Subgroups — #13123749

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88204	C₂	1	1	yes	maximal, characteristic
2	#11719524	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88062	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718666	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718658	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718485	C₂	1	4	yes	maximal, complemented
9	#11718484	C₂	1	4	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂×#6127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#117	C₂²	1	1	yes	characteristic
14	#88204	C₄	1	2	yes	complemented
15	#88204	C₄	1	2	yes	complemented
16	#88204	C₂²	1	2	yes	—
17	#88204	C₂²	1	2	yes	—
18	#88202	—	2	2	no	—
19	#88202	C₂²	1	4	yes	complemented
20	C₂×#6107	C₄	1	1	yes	characteristic, complemented
21	C₂×#6106	C₄	1	1	yes	characteristic, complemented
22	C₂×#6105	C₄	1	1	yes	characteristic, complemented
23	C₂×#6104	C₄	1	1	yes	characteristic, complemented
24	C₂×#6099	—	2	2	no	—
25	C₂×#6099	C₂²	1	1	yes	characteristic
26	C₂×#6099	C₂²	1	1	yes	characteristic
27	C₂²×#1055	C₂²	1	1	yes	characteristic
28	#88161	C₂²	1	4	yes	complemented
29	M₁₆×D₁₃	—	2	2	no	—
30	#88155	—	2	2	no	—
31	C₁₃×#104	C₂²	1	4	yes	complemented
32	#88062	C₂²	1	2	yes	complemented
33	#88062	C₂²	1	2	yes	complemented
34	#88045	C₂²	1	4	yes	complemented
35	#88043	C₂²	1	2	yes	complemented
36	C₄×#1055	C₂²	1	2	yes	complemented
37	C₁₃×#41	—	2	1	no	—
38	C₁₃×#41	—	2	1	no	—
39	C₁₃×#41	C₂×C₄	1	2	yes	—
40	C₁₃×#41	C₂³	1	2	yes	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	—	2	1	no	—
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃	—	2	1	no	—
47	#6127	—	2	1	no	—
48	#6127	—	2	1	no	—
49	#6127	C₂×C₄	1	2	yes	—
50	#6127	C₂³	1	2	yes	—
51	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃	—	2	1	no	—
53	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
54	C₁₃×M₁₆	—	2	1	no	—
55	C₁₃×M₁₆	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₂²⋊C₄	—	2	1	no	—
59	C₄²×C₁₃	C₂³	1	2	yes	—
60	#6107	C₂×C₄	1	2	yes	—
61	#6107	C₂×C₄	1	2	yes	—
62	#6106	C₂×C₄	1	2	yes	—
63	#6106	C₂×C₄	1	2	yes	—
64	#6105	C₂×C₄	1	2	yes	—
65	#6105	C₂×C₄	1	2	yes	—
66	#6104	C₂×C₄	1	2	yes	—
67	#6104	C₂×C₄	1	2	yes	—
68	#6099	—	2	1	no	—
69	#6099	—	2	4	no	—
70	#6099	D₄	1	2	yes	—
71	#6099	D₄	1	2	yes	—
72	#6099	C₂³	1	2	yes	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	—	2	4	no	—
75	C₂×#1055	C₂×C₄	1	1	yes	characteristic
76	C₂×#1055	C₂×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂×C₄	1	1	yes	characteristic
78	C₂×#1055	C₂×C₄	1	1	yes	characteristic
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	#2957	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	#41	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	1	no	—
88	Q₈×C₁₃	—	2	1	no	—
89	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	2	1	no	—
93	D₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₃	—	2	2	no	—
105	C₂²⋊D₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	D₅₂	—	2	1	no	—
108	D₅₂	C₂²×C₄	1	2	yes	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	Dic₂₆	—	2	1	no	—
112	Dic₂₆	C₂²×C₄	1	2	yes	—
113	C₈×C₁₃	—	2	2	no	—
114	#1055	—	2	2	no	—
115	#1055	C₂²×C₄	1	1	yes	characteristic
116	#1055	C₂²×C₄	1	1	yes	characteristic
117	#1055	C₂²×C₄	1	1	yes	characteristic
118	#1055	C₂²×C₄	1	1	yes	characteristic
119	#1055	C₂×D₄	1	4	yes	—
120	C₂×#135	—	13	1	no	—
121	#568	—	13	1	no	—
122	#568	—	13	1	no	—
123	#444	—	13	4	no	—
124	#443	—	13	4	no	—
125	C₂×#108	—	13	1	no	—
126	#421	—	13	1	no	—
127	#421	D₁₃	1	1	yes	characteristic, complemented
128	#405	—	13	1	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	8	1	no	—
132	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	8	1	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	Dic₁₃	—	2	2	no	—
144	C₂×#41	—	13	1	no	—
145	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
146	#135	—	13	2	no	—
147	#135	—	13	2	no	—
