Subgroups — #16015956

Order 3328.

674 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	C₂²×#88038	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#427	C₂	1	2	yes	maximal
3	#13116623	C₂	1	4	yes	maximal, complemented
4	#13116621	C₂	1	4	yes	maximal, complemented
5	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116618	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	D₁₃×#117	—	2	2	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#427	C₂²	1	2	yes	—
12	#11718767	C₂²	1	2	yes	—
13	C₂²×#6090	C₂²	1	1	yes	characteristic
14	#11718659	—	2	2	no	—
15	C₂×#88038	C₄	1	8	yes	complemented
16	C₂×#88038	C₂²	1	2	yes	—
17	C₂×#88038	C₂²	1	2	yes	—
18	#11718541	C₂²	1	4	yes	—
19	D₁₃×#107	C₂²	1	4	yes	—
20	#11718521	—	2	2	no	—
21	D₁₃×#106	—	2	2	no	—
22	C₁₃×#367	C₂²	1	2	yes	complemented
23	C₁₃×#366	C₂²	1	2	yes	complemented
24	#11718097	C₂²	1	2	yes	complemented
25	#11718096	C₂²	1	2	yes	complemented
26	#11718082	C₂²	1	4	yes	complemented
27	#11718081	C₂²	1	4	yes	complemented
28	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#117	—	2	2	no	—
31	#88180	—	2	2	no	—
32	C₂²×#1055	D₄	1	1	yes	characteristic
33	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₃	D₄	1	2	yes	—
35	C₂×C₈×D₁₃	Q₈	1	4	yes	—
36	C₂×C₈×D₁₃	C₂³	1	2	yes	—
37	#88146	—	2	4	no	—
38	C₄⋊C₄×D₁₃	—	2	4	no	—
39	C₂²⋊C₄×D₁₃	—	4	2	no	—
40	#88131	—	4	2	no	—
41	#88126	—	4	2	no	—
42	C₄²×D₁₃	—	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	4	yes	—
47	C₁₃×#106	—	2	2	no	—
48	#88058	C₂³	1	4	yes	—
49	#88057	—	2	2	no	—
50	C₂×#6090	C₂×C₄	1	4	yes	complemented
51	C₂×#6090	C₂³	1	2	yes	—
52	#88048	—	2	4	no	—
53	#88047	—	2	4	no	—
54	#88038	C₂×C₄	1	8	yes	complemented
55	#88038	C₂×C₄	1	8	yes	complemented
56	#33409	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
60	C₂×C₄×D₁₃	—	4	4	no	—
61	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×D₁₃	D₈	1	2	yes	—
63	C₂×C₄×D₁₃	Q₁₆	1	4	yes	—
64	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₁₃	—	4	2	no	—
72	C₂²⋊Dic₁₃	—	4	2	no	—
73	#6103	—	4	4	no	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	#6101	—	4	4	no	—
76	C₄×Dic₁₃	—	4	2	no	—
77	C₄×Dic₁₃	—	4	4	no	—
78	C₂×#1055	C₄⋊C₄	1	4	yes	—
79	C₂×#1055	C₂×D₄	1	2	yes	—
80	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
81	C₈×D₁₃	C₂²×C₄	1	4	yes	—
82	C₈×D₁₃	C₂×Q₈	1	4	yes	—
83	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
84	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
85	#6090	C₂²×C₄	1	4	yes	—
86	C₂²×M₃₂	—	13	1	no	—
87	C₂×#427	—	13	2	no	—
88	#2159	—	13	4	no	—
89	#2159	—	13	4	no	—
90	#2158	—	13	1	no	—
91	#2158	—	13	2	no	—
92	#2158	D₁₃	1	1	yes	characteristic, complemented
93	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₁₃	—	4	2	no	—
95	C₂²×D₁₃	#107	1	4	yes	—
96	C₂²×D₁₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₁₃	—	4	4	no	—
103	C₂×Dic₁₃	#107	1	4	yes	—
104	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
105	C₄×D₁₃	—	8	8	no	—
106	C₄×D₁₃	#107	1	8	yes	—
107	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
108	C₄×D₁₃	C₂×Q₁₆	1	4	yes	—
109	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
112	#1055	C₂×C₄⋊C₄	1	4	yes	—
113	C₂³×C₈	—	13	1	no	—
114	C₂×#117	—	26	2	no	—
115	C₂×M₃₂	—	13	8	no	—
116	C₂×M₃₂	—	13	2	no	—
117	C₂×M₃₂	—	13	2	no	—
118	C₂²×C₁₆	—	13	1	no	—
119	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
120	#427	—	13	2	no	—
121	#427	—	13	4	no	—
122	#427	—	26	2	no	—
123	#427	—	26	2	no	—
124	#427	C₂×D₁₃	1	2	yes	—
125	C₂×#107	—	13	4	no	—
126	C₂×#106	—	26	2	no	—
127	#367	—	13	2	no	—
128	#367	—	13	4	no	—
129	#367	C₂×D₁₃	1	2	yes	complemented
130	#366	—	13	2	no	—
131	#366	—	13	4	no	—
132	#366	C₂×D₁₃	1	2	yes	complemented
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	C₂×#107	1	2	yes	—
135	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₁₃	—	2	4	no	—
137	C₂×D₁₃	—	4	4	no	—
138	C₂×D₁₃	C₂×#107	1	4	yes	—
139	C₄×C₁₃	—	8	4	no	—
