Subgroups — #13124647

Order 1664.

416 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#417	C₂	1	1	yes	maximal, characteristic
4	#11718893	C₂	1	1	yes	maximal, characteristic
5	#11718782	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718779	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718625	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718455	C₂	1	2	yes	maximal, complemented
9	#11718452	C₂	1	2	yes	maximal, complemented
10	#11718422	C₂	1	2	yes	maximal, complemented
11	#11718420	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₆	—	2	1	no	—
14	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	D₄×Dic₁₃	—	2	1	no	—
19	D₄×Dic₁₃	—	2	1	no	—
20	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	#88180	—	2	1	no	—
23	#88180	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×#6101	—	2	1	no	—
26	C₂×C₄×Dic₁₃	—	2	1	no	—
27	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×#6099	C₂²	1	1	yes	characteristic
29	#88142	—	2	1	no	—
30	#88142	—	2	1	no	—
31	#88142	C₂²	1	2	yes	—
32	#88142	C₂²	1	2	yes	—
33	C₁₃×#103	C₂²	1	2	yes	—
34	C₁₃×#102	C₂²	1	2	yes	—
35	#88075	C₂²	1	2	yes	—
36	#88072	C₂²	1	2	yes	—
37	#88058	C₂²	1	2	yes	—
38	#88057	C₂²	1	2	yes	—
39	#88054	C₂²	1	2	yes	—
40	#88054	C₂²	1	2	yes	—
41	#88048	C₂²	1	2	yes	—
42	#88047	C₂²	1	2	yes	—
43	C₁₃×#41	—	2	2	no	—
44	C₁₃×#41	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	2	1	no	—
52	C₂²×Dic₁₃	—	2	1	no	—
53	C₂²×Dic₁₃	—	2	1	no	—
54	C₂²×Dic₁₃	D₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	D₄	1	1	yes	characteristic
56	C₂×Dic₂₆	—	2	2	no	—
57	C₂×Dic₂₆	—	2	2	no	—
58	C₂×Dic₂₆	—	4	4	no	—
59	C₁₃×M₁₆	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂³	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₂²⋊Dic₁₃	—	2	1	no	—
64	C₂²⋊Dic₁₃	—	2	1	no	—
65	C₂²⋊Dic₁₃	—	2	1	no	—
66	C₄⋊Dic₁₃	—	2	1	no	—
67	C₄⋊Dic₁₃	—	2	1	no	—
68	C₄⋊Dic₁₃	—	2	1	no	—
69	C₄⋊Dic₁₃	C₂³	1	2	yes	—
70	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
72	#6101	—	2	2	no	—
73	#6101	—	2	2	no	—
74	C₄×Dic₁₃	—	2	1	no	—
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	—	2	1	no	—
77	C₄×Dic₁₃	D₄	1	1	yes	characteristic
78	C₄×Dic₁₃	D₄	1	1	yes	characteristic
79	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
81	#6099	—	2	2	no	—
82	C₂×#1055	C₂³	1	2	yes	—
83	#3079	—	13	1	no	maximal, Sylow
84	C₂³×C₁₃	—	2	1	no	—
85	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₁₃	—	2	1	no	—
87	Q₈×C₁₃	C₂×Q₈	1	2	yes	complemented
88	D₄×C₁₃	—	2	4	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	C₂×Q₈	1	2	yes	complemented
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₁₃	#41	1	2	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	4	2	no	—
105	C₂×Dic₁₃	—	4	2	no	—
106	C₂×Dic₁₃	—	4	2	no	—
107	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	Dic₂₆	—	4	4	no	—
112	Dic₂₆	—	4	4	no	—
113	C₈×C₁₃	—	2	2	no	—
114	#1055	—	2	2	no	—
115	C₂×C₄⋊Q₈	—	13	1	no	—
116	C₄×#41	—	13	1	no	—
117	#477	—	13	2	no	—
118	#476	—	13	2	no	—
119	#475	—	13	2	no	—
120	#474	—	13	2	no	—
121	#428	—	13	1	no	—
122	#428	—	13	1	no	—
123	#423	—	13	1	no	—
124	#417	—	13	1	no	—
125	#417	D₁₃	1	1	yes	characteristic
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
136	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	Dic₁₃	—	2	1	no	—
139	Dic₁₃	—	2	1	no	—
140	Dic₁₃	—	4	2	no	—
141	Dic₁₃	—	4	1	no	—
142	Dic₁₃	—	4	1	no	—
143	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	Dic₁₃	#137	1	2	yes	—
145	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
146	C₂²×Q₈	—	26	1	no	—
147	C₂×M₁₆	—	13	1	no	—
148	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
149	C₄⋊Q₈	—	13	2	no	—
150	C₄⋊Q₈	—	13	2	no	—
151	C₄⋊Q₈	—	26	1	no	—
152	C₄⋊Q₈	—	26	1	no	—