148	#135	—	13	2	no	—
149	#135	—	13	2	no	—
150	C₂×Q₁₆	—	13	1	no	—
151	C₂×SD₁₆	—	13	1	no	—
152	C₂×SD₁₆	—	13	1	no	—
153	C₂×D₈	—	13	1	no	—
154	#131	—	13	4	no	—
155	#131	—	13	4	no	—
156	#131	—	26	2	no	—
157	#131	—	26	2	no	—
158	C₂×M₁₆	—	13	1	no	—
159	C₂×M₁₆	—	13	1	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂×M₁₆	—	26	2	no	—
162	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
163	C₂²×C₈	—	13	1	no	—
164	#117	C₂×D₁₃	1	1	yes	characteristic
165	#108	—	13	2	no	—
166	#108	—	13	2	no	—
167	#104	—	13	4	no	—
168	#104	C₂×D₁₃	1	4	yes	complemented
169	#97	—	13	2	no	—
170	C₄×C₈	—	13	2	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	1	no	—
173	C₂×C₁₃	—	2	1	no	—
174	C₂×C₁₃	—	4	2	no	—
175	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
176	D₁₃	—	4	2	no	—
177	#41	—	2	1	no	—
178	#41	—	2	1	no	—
179	#41	—	13	2	no	—
180	#41	—	13	2	no	—
181	#41	—	26	1	no	—
182	#41	—	26	1	no	—
183	#41	C₄×D₁₃	1	2	yes	—
184	#41	C₂²×D₁₃	1	2	yes	—
185	C₂×Q₈	—	13	1	no	—
186	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	13	1	no	—
189	C₂×D₄	—	26	1	no	—
190	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	26	1	no	—
193	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	Q₁₆	—	13	2	no	—
195	Q₁₆	—	13	2	no	—
196	SD₁₆	—	13	2	no	—
197	SD₁₆	—	13	2	no	—
198	SD₁₆	—	13	2	no	—
199	SD₁₆	—	13	2	no	—
200	D₈	—	13	2	no	—
201	D₈	—	13	2	no	—
202	M₁₆	—	2	1	no	—
203	M₁₆	—	13	2	no	—
204	M₁₆	—	13	2	no	—
205	M₁₆	—	13	2	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	26	4	no	—
208	M₁₆	—	26	1	no	—
209	M₁₆	C₂²×D₁₃	1	2	yes	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	—	26	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₄²	C₂²×D₁₃	1	2	yes	—
224	C₁₃	#2957	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	26	1	no	—
227	C₂³	#6129	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	13	2	no	—
230	Q₈	—	26	1	no	—
231	Q₈	C₂×C₄×D₁₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	13	2	no	—
236	D₄	—	26	2	no	—
237	D₄	—	26	2	no	—
238	D₄	—	26	1	no	—
239	D₄	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	26	1	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
250	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
251	C₂×C₄	#6129	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	13	1	no	—
255	C₈	—	13	1	no	—
256	C₈	—	13	1	no	—
257	C₈	—	13	1	no	—
258	C₈	—	13	4	no	—
259	C₈	—	26	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	26	2	no	—
264	C₂²	—	104	1	no	—
265	C₂²	#88135	1	1	yes	characteristic
266	C₂²	#88135	1	1	yes	characteristic
267	C₂²	C₂×#6129	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	26	2	no	—
271	C₄	#88135	1	1	yes	characteristic
272	C₄	#88135	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
274	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	52	2	no	—
280	C₂	C₂×#88135	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#421	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88204	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#421	D₁₃
hypocenter	C₁₃	#2957
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #11719524	C₂	semidirect
3. #11719450	C₂	semidirect
4. C₁₃×#421	C₂	semidirect
5. C₂×#88062	C₂	semidirect
6. #11718666	C₂	semidirect
7. #11718658	C₂	semidirect
8. #11718485	C₂	semidirect
9. #11718484	C₂	semidirect
14. #88204	C₄	semidirect
15. #88204	C₄	semidirect
19. #88202	C₂²	semidirect
20. C₂×#6107	C₄	semidirect
21. C₂×#6106	C₄	semidirect
22. C₂×#6105	C₄	semidirect
23. C₂×#6104	C₄	semidirect
28. #88161	C₂²	semidirect
31. C₁₃×#104	C₂²	semidirect
32. #88062	C₂²	semidirect
33. #88062	C₂²	semidirect
34. #88045	C₂²	semidirect
35. #88043	C₂²	semidirect
36. C₄×#1055	C₂²	semidirect
127. #421	D₁₃	semidirect
168. #104	C₂×D₁₃	semidirect
224. C₁₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88204	2
#11719524	2
#11719450	2
C₁₃×#421	2
C₂×#88062	2
#11718666	2
#11718658	2
#11718485	2
#11718484	2
#2957	13