Subgroups — #13123566

Order 1664.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719533	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718780	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88060	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718762	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	2	yes	maximal, complemented
9	#11718390	C₂	1	2	yes	maximal, complemented
10	#11718389	C₂	1	2	yes	maximal, complemented
11	#11718386	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#88202	—	2	4	no	—
17	#88200	C₄	1	1	yes	characteristic
18	#88200	C₄	1	1	yes	characteristic
19	#88192	C₄	1	1	yes	characteristic
20	#88192	C₄	1	1	yes	characteristic
21	#88191	C₄	1	2	yes	—
22	#88189	C₄	1	2	yes	—
23	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
25	#88180	C₂²	1	1	yes	characteristic
26	C₂×#6099	C₂²	1	1	yes	characteristic
27	C₂×#6099	C₂²	1	2	yes	complemented
28	C₂²×#1055	C₂²	1	1	yes	characteristic
29	C₂²×#1055	C₂²	1	2	yes	complemented
30	M₁₆×D₁₃	—	2	4	no	—
31	C₂×#6094	C₂²	1	2	yes	—
32	C₂×C₈×D₁₃	C₂²	1	2	yes	—
33	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	#88060	C₂²	1	1	yes	characteristic
36	#88060	C₂²	1	1	yes	characteristic
37	#88060	C₂²	1	1	yes	characteristic
38	#88060	C₂²	1	1	yes	characteristic
39	#88055	C₂²	1	2	yes	complemented
40	#88054	C₂²	1	2	yes	complemented
41	#88054	C₂²	1	2	yes	complemented
42	C₈×Dic₁₃	C₂²	1	2	yes	complemented
43	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₁₃×#41	—	2	2	no	—
45	C₁₃×#41	—	2	2	no	—
46	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
48	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
53	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	—	2	2	no	—
55	C₂×C₄×D₁₃	—	2	2	no	—
56	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	C₂³	1	2	yes	—
63	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
68	#6103	C₂×C₄	1	2	yes	—
69	#6103	C₂×C₄	1	2	yes	—
70	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	#6101	C₂×C₄	1	2	yes	—
73	#6101	C₂×C₄	1	2	yes	—
74	C₄×Dic₁₃	C₂³	1	2	yes	—
75	#6099	—	2	4	no	—
76	#6099	D₄	1	4	yes	complemented
77	C₂×#1055	—	2	2	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	C₂³	1	1	yes	characteristic
80	C₂×#1055	C₂³	1	1	yes	characteristic
81	C₂×#1055	C₂³	1	2	yes	—
82	#6094	—	2	4	no	—
83	C₈×D₁₃	—	2	4	no	—
84	#2933	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	2	no	—
88	C₂²×D₁₃	—	2	2	no	—
89	C₂²×D₁₃	—	4	1	no	—
90	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
92	Q₈×C₁₃	—	2	2	no	—
93	D₄×C₁₃	—	2	4	no	—
94	D₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	#41	1	2	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₁₃	—	2	4	no	—
103	C₂²⋊D₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
106	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
107	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	—	2	1	no	—
112	C₄×D₁₃	—	2	1	no	—
113	C₄×D₁₃	—	2	1	no	—
114	C₄×D₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	1	no	—
118	#1055	—	2	1	no	—
119	#1055	—	2	1	no	—
120	#1055	C₂×D₄	1	4	yes	—
121	C₂×#131	—	13	1	no	—
122	C₂²×M₁₆	—	13	1	no	—
123	#525	—	13	1	no	—
124	#436	—	13	2	no	—
125	#435	—	13	2	no	—
126	#435	—	13	2	no	—
127	C₈×D₄	—	13	2	no	—
128	#424	—	13	1	no	—
129	#408	D₁₃	1	1	yes	characteristic, complemented
130	C₂×C₂²⋊C₈	—	13	1	no	—
131	#405	—	13	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	C₄×D₄	1	2	yes	—
138	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₁₃	—	2	1	no	—
140	C₂×D₁₃	—	2	1	no	—
141	C₂×D₁₃	—	2	1	no	—
142	C₂×D₁₃	—	2	1	no	—
143	C₂×D₁₃	—	2	1	no	—
144	C₂×D₁₃	—	4	4	no	—
145	C₂×D₁₃	#131	1	2	yes	complemented
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	C₄×D₄	1	2	yes	—
148	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	—	2	1	no	—
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	2	no	—
154	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
155	C₂³×C₄	—	13	1	no	—
