Subgroups — #15834956

Order 3328.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 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	#13123902	C₂	1	2	yes	maximal, complemented
3	#13113749	C₂	1	2	yes	maximal, complemented
4	#13113664	C₂	1	4	yes	maximal, complemented
5	#13113659	C₂	1	2	yes	maximal, complemented
6	#13113252	C₂	1	2	yes	maximal, complemented
7	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719182	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718776	C₄	1	2	yes	—
13	#11718776	C₄	1	2	yes	—
14	#11718586	C₂²	1	1	yes	characteristic
15	C₄×#6099	C₂²	1	1	yes	characteristic
16	#11718499	C₂²	1	4	yes	—
17	#11718495	C₂²	1	4	yes	—
18	#11718388	—	2	4	no	—
19	#11718356	C₄	1	1	yes	characteristic
20	#11718355	C₄	1	2	yes	—
21	C₄×D₁₀₄	C₂²	1	2	yes	—
22	C₄×D₁₀₄	C₂²	1	2	yes	—
23	#11718349	—	2	2	no	—
24	C₈×D₅₂	—	2	2	no	—
25	#11718343	C₄	1	1	yes	characteristic
26	C₁₃×#335	C₂²	1	2	yes	—
27	C₁₃×#334	C₂²	1	1	yes	characteristic
28	C₁₃×#334	C₂²	1	1	yes	characteristic
29	#11718075	C₂²	1	4	yes	—
30	#11718075	C₂²	1	4	yes	—
31	#11718047	C₂²	1	2	yes	—
32	#11718046	C₂²	1	2	yes	—
33	C₂×#6127	—	2	2	no	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	#88184	—	2	2	no	—
38	#88184	C₂×C₄	1	4	yes	—
39	C₂×#6099	—	2	1	no	—
40	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
41	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
42	C₂×#6094	—	2	4	no	—
43	C₂×C₈×D₁₃	—	2	4	no	—
44	#88129	—	2	2	no	—
45	#88128	C₂×C₄	1	2	yes	—
46	C₄×D₅₂	—	2	2	no	—
47	C₄×D₅₂	C₂³	1	4	yes	—
48	#88123	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
50	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
51	C₁₃×C₂²⋊C₈	—	2	2	no	—
52	C₄×C₈×C₁₃	—	2	2	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²⋊#1055	D₄	1	2	yes	—
58	#88061	C₂×C₄	1	4	yes	—
59	#88061	C₂×C₄	1	4	yes	—
60	#88060	—	2	4	no	—
61	#88060	—	2	4	no	—
62	#88058	C₂×C₄	1	2	yes	—
63	#88058	C₂×C₄	1	2	yes	—
64	C₄⋊#1055	D₄	1	2	yes	—
65	#88043	—	2	1	no	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₄×#1055	C₂³	1	2	yes	—
68	#18091	—	13	1	no	maximal, Sylow
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₃	—	2	4	no	—
73	#6127	—	8	2	no	—
74	#6127	—	8	2	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₁₃	—	2	4	no	—
79	C₂×Dic₂₆	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
92	#6103	—	2	4	no	—
93	#6103	—	2	4	no	—
94	C₄⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
96	#6099	—	4	2	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	C₂×D₄	1	4	yes	—
99	D₁₀₄	—	2	4	no	—
100	D₁₀₄	—	4	2	no	—
101	#6094	—	8	4	no	—
102	C₈×D₁₃	—	8	4	no	—
103	#3169	—	13	1	no	—
104	#2979	—	13	2	no	—
105	#1594	—	13	2	no	—
106	#1581	—	13	4	no	—
107	#1580	—	13	2	no	—
108	#1564	D₁₃	1	1	yes	characteristic, complemented
109	#1484	—	13	2	no	—
110	#1451	—	13	1	no	—
111	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂²⋊D₁₃	—	8	4	no	—
124	C₂×Dic₁₃	—	2	4	no	—
125	D₅₂	—	2	4	no	—
126	D₅₂	—	2	4	no	—
127	D₅₂	—	4	4	no	—
128	D₅₂	—	8	2	no	—
129	C₄×D₁₃	—	4	4	no	—
130	C₄×D₁₃	—	4	4	no	—
131	Dic₂₆	—	8	2	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	4	2	no	—
136	C₈×C₁₃	—	4	2	no	—
137	C₈×C₁₃	#131	1	2	yes	—
138	#1055	—	2	4	no	—
139	#1055	—	4	2	no	—
140	#532	—	13	2	no	—
141	#500	—	13	1	no	—
142	#493	—	13	1	no	—
143	#486	—	13	2	no	—
144	#466	—	13	2	no	—
145	#466	—	13	2	no	—
146	C₄×D₈	—	13	2	no	—
147	C₄×D₈	—	13	2	no	—
148	#436	—	13	4	no	—
149	#436	—	26	2	no	—
150	C₈×D₄	—	13	4	no	—
151	C₈×D₄	—	26	2	no	—
152	#431	—	13	1	no	—
153	#431	C₂×D₁₃	1	2	yes	—
154	#408	—	26	4	no	—
155	C₄×M₁₆	—	13	1	no	—
156	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	#335	C₂×D₁₃	1	2	yes	—
158	#334	C₂×D₁₃	1	1	yes	characteristic
159	#334	C₂×D₁₃	1	1	yes	characteristic
160	D₄⋊C₈	—	13	4	no	—
161	D₄⋊C₈	—	13	4	no	—
162	#322	—	13	2	no	—
