Subgroups — #13123742

Order 1664.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719527	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	C₂×C₁₃×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718785	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88045	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6099	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	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	#88205	C₄	1	2	yes	complemented
15	#88204	C₂²	1	4	yes	—
16	#88203	C₄	1	2	yes	complemented
17	#88202	—	2	2	no	—
18	#88202	C₂²	1	4	yes	complemented
19	#88196	C₄	1	2	yes	complemented
20	#88186	C₄	1	2	yes	complemented
21	C₂×#6099	—	2	2	no	—
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	1	yes	characteristic
24	C₂×#6099	C₂²	1	1	yes	characteristic
25	#88161	C₂²	1	4	yes	complemented
26	M₁₆×D₁₃	—	2	2	no	—
27	#88155	—	2	2	no	—
28	C₁₃×#104	C₂²	1	2	yes	complemented
29	C₁₃×#104	C₂²	1	2	yes	complemented
30	#88062	C₂²	1	4	yes	complemented
31	#88045	C₂²	1	2	yes	complemented
32	#88045	C₂²	1	2	yes	complemented
33	#88043	C₂²	1	2	yes	complemented
34	C₄×#1055	C₂²	1	2	yes	complemented
35	C₁₃×#41	—	2	1	no	—
36	C₁₃×#41	—	2	1	no	—
37	C₁₃×#41	C₂×C₄	1	2	yes	—
38	C₁₃×#41	C₂³	1	2	yes	—
39	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₁₃	—	2	1	no	—
41	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	—	2	1	no	—
46	#6127	—	2	1	no	—
47	#6127	—	2	1	no	—
48	#6127	C₂×C₄	1	2	yes	—
49	#6127	C₂³	1	2	yes	—
50	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
53	C₁₃×M₁₆	—	2	1	no	—
54	C₁₃×M₁₆	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	#6107	C₂×C₄	1	4	yes	complemented
60	#6106	C₂×C₄	1	4	yes	complemented
61	#6105	C₂×C₄	1	4	yes	complemented
62	#6104	C₂×C₄	1	4	yes	complemented
63	#6099	—	2	1	no	—
64	#6099	—	2	4	no	—
65	#6099	C₂×C₄	1	2	yes	—
66	#6099	C₂×C₄	1	2	yes	—
67	#6099	D₄	1	2	yes	—
68	#6099	D₄	1	2	yes	—
69	#6099	C₂³	1	2	yes	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	4	no	—
72	C₂×#1055	C₂³	1	2	yes	—
73	C₂×#1055	C₂³	1	2	yes	—
74	#6094	—	2	4	no	—
75	C₈×D₁₃	—	2	4	no	—
76	#2956	—	13	1	no	maximal, Sylow
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	#41	1	1	yes	characteristic
79	C₂²×D₁₃	—	2	1	no	—
80	Q₈×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	—	2	2	no	—
84	D₄×C₁₃	—	2	1	no	—
85	D₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₁₃	—	2	2	no	—
99	C₂²⋊D₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	D₅₂	—	2	1	no	—
102	D₅₂	C₂²×C₄	1	2	yes	—
103	C₄×D₁₃	—	2	2	no	—
104	C₄×D₁₃	—	2	2	no	—
105	Dic₂₆	—	2	1	no	—
106	Dic₂₆	C₂²×C₄	1	2	yes	—
107	C₈×C₁₃	—	2	2	no	—
108	#1055	—	2	2	no	—
109	#1055	C₂²×C₄	1	4	yes	—
110	#1055	C₂×D₄	1	4	yes	—
111	#575	—	13	1	no	—
112	#568	—	13	1	no	—
113	#568	—	13	1	no	—
114	#444	—	13	4	no	—
115	#443	—	13	4	no	—
116	#430	—	13	1	no	—
117	C₂×#104	—	13	1	no	—
118	C₂×#104	D₁₃	1	1	yes	characteristic, complemented
119	C₄×M₁₆	—	13	1	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	—	8	1	no	—
123	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×D₁₃	—	2	2	no	—
127	C₂×D₁₃	—	8	1	no	—
128	C₄×C₁₃	—	2	2	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	4	1	no	—
131	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	Dic₁₃	—	2	2	no	—
136	C₂×#41	—	13	1	no	—
137	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
138	#137	—	13	2	no	—
139	#137	—	13	2	no	—
140	AGL(1,ℤ/8)	—	13	2	no	—
141	AGL(1,ℤ/8)	—	13	2	no	—
142	#135	—	13	4	no	—
143	#131	—	13	4	no	—
144	#131	—	13	4	no	—
145	#131	—	26	2	no	—
146	#131	—	26	2	no	—
147	C₂×M₁₆	—	13	1	no	—
148	C₂×M₁₆	—	13	1	no	—
149	C₂×M₁₆	—	13	1	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂×M₁₆	—	26	2	no	—