153	C₄×Q₈	—	13	1	no	—
154	C₄×Q₈	—	13	1	no	—
155	C₄×D₄	—	13	1	no	—
156	C₄×D₄	—	13	1	no	—
157	C₄×D₄	—	26	1	no	—
158	C₄×D₄	—	26	1	no	—
159	#117	—	13	1	no	—
160	#117	—	26	1	no	—
161	C₂×C₄⋊C₄	—	13	1	no	—
162	C₂×C₄⋊C₄	—	26	1	no	—
163	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	26	1	no	—
166	#107	—	13	2	no	—
167	#107	—	13	2	no	—
168	#106	—	13	2	no	—
169	#106	—	13	2	no	—
170	C₄⋊C₈	—	13	2	no	—
171	C₄⋊C₈	—	13	2	no	—
172	#103	—	13	2	no	—
173	#103	C₂×D₁₃	1	2	yes	—
174	#102	—	13	2	no	—
175	#102	C₂×D₁₃	1	2	yes	—
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
181	#41	—	2	2	no	—
182	#41	Dic₂₆	1	4	yes	complemented
183	C₂×Q₈	—	26	2	no	—
184	C₂×Q₈	—	26	2	no	—
185	C₂×Q₈	—	52	4	no	—
186	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	26	1	no	—
194	C₂²×C₄	—	26	1	no	—
195	C₂²×C₄	—	26	1	no	—
196	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
197	M₁₆	—	2	2	no	—
198	M₁₆	—	26	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	C₂²×D₁₃	1	2	yes	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	13	2	no	—
203	C₄⋊C₄	—	13	1	no	—
204	C₄⋊C₄	—	13	1	no	—
205	C₄⋊C₄	—	26	2	no	—
206	C₄⋊C₄	—	26	2	no	—
207	C₄⋊C₄	—	26	1	no	—
208	C₄⋊C₄	—	26	1	no	—
209	C₄⋊C₄	—	26	1	no	—
210	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
211	C₂²⋊C₄	—	26	1	no	—
212	C₂²⋊C₄	—	26	1	no	—
213	C₂²⋊C₄	—	26	1	no	—
214	C₄²	—	13	1	no	—
215	C₄²	—	13	1	no	—
216	C₄²	—	13	1	no	—
217	C₄²	—	13	1	no	—
218	C₄²	—	26	1	no	—
219	C₄²	—	26	1	no	—
220	C₄²	—	26	1	no	—
221	C₁₃	#3079	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	2	1	no	—
223	C₂³	D₄×D₁₃	1	1	yes	characteristic
224	Q₈	—	2	1	no	—
225	Q₈	—	52	4	no	—
226	Q₈	—	52	4	no	—
227	Q₈	C₂×Dic₂₆	1	2	yes	complemented
228	D₄	—	2	4	no	—
229	D₄	—	2	1	no	—
230	D₄	C₂×Dic₂₆	1	2	yes	complemented
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	13	1	no	—
236	C₂×C₄	—	13	1	no	—
237	C₂×C₄	—	13	1	no	—
238	C₂×C₄	—	13	1	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	2	no	—
241	C₂×C₄	—	26	1	no	—
242	C₂×C₄	—	26	1	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	52	2	no	—
245	C₂×C₄	—	52	2	no	—
246	C₂×C₄	—	52	2	no	—
247	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
248	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
249	C₂×C₄	#6129	1	2	yes	—
250	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	26	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	#88132	1	2	yes	—
259	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	13	2	no	—
263	C₄	—	13	2	no	—
264	C₄	—	26	1	no	—
265	C₄	—	26	1	no	—
266	C₄	—	52	1	no	—
267	C₄	—	52	2	no	—
268	C₄	—	52	1	no	—
269	C₄	#88132	1	2	yes	—
270	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
271	C₄	C₂×#6129	1	1	yes	characteristic
272	C₂	—	2	2	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#88132	1	1	yes	characteristic
275	C₂	D₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₂	D₁₃×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#417	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×#41	C₂²
hypercenter	#417	D₁₃
hypocenter	C₁₃	#3079
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. C₂×#88142	C₂	semidirect
5. #11718782	C₂	semidirect
6. #11718779	C₂	semidirect
7. #11718625	C₂	semidirect
8. #11718455	C₂	semidirect
9. #11718452	C₂	semidirect
10. #11718422	C₂	semidirect
11. #11718420	C₂	semidirect
44. C₁₃×#41	Q₈	semidirect
87. Q₈×C₁₃	C₂×Q₈	semidirect
90. D₄×C₁₃	C₂×Q₈	semidirect
182. #41	Dic₂₆	semidirect
221. C₁₃	#3079	semidirect
227. Q₈	C₂×Dic₂₆	semidirect
230. D₄	C₂×Dic₂₆	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₁₃	2
C₂×#88142	2
C₁₃×#417	2
#11718893	2
#11718782	2
#11718779	2
#11718625	2
#11718455	2
#11718452	2
#11718422	2
#11718420	2
#3079	13