156	#131	—	26	4	no	—
157	C₂×M₁₆	—	13	1	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	26	4	no	—
161	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#123	—	13	2	no	—
167	C₂²⋊Q₈	—	13	1	no	—
168	C₂²⋊Q₈	—	13	1	no	—
169	#121	—	13	1	no	—
170	#121	—	13	1	no	—
171	#120	—	13	2	no	—
172	C₄×D₄	—	13	2	no	—
173	C₄×D₄	—	13	2	no	—
174	#117	—	13	1	no	—
175	C₄⋊C₈	—	13	2	no	—
176	C₄⋊C₈	—	13	2	no	—
177	C₂²⋊C₈	—	13	1	no	—
178	C₂²⋊C₈	—	13	1	no	—
179	C₂²⋊C₈	—	13	1	no	—
180	C₂²⋊C₈	—	13	1	no	—
181	C₂²⋊C₈	C₂×D₁₃	1	2	yes	complemented
182	C₂²⋊C₈	C₂×D₁₃	1	2	yes	complemented
183	#97	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
189	C₂×C₁₃	#568	1	1	yes	characteristic
190	D₁₃	—	2	2	no	—
191	D₁₃	—	4	1	no	—
192	C₂⁴	—	13	1	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
196	C₂×D₄	—	13	2	no	—
197	C₂×D₄	—	13	2	no	—
198	C₂×D₄	C₄×D₁₃	1	2	yes	—
199	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	26	2	no	—
206	C₂²×C₄	—	26	2	no	—
207	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	13	4	no	—
211	M₁₆	—	26	4	no	—
212	M₁₆	—	26	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	13	2	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₃	1	2	yes	—
223	C₄⋊C₄	—	13	1	no	—
224	C₄⋊C₄	—	13	1	no	—
225	C₄⋊C₄	—	13	2	no	—
226	C₄⋊C₄	—	13	2	no	—
227	C₂²⋊C₄	—	13	2	no	—
228	C₂²⋊C₄	—	13	2	no	—
229	C₂²⋊C₄	—	13	2	no	—
230	C₂²⋊C₄	—	13	2	no	—
231	C₂²⋊C₄	—	13	2	no	—
232	C₄²	—	13	2	no	—
233	C₁₃	#2933	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	13	1	no	—
235	C₂³	—	13	1	no	—
236	C₂³	—	26	2	no	—
237	C₂³	—	26	2	no	—
238	C₂³	—	52	1	no	—
239	C₂³	C₂×C₄×D₁₃	1	2	yes	—
240	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	26	4	no	—
245	D₄	—	52	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	13	2	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	26	2	no	—
253	C₂×C₄	—	26	2	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	26	1	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	1	no	—
259	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₄	D₄×D₁₃	1	2	yes	—
262	C₂×C₄	#6129	1	2	yes	—
263	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	13	4	no	—
268	C₈	—	26	1	no	—
269	C₈	—	26	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	13	2	no	—
276	C₂²	—	26	1	no	—
277	C₂²	—	26	1	no	—
278	C₂²	—	26	1	no	—
279	C₂²	—	26	1	no	—
280	C₂²	—	26	1	no	—
281	C₂²	—	52	4	no	—
282	C₂²	#88135	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	26	1	no	—
286	C₄	—	26	1	no	—
287	C₄	—	26	2	no	—
288	C₄	—	26	2	no	—
289	C₄	#88135	1	2	yes	—
290	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
291	C₄	C₂×#6129	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	4	2	no	—
294	C₂	—	26	2	no	—
295	C₂	—	52	1	no	—
296	C₂	C₂×#88135	1	1	yes	characteristic
297	C₂	D₁₃×#131	1	1	yes	characteristic
298	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#408	C₂
prime radical	#11719533	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#408	D₁₃
hypocenter	C₁₃	#2933
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×M₁₆×D₁₃	C₂	semidirect
4. C₁₃×#408	C₂	semidirect
5. #11718780	C₂	semidirect
6. C₂×#88060	C₂	semidirect
7. #11718762	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718390	C₂	semidirect
10. #11718389	C₂	semidirect
11. #11718386	C₂	semidirect
27. C₂×#6099	C₂²	semidirect
29. C₂²×#1055	C₂²	semidirect
33. C₁₃×C₂²⋊C₈	C₂²	semidirect
34. C₁₃×C₂²⋊C₈	C₂²	semidirect
39. #88055	C₂²	semidirect
40. #88054	C₂²	semidirect
41. #88054	C₂²	semidirect
42. C₈×Dic₁₃	C₂²	semidirect
76. #6099	D₄	semidirect
129. #408	D₁₃	semidirect
145. C₂×D₁₃	#131	semidirect
181. C₂²⋊C₈	C₂×D₁₃	semidirect
182. C₂²⋊C₈	C₂×D₁₃	semidirect
233. C₁₃	#2933	semidirect

Maximal subgroups

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

subgroup	index
#11719533	2
C₂×#88202	2
C₂×M₁₆×D₁₃	2
C₁₃×#408	2
#11718780	2
C₂×#88060	2
#11718762	2
#11718391	2
#11718390	2
#11718389	2
#11718386	2
#2933	13