Subgroups — #15768604

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	#13114568	C₂	1	2	yes	maximal
2	#13114547	C₂	1	2	yes	maximal
3	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2873	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂×#88180	C₂²	1	2	yes	—
11	C₁₃×#405	—	2	2	no	—
12	C₁₃×#405	C₄	1	4	yes	complemented
13	C₄×C₁₃×M₁₆	—	2	2	no	—
14	#11718149	—	2	2	no	—
15	#11718149	—	2	2	no	—
16	#11718147	C₄	1	8	yes	complemented
17	#11718119	—	2	4	no	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	1	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
26	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₁₃×#117	—	2	1	no	—
30	C₁₃×#117	—	2	1	no	—
31	C₁₃×#117	C₂×C₄	1	2	yes	—
32	C₁₃×#117	Q₈	1	2	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
37	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
38	#88180	—	2	2	no	—
39	#88180	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	—	2	2	no	—
44	#88180	—	2	2	no	—
45	#88180	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄×Dic₁₃	—	2	2	no	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
50	C₁₃×#97	—	2	4	no	—
51	C₄×C₈×C₁₃	—	2	4	no	—
52	#10811	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×Dic₁₃	—	2	4	no	—
66	C₂²×Dic₁₃	—	4	2	no	—
67	C₁₃×M₁₆	—	2	2	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	C₄²	1	4	yes	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₂²⋊C₄	—	2	2	no	—
80	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₂²⋊Dic₁₃	—	2	4	no	—
84	C₂²⋊Dic₁₃	—	4	2	no	—
85	C₂²⋊Dic₁₃	—	4	2	no	—
86	C₂²⋊Dic₁₃	—	4	2	no	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	C₄⋊Dic₁₃	—	4	2	no	—
89	C₄⋊Dic₁₃	—	4	2	no	—
90	C₄⋊Dic₁₃	—	4	2	no	—
91	C₄×Dic₁₃	—	2	4	no	—
92	C₄×Dic₁₃	—	2	4	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	4	2	no	—
95	#2873	D₁₃	1	1	yes	characteristic
96	#1740	—	13	2	no	—
97	#1733	—	13	2	no	—
98	C₂×H(3,ℤ/4)	—	13	1	no	—
99	C₂×H(3,ℤ/4)	—	13	1	no	—
100	C₂³×C₁₃	—	8	1	no	—
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₄×D₄	1	2	yes	—
103	C₂³×C₁₃	#121	1	1	yes	characteristic
104	C₂³×C₁₃	#121	1	1	yes	characteristic
105	C₂³×C₁₃	#124	1	2	yes	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	2	yes	—
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	#123	1	2	yes	—
123	C₂×C₄×C₁₃	#125	1	2	yes	—
124	C₂×C₄×C₁₃	#126	1	2	yes	—
125	C₂×C₄×C₁₃	#126	1	2	yes	—
126	C₂×Dic₁₃	—	4	4	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	8	4	no	—
129	C₂×Dic₁₃	—	8	4	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	2	4	no	—
132	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
133	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#117	—	13	2	no	—
135	C₂×#117	—	13	2	no	—
136	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
137	#405	—	2	2	no	—
138	#405	Dic₁₃	1	4	yes	complemented
139	C₄×M₁₆	—	2	2	no	—
140	#342	—	26	4	no	—
141	#339	—	13	8	no	—
142	H(3,ℤ/4)	—	26	2	no	—
143	H(3,ℤ/4)	—	26	2	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	#380	1	1	yes	characteristic
151	C₂²×C₁₃	#382	1	1	yes	characteristic
152	C₂²×C₁₃	#388	1	2	yes	—
153	C₂²×C₁₃	#388	1	2	yes	—
154	C₄×C₁₃	—	4	4	no	—
155	C₄×C₁₃	#376	1	1	yes	characteristic
156	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₁₃	#381	1	1	yes	characteristic
158	C₄×C₁₃	#383	1	1	yes	characteristic
159	C₄×C₁₃	#387	1	2	yes	—
160	C₄×C₁₃	#387	1	2	yes	—
161	Dic₁₃	—	8	4	no	—
162	Dic₁₃	—	8	4	no	—
163	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
169	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
