic
254	C₂	#13127986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#6127	D₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33410
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #13116625	C₂	semidirect
4. #13116623	C₂	semidirect
5. #13116617	C₂	semidirect
6. C₂×#11718096	C₂	semidirect
7. D₁₃×#366	C₂	semidirect
8. C₁₃×#2158	C₂	semidirect
16. #11718562	C₄	semidirect
43. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
51. #88052	C₂×C₄	semidirect
55. #88038	C₂×C₄	semidirect
56. C₁₆×D₁₃	C₂×C₄	semidirect
93. #2158	D₁₃	semidirect
169. C₂×C₁₆	C₄×D₁₃	semidirect
214. C₁₃	#33410	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718562	2
#13123433	2
#13116625	2
#13116623	2
#13116617	2
C₂×#11718096	2
D₁₃×#366	2
C₁₃×#2158	2
#33410	13