163	#322	—	13	2	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₁₃	—	4	4	no	—
169	C₂×D₁₃	—	4	4	no	—
170	C₂×D₁₃	—	4	4	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	#444	1	2	yes	—
175	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
176	C₄×C₁₃	#568	1	1	yes	characteristic
177	C₄×C₁₃	#576	1	2	yes	—
178	Dic₁₃	—	4	4	no	—
179	C₂×#41	—	26	2	no	—
180	C₂×D₈	—	13	2	no	—
181	C₂×D₈	—	13	2	no	—
182	C₂×M₁₆	—	26	4	no	—
183	C₂×M₁₆	—	26	1	no	—
184	C₂²×C₈	—	26	4	no	—
185	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	13	2	no	—
188	#127	—	13	2	no	—
189	#124	—	26	2	no	—
190	#121	—	13	4	no	—
191	#121	—	26	2	no	—
192	C₄×D₄	—	13	4	no	—
193	C₄×D₄	—	26	2	no	—
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	#107	—	13	2	no	—
196	#107	—	13	2	no	—
197	C₄⋊C₈	—	13	2	no	—
198	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
199	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
200	#102	—	13	4	no	—
201	#102	—	13	4	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	13	2	no	—
204	C₂²⋊C₈	—	26	4	no	—
205	C₂²⋊C₈	—	26	4	no	—
206	#97	—	26	1	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
211	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂×C₁₃	—	4	1	no	—
215	C₂×C₁₃	#2935	1	1	yes	characteristic
216	C₂×C₁₃	#2948	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
218	D₁₃	—	8	4	no	—
219	#41	—	104	2	no	—
220	#41	—	104	2	no	—
221	C₂×Q₈	—	26	2	no	—
222	C₂×D₄	—	13	4	no	—
223	C₂×D₄	—	26	4	no	—
224	C₂×D₄	—	26	2	no	—
225	C₂×D₄	—	26	2	no	—
226	C₂²×C₄	—	26	4	no	—
227	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
229	C₂²×C₄	#6131	1	1	yes	characteristic
230	D₈	—	26	4	no	—
231	D₈	—	52	2	no	—
232	M₁₆	—	52	2	no	—
233	M₁₆	—	104	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	26	2	no	—
240	C₂×C₈	—	104	4	no	—
241	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₄⋊C₄	—	13	4	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₂²⋊C₄	—	26	4	no	—
248	C₂²⋊C₄	—	26	4	no	—
249	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₄²	D₄×D₁₃	1	1	yes	characteristic
251	C₄²	#6131	1	1	yes	characteristic
252	C₄²	C₂³×D₁₃	1	1	yes	characteristic
253	C₁₃	#18091	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	26	4	no	—
255	C₂³	#88147	1	1	yes	characteristic
256	Q₈	—	104	2	no	—
257	D₄	—	26	4	no	—
258	D₄	—	26	4	no	—
259	D₄	—	52	4	no	—
260	D₄	—	104	2	no	—
261	D₄	—	104	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	26	4	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	—	52	4	no	—
269	C₂×C₄	#88147	1	1	yes	characteristic
270	C₂×C₄	#88147	1	1	yes	characteristic
271	C₂×C₄	#88147	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#6131	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	26	4	no	—
281	C₈	—	52	2	no	—
282	C₈	#88155	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	52	4	no	—
287	C₂²	—	52	4	no	—
288	C₂²	—	52	4	no	—
289	C₂²	C₂×#88147	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	52	4	no	—
294	C₄	#11718555	1	2	yes	—
295	C₄	C₂×#88155	1	1	yes	characteristic
296	C₄	#11719450	1	1	yes	characteristic
297	C₄	#11719474	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	104	4	no	—
300	C₂	#13123579	1	1	yes	characteristic
301	C₂	#13123687	1	1	yes	characteristic
302	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×#1564	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#18091
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #13123902	C₂	semidirect
3. #13113749	C₂	semidirect
4. #13113664	C₂	semidirect
5. #13113659	C₂	semidirect
6. #13113252	C₂	semidirect
7. #13113125	C₂	semidirect
8. C₁₃×#1564	C₂	semidirect
108. #1564	D₁₃	semidirect
253. C₁₃	#18091	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123902	2
#13113749	2
#13113664	2
#13113659	2
#13113252	2
#13113125	2
C₁₃×#1564	2
#18091	13