170	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
171	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
174	#117	—	2	1	no	—
175	#117	—	2	1	no	—
176	#117	—	13	4	no	—
177	#117	—	26	2	no	—
178	#117	—	26	2	no	—
179	#117	—	26	2	no	—
180	#117	—	26	2	no	—
181	#117	—	26	2	no	—
182	#117	—	26	2	no	—
183	#117	—	26	2	no	—
184	#117	Dic₂₆	1	2	yes	—
185	#117	C₂×Dic₁₃	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	26	2	no	—
188	C₂×C₄⋊C₄	—	26	2	no	—
189	C₂×C₂²⋊C₄	—	26	2	no	—
190	C₂×C₂²⋊C₄	—	26	2	no	—
191	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	13	4	no	—
194	C₂×C₄²	—	26	2	no	—
195	#97	—	2	4	no	—
196	C₄×C₈	—	2	4	no	—
197	C₂×C₁₃	—	2	2	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	#1434	1	1	yes	characteristic
203	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	26	4	no	—
206	C₂²×C₄	—	52	2	no	—
207	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
209	C₂²×C₄	#6127	1	2	yes	—
210	C₂²×C₄	#6127	1	2	yes	—
211	C₂²×C₄	#6127	1	2	yes	—
212	C₂²×C₄	#6127	1	1	yes	characteristic
213	C₂²×C₄	#6127	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	C₄×Dic₁₃	1	4	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	26	4	no	—
230	C₄⋊C₄	—	52	2	no	—
231	C₄⋊C₄	—	52	2	no	—
232	C₄⋊C₄	—	52	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	26	4	no	—
235	C₂²⋊C₄	—	52	2	no	—
236	C₂²⋊C₄	—	52	2	no	—
237	C₂²⋊C₄	—	52	2	no	—
238	C₂²⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	26	4	no	—
242	C₄²	—	26	4	no	—
243	C₄²	—	26	4	no	—
244	C₄²	—	52	2	no	—
245	C₁₃	#10811	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	8	1	no	—
247	C₂³	C₄×D₅₂	1	2	yes	—
248	C₂³	#88129	1	2	yes	—
249	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
250	C₂³	#88184	1	1	yes	characteristic
251	C₂³	#88184	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	—	104	4	no	—
257	C₂×C₄	—	104	4	no	—
258	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
259	C₂×C₄	#88124	1	2	yes	—
260	C₂×C₄	#88126	1	2	yes	—
261	C₂×C₄	#88126	1	2	yes	—
262	C₂×C₄	#88130	1	2	yes	—
263	C₂×C₄	#88130	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
265	C₂×C₄	#88178	1	1	yes	characteristic
266	C₂×C₄	#88178	1	1	yes	characteristic
267	C₂×C₄	#88180	1	1	yes	characteristic
268	C₂×C₄	#88180	1	1	yes	characteristic
269	C₂×C₄	#88180	1	1	yes	characteristic
270	C₂×C₄	#88180	1	1	yes	characteristic
271	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
272	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
273	C₂×C₄	#88183	1	2	yes	—
274	C₂×C₄	#88183	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
284	C₂²	#11718597	1	1	yes	characteristic
285	C₂²	#11718601	1	2	yes	—
286	C₂²	#11718601	1	2	yes	—
287	C₂²	#11718873	1	1	yes	characteristic
288	C₄	—	4	4	no	—
289	C₄	—	104	4	no	—
290	C₄	—	104	4	no	—
291	C₄	#11718593	1	1	yes	characteristic
292	C₄	#11718594	1	2	yes	—
293	C₄	#11718594	1	2	yes	—
294	C₄	#11718598	1	1	yes	characteristic
295	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
296	C₄	#11718872	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#13113063	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#2873	C₂
socle	C₂×C₁₃	#1434
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#2873	D₁₃
hypocenter	C₁₃	#10811
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
12. C₁₃×#405	C₄	semidirect
16. #11718147	C₄	semidirect
138. #405	Dic₁₃	semidirect
245. C₁₃	#10811	semidirect

Maximal subgroups

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

subgroup	index
#13114568	2
#13114547	2
C₂×#11718149	2
C₁₃×#2873	2
#10811	13