152	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
154	#108	—	13	4	no	—
155	#104	—	13	2	no	—
156	#104	—	13	2	no	—
157	#104	C₂×D₁₃	1	2	yes	complemented
158	#104	C₂×D₁₃	1	2	yes	complemented
159	#97	—	13	2	no	—
160	C₄×C₈	—	13	2	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	2	1	no	—
163	C₂×C₁₃	—	2	1	no	—
164	C₂×C₁₃	—	4	2	no	—
165	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
166	D₁₃	—	4	2	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	13	2	no	—
170	#41	—	13	2	no	—
171	#41	—	26	1	no	—
172	#41	—	26	1	no	—
173	#41	C₄×D₁₃	1	2	yes	—
174	#41	C₂²×D₁₃	1	2	yes	—
175	C₂×Q₈	—	13	1	no	—
176	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	13	1	no	—
179	C₂×D₄	—	26	1	no	—
180	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	26	1	no	—
184	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	Q₁₆	—	13	4	no	—
186	SD₁₆	—	13	4	no	—
187	SD₁₆	—	13	4	no	—
188	D₈	—	13	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	13	2	no	—
191	M₁₆	—	13	2	no	—
192	M₁₆	—	13	2	no	—
193	M₁₆	—	13	2	no	—
194	M₁₆	—	13	2	no	—
195	M₁₆	—	26	4	no	—
196	M₁₆	—	26	4	no	—
197	M₁₆	—	26	1	no	—
198	M₁₆	C₂²×D₁₃	1	2	yes	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	2	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	26	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	C₂²×D₁₃	1	1	yes	characteristic
207	C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	C₁₃	#2956	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	1	no	—
210	C₂³	—	26	1	no	—
211	C₂³	#6129	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	13	2	no	—
214	Q₈	—	26	1	no	—
215	Q₈	C₂×C₄×D₁₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	13	2	no	—
220	D₄	—	26	2	no	—
221	D₄	—	26	2	no	—
222	D₄	—	26	1	no	—
223	D₄	C₂×C₄×D₁₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	26	1	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
236	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
237	C₂×C₄	#6129	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	13	4	no	—
241	C₈	—	13	4	no	—
242	C₈	—	26	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	26	2	no	—
247	C₂²	—	104	1	no	—
248	C₂²	#88135	1	1	yes	characteristic
249	C₂²	#88135	1	1	yes	characteristic
250	C₂²	C₂×#6129	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	26	2	no	—
255	C₄	#88135	1	1	yes	characteristic
256	C₄	#88135	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
258	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	52	2	no	—
264	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	C₂×C₁₃×#104	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#11719527	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×#104	D₁₃
hypocenter	C₁₃	#2956
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

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. C₂×C₁₃×#104	C₂	semidirect
5. #11718785	C₂	semidirect
6. C₂×#88045	C₂	semidirect
7. C₄×#6099	C₂	semidirect
8. #11718485	C₂	semidirect
9. #11718484	C₂	semidirect
14. #88205	C₄	semidirect
16. #88203	C₄	semidirect
18. #88202	C₂²	semidirect
19. #88196	C₄	semidirect
20. #88186	C₄	semidirect
25. #88161	C₂²	semidirect
28. C₁₃×#104	C₂²	semidirect
29. C₁₃×#104	C₂²	semidirect
30. #88062	C₂²	semidirect
31. #88045	C₂²	semidirect
32. #88045	C₂²	semidirect
33. #88043	C₂²	semidirect
34. C₄×#1055	C₂²	semidirect
59. #6107	C₂×C₄	semidirect
60. #6106	C₂×C₄	semidirect
61. #6105	C₂×C₄	semidirect
62. #6104	C₂×C₄	semidirect
118. C₂×#104	D₁₃	semidirect
157. #104	C₂×D₁₃	semidirect
158. #104	C₂×D₁₃	semidirect
208. C₁₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#11719527	2
#11719524	2
#11719450	2
C₂×C₁₃×#104	2
#11718785	2
C₂×#88045	2
C₄×#6099	2
#11718485	2
#11718484	2
#2956	13