Subgroups — #15833626

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13123547	C₂	1	2	yes	maximal, complemented
3	#13113664	C₂	1	8	yes	maximal, complemented
4	#13113286	C₂	1	2	yes	maximal, complemented
5	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88161	—	2	2	no	—
8	#11719182	C₂²	1	2	yes	—
9	C₁₃×#423	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718790	—	2	2	no	—
12	#11718776	C₄	1	4	yes	—
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	#11718499	—	2	2	no	—
16	#11718499	C₂²	1	4	yes	—
17	#11718495	—	2	2	no	—
18	#11718495	C₂²	1	4	yes	—
19	#11718356	C₄	1	1	yes	characteristic
20	#11718355	C₄	1	2	yes	—
21	C₄×D₁₀₄	C₂²	1	4	yes	—
22	#11718343	C₄	1	1	yes	characteristic
23	C₁₃×#335	C₂²	1	4	yes	—
24	#11718075	C₂²	1	8	yes	—
25	#11718046	C₂²	1	4	yes	—
26	C₂×#6127	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	#88184	—	2	2	no	—
31	#88184	C₂×C₄	1	4	yes	—
32	C₂²×#1055	—	2	1	no	—
33	C₂²×#1055	D₄	1	1	yes	characteristic
34	C₂²×#1055	D₄	1	1	yes	characteristic
35	#88161	—	4	8	no	—
36	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
37	C₂×#6094	—	2	4	no	—
38	C₂×C₈×D₁₃	—	2	4	no	—
39	#88129	—	2	2	no	—
40	#88128	C₂×C₄	1	2	yes	—
41	C₄×D₅₂	—	2	2	no	—
42	C₄×D₅₂	C₂³	1	4	yes	—
43	#88123	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₈	—	2	2	no	—
45	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	#88061	C₂×C₄	1	8	yes	—
50	#88060	—	2	4	no	—
51	#88060	—	2	4	no	—
52	#88058	C₂×C₄	1	4	yes	—
53	C₄×#1055	—	2	1	no	—
54	C₄×#1055	D₄	1	2	yes	—
55	C₄×#1055	C₂³	1	2	yes	—
56	C₄×#1055	C₂³	1	2	yes	—
57	#17975	—	13	1	no	maximal, Sylow
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	2	4	no	—
62	#6127	—	4	4	no	—
63	#6127	—	4	4	no	—
64	C₂×D₅₂	—	2	2	no	—
65	C₂×D₅₂	—	2	2	no	—
66	C₂×D₅₂	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₁₃	—	2	4	no	—
68	C₂×Dic₂₆	—	2	2	no	—
69	C₁₃×M₁₆	—	4	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃	#41	1	1	yes	characteristic
77	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
78	#6103	—	2	4	no	—
79	#6103	—	2	4	no	—
80	C₄⋊Dic₁₃	—	2	2	no	—
81	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
82	C₂×#1055	—	2	2	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	4	2	no	—
85	C₂×#1055	D₈	1	4	yes	complemented
86	C₂×#1055	C₂×D₄	1	2	yes	—
87	C₂×#1055	C₂×D₄	1	2	yes	—
88	D₁₀₄	—	4	4	no	—
89	#6094	—	4	8	no	—
90	C₈×D₁₃	—	4	8	no	—
91	#3169	—	13	1	no	—
92	#2931	—	13	2	no	—
93	#1581	—	13	8	no	—
94	#1567	D₁₃	1	1	yes	characteristic, complemented
95	#1492	—	13	2	no	—
96	C₂×#322	—	13	1	no	—
97	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂²×D₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
109	C₂²⋊D₁₃	—	4	8	no	—
110	C₂×Dic₁₃	—	2	4	no	—
111	D₅₂	—	2	8	no	—
112	D₅₂	—	4	4	no	—
113	D₅₂	—	4	4	no	—
114	C₄×D₁₃	—	4	4	no	—
115	C₄×D₁₃	—	4	4	no	—
116	Dic₂₆	—	4	4	no	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	4	4	no	—
120	#1055	—	2	2	no	—
121	#1055	—	4	2	no	—
122	#1055	C₂×D₈	1	4	yes	—
123	C₂×#131	—	26	2	no	—
124	#532	—	13	2	no	—
125	#500	—	13	1	no	—
126	#493	—	13	1	no	—
127	#486	—	13	2	no	—
128	#466	—	13	4	no	—
129	C₄×D₈	—	13	4	no	—
130	#436	—	13	4	no	—
131	#436	—	26	2	no	—
132	C₈×D₄	—	13	4	no	—
133	C₈×D₄	—	26	2	no	—
134	#423	C₂×D₁₃	1	2	yes	—
135	#408	—	26	2	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#335	C₂×D₁₃	1	4	yes	—
140	D₄⋊C₈	—	13	8	no	—
141	#322	—	13	4	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	C₄×D₈	1	2	yes	—
145	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×D₁₃	—	4	4	no	—
147	C₂×D₁₃	—	4	4	no	—
148	C₂×D₁₃	—	4	4	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	C₄×D₈	1	2	yes	—