140	C₄×C₁₃	C₂×#107	1	2	yes	—
141	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
143	Dic₁₃	—	8	4	no	—
144	Dic₁₃	C₂×#107	1	4	yes	—
145	C₂³×C₄	—	13	1	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	4	no	—
148	C₂²×C₈	—	13	2	no	—
149	C₂²×C₈	—	13	4	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
152	#117	—	2	2	no	—
153	#117	—	26	4	no	—
154	#117	—	26	2	no	—
155	#117	—	52	2	no	—
156	#117	—	52	2	no	—
157	C₂×C₄⋊C₄	—	26	4	no	—
158	C₂×C₂²⋊C₄	—	52	2	no	—
159	C₂×C₄²	—	52	2	no	—
160	M₃₂	—	13	8	no	—
161	M₃₂	—	13	8	no	—
162	C₂×C₁₆	—	13	4	no	—
163	C₂×C₁₆	—	13	2	no	—
164	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
165	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
167	#107	—	13	4	no	—
168	#107	—	26	4	no	—
169	#107	C₂²×D₁₃	1	4	yes	—
170	#106	—	2	2	no	—
171	#106	—	26	4	no	—
172	#106	—	26	2	no	—
173	C₂×C₁₃	—	2	2	no	—
174	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
175	C₂×C₁₃	#3414	1	2	yes	—
176	D₁₃	—	2	4	no	—
177	C₂⁴	—	13	1	no	—
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	4	no	—
180	C₂²×C₄	—	13	2	no	—
181	C₂²×C₄	—	13	4	no	—
182	C₂²×C₄	—	13	2	no	—
183	C₂²×C₄	—	52	4	no	—
184	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
185	C₂×C₈	—	13	4	no	—
186	C₂×C₈	—	13	2	no	—
187	C₂×C₈	—	13	8	no	—
188	C₂×C₈	—	13	4	no	—
189	C₂×C₈	—	13	4	no	—
190	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
191	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
192	C₂×C₈	Q₈×D₁₃	1	2	yes	—
193	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	26	4	no	—
196	C₄⋊C₄	—	52	4	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	52	4	no	—
199	C₂²⋊C₄	—	52	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	52	4	no	—
202	C₄²	—	52	2	no	—
203	C₁₆	—	13	4	no	—
204	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
205	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
206	C₁₃	#33409	1	1	yes	characteristic, complemented, Sylow
207	C₂³	—	13	2	no	—
208	C₂³	—	13	4	no	—
209	C₂³	—	52	2	no	—
210	C₂³	D₈×D₁₃	1	1	yes	characteristic
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	13	2	no	—
213	C₂×C₄	—	13	4	no	—
214	C₂×C₄	—	13	4	no	—
215	C₂×C₄	—	13	4	no	—
216	C₂×C₄	—	13	8	no	—
217	C₂×C₄	—	52	4	no	—
218	C₂×C₄	—	104	8	no	—
219	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
220	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
221	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
223	C₈	—	13	4	no	—
224	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
225	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
226	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	13	4	no	—
229	C₂²	—	26	4	no	—
230	C₂²	—	52	4	no	—
231	C₂²	D₁₃×#107	1	2	yes	—
232	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	13	4	no	—
235	C₄	—	104	4	no	—
236	C₄	D₁₃×#107	1	2	yes	—
237	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
238	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	26	4	no	—
241	C₂	C₂×D₁₃×#107	1	1	yes	characteristic
242	C₂	#13127969	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×D₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33409
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #13116623	C₂	semidirect
4. #13116621	C₂	semidirect
5. #13116620	C₂	semidirect
6. #13116618	C₂	semidirect
7. C₁₃×#2158	C₂	semidirect
15. C₂×#88038	C₄	semidirect
22. C₁₃×#367	C₂²	semidirect
23. C₁₃×#366	C₂²	semidirect
24. #11718097	C₂²	semidirect
25. #11718096	C₂²	semidirect
26. #11718082	C₂²	semidirect
27. #11718081	C₂²	semidirect
43. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
50. C₂×#6090	C₂×C₄	semidirect
54. #88038	C₂×C₄	semidirect
55. #88038	C₂×C₄	semidirect
92. #2158	D₁₃	semidirect
129. #367	C₂×D₁₃	semidirect
132. #366	C₂×D₁₃	semidirect
164. C₂×C₁₆	C₄×D₁₃	semidirect
206. C₁₃	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88038	2
D₁₃×#427	2
#13116623	2
#13116621	2
#13116620	2
#13116618	2
C₁₃×#2158	2
#33409	13