152	C₄×C₁₃	#444	1	2	yes	—
153	C₄×C₁₃	#568	1	2	yes	—
154	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
156	Dic₁₃	—	4	4	no	—
157	C₂×#41	—	26	2	no	—
158	C₂×D₈	—	13	4	no	—
159	#131	—	52	8	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	26	4	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	26	4	no	—
165	C₂²×C₈	—	26	1	no	—
166	C₂²×C₈	C₄×D₁₃	1	2	yes	—
167	C₄⋊Q₈	—	13	2	no	—
168	#127	—	13	2	no	—
169	#124	—	26	2	no	—
170	#121	—	13	4	no	—
171	#121	—	26	2	no	—
172	C₄×D₄	—	13	4	no	—
173	C₄×D₄	—	26	2	no	—
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	#107	—	13	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
178	#102	—	13	8	no	—
179	C₂²⋊C₈	—	26	4	no	—
180	C₂²⋊C₈	—	26	4	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	26	1	no	—
185	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
186	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	#2932	1	1	yes	characteristic
190	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
191	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
192	D₁₃	—	8	4	no	—
193	#41	—	52	4	no	—
194	#41	—	52	4	no	—
195	C₂×Q₈	—	26	2	no	—
196	C₂×D₄	—	13	4	no	—
197	C₂×D₄	—	26	4	no	—
198	C₂×D₄	—	26	2	no	—
199	C₂×D₄	—	26	2	no	—
200	C₂²×C₄	—	26	4	no	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	#6131	1	1	yes	characteristic
204	D₈	—	52	4	no	—
205	M₁₆	—	4	4	no	—
206	M₁₆	—	52	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	—	26	2	no	—
214	C₂×C₈	—	52	8	no	—
215	C₂×C₈	—	52	2	no	—
216	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
218	C₄⋊C₄	—	13	4	no	—
219	C₄⋊C₄	—	26	2	no	—
220	C₂²⋊C₄	—	26	4	no	—
221	C₂²⋊C₄	—	26	4	no	—
222	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
223	C₄²	D₄×D₁₃	1	1	yes	characteristic
224	C₄²	#6131	1	1	yes	characteristic
225	C₄²	C₂³×D₁₃	1	1	yes	characteristic
226	C₁₃	#17975	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	26	4	no	—
228	C₂³	#88147	1	1	yes	characteristic
229	Q₈	—	52	4	no	—
230	D₄	—	26	8	no	—
231	D₄	—	52	8	no	—
232	D₄	—	52	4	no	—
233	D₄	—	52	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	26	4	no	—
237	C₂×C₄	—	52	4	no	—
238	C₂×C₄	—	52	4	no	—
239	C₂×C₄	#88147	1	1	yes	characteristic
240	C₂×C₄	#88147	1	1	yes	characteristic
241	C₂×C₄	#88147	1	1	yes	characteristic
242	C₂×C₄	D₈×D₁₃	1	2	yes	—
243	C₂×C₄	#88173	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#6131	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	4	no	—
250	C₈	—	13	4	no	—
251	C₈	—	26	2	no	—
252	C₈	—	52	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	52	4	no	—
256	C₂²	—	52	4	no	—
257	C₂²	—	52	4	no	—
258	C₂²	#11718533	1	2	yes	—
259	C₂²	C₂×#88147	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	52	4	no	—
263	C₄	#11718533	1	2	yes	—
264	C₄	#11718555	1	2	yes	—
265	C₄	#11719450	1	2	yes	—
266	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
267	C₄	C₂×#88173	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	104	4	no	—
270	C₂	#13123553	1	1	yes	characteristic
271	C₂	C₂×#11718533	1	1	yes	characteristic
272	C₂	C₂×#11718555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1567	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1567	D₁₃
hypocenter	C₁₃	#17975
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
2. #13123547	C₂	semidirect
3. #13113664	C₂	semidirect
4. #13113286	C₂	semidirect
5. C₂×#11718046	C₂	semidirect
6. C₁₃×#1567	C₂	semidirect
85. C₂×#1055	D₈	semidirect
94. #1567	D₁₃	semidirect
226. C₁₃	#17975	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123547	2
#13113664	2
#13113286	2
C₂×#11718046	2
C₁₃×#1567	2
#17975	13