Subgroups — #15768548

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718149	C₂	1	2	yes	maximal
4	#13114534	C₂	1	2	yes	maximal
5	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	2	yes	—
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	#11718783	—	2	2	no	—
12	#11718783	—	2	2	no	—
13	C₂×#88055	C₄	1	2	yes	complemented
14	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
15	C₁₃×#338	—	2	2	no	—
16	#11718149	—	2	2	no	—
17	#11718149	—	2	4	no	—
18	#11718142	C₄	1	8	yes	complemented
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
33	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
34	#88180	—	2	1	no	—
35	#88180	—	2	1	no	—
36	#88180	—	2	2	no	—
37	#88180	—	2	2	no	—
38	#88180	—	2	2	no	—
39	#88180	—	2	2	no	—
40	#88180	D₄	1	2	yes	—
41	#88180	Q₈	1	2	yes	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×C₄⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
46	C₂²×#1055	—	2	1	no	—
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	—	2	2	no	—
49	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
50	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
51	C₂²×#1055	C₂×C₄	1	4	yes	—
52	#88055	—	2	4	no	—
53	C₈×Dic₁₃	—	2	4	no	—
54	#10800	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×Dic₁₃	—	2	2	no	—
67	C₂²×Dic₁₃	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₁₃×M₁₆	—	4	2	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₁₃×M₁₆	—	4	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
80	C₂²⋊Dic₁₃	—	2	2	no	—
81	C₂²⋊Dic₁₃	—	2	2	no	—
82	C₂²⋊Dic₁₃	—	4	2	no	—
83	C₂²⋊Dic₁₃	—	4	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	4	2	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	C₄×Dic₁₃	—	2	2	no	—
89	C₄×Dic₁₃	—	2	4	no	—
90	C₄×Dic₁₃	—	2	4	no	—
91	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	C₄²	1	2	yes	—
99	C₂×#1055	C₄²	1	2	yes	—
100	C₂×#405	—	13	1	no	—
101	C₂×#338	D₁₃	1	1	yes	characteristic
102	C₂×H(3,ℤ/4)	—	13	2	no	—
103	C₂×H(3,ℤ/4)	—	13	1	no	—
104	#1729	—	13	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	#117	1	1	yes	characteristic
107	C₂³×C₁₃	#117	1	2	yes	—
108	C₂³×C₁₃	#121	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₁₃	#126	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#117	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#123	1	2	yes	—
125	C₂×C₄×C₁₃	#123	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#125	1	2	yes	—
128	C₂×C₄×C₁₃	#126	1	2	yes	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	4	8	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₈×C₁₃	—	2	4	no	—
134	C₈×C₁₃	—	4	4	no	—
135	#1055	—	2	2	no	—
136	#1055	—	2	2	no	—
137	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
138	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
139	C₂³×C₈	—	13	1	no	—
140	C₂×#117	—	13	1	no	—
141	C₂×#117	—	13	2	no	—
142	#405	—	26	2	no	—
143	#405	—	26	2	no	—
144	C₂×#97	—	13	2	no	—
145	C₂×C₄×C₈	—	13	2	no	—
146	#338	—	2	2	no	—
147	H(3,ℤ/4)	—	26	4	no	—
148	H(3,ℤ/4)	—	26	2	no	—
149	#336	—	13	8	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	#376	1	1	yes	characteristic
156	C₂²×C₁₃	#380	1	1	yes	characteristic
157	C₂²×C₁₃	#383	1	1	yes	characteristic
158	C₂²×C₁₃	#387	1	2	yes	—
159	C₂²×C₁₃	#388	1	2	yes	—
160	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₁₃	#381	1	1	yes	characteristic
163	C₄×C₁₃	#382	1	1	yes	characteristic
164	C₄×C₁₃	#387	1	2	yes	—
165	C₄×C₁₃	#388	1	2	yes	—
166	Dic₁₃	—	4	4	no	—
167	Dic₁₃	—	8	4	no	—
168	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	D₅₂	1	2	yes	—
178	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
179	C₂²×C₈	—	13	4	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	26	1	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	—	26	1	no	—
185	C₂²×C₈	C₄×D₁₃	1	4	yes	—
186	C₂²×C₈	C₄×D₁₃	1	2	yes	—
187	#117	—	13	2	no	—
188	#117	—	13	2	no	—
189	#117	—	26	1	no	—
190	#117	—	26	1	no	—
191	#117	—	26	2	no	—
192	#117	—	26	2	no	—
193	#117	—	26	2	no	—
194	#117	—	26	2	no	—
195	C₂×C₄⋊C₄	—	26	1	no	—
196	C₂×C₄⋊C₄	—	26	2	no	—
197	C₂×C₂²⋊C₄	—	26	1	no	—
198	C₂×C₂²⋊C₄	—	26	2	no	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	—	13	4	no	—
201	#97	—	26	4	no	—
202	C₄×C₈	—	26	4	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	2	2	no	—
207	C₂×C₁₃	#1434	1	1	yes	characteristic
208	C₂×C₁₃	#1747	1	2	yes	—
209	C₂⁴	#6131	1	1	yes	characteristic
210	C₂²×C₄	—	26	2	no	—
211	C₂²×C₄	—	26	4	no	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
214	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
215	C₂²×C₄	#6127	1	2	yes	—
216	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
217	C₂²×C₄	#6129	1	1	yes	characteristic
218	C₂²×C₄	#6129	1	2	yes	—
219	C₂²×C₄	#6129	1	2	yes	—
220	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	13	2	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	—	26	1	no	—
236	C₂×C₈	—	26	1	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	1	no	—
239	C₂×C₈	—	26	1	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	#6103	1	4	yes	—
242	C₄⋊C₄	—	26	2	no	—
243	C₄⋊C₄	—	26	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₄⋊C₄	—	52	2	no	—
246	C₂²⋊C₄	—	26	2	no	—
247	C₂²⋊C₄	—	26	2	no	—
248	C₂²⋊C₄	—	52	2	no	—
249	C₂²⋊C₄	—	52	2	no	—
250	C₄²	—	13	4	no	—
251	C₄²	—	26	2	no	—
252	C₄²	—	26	4	no	—
253	C₄²	—	26	4	no	—
254	C₁₃	#10800	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	4	2	no	—
256	C₂³	#88146	1	1	yes	characteristic
257	C₂³	#88146	1	2	yes	—
258	C₂³	#88149	1	1	yes	characteristic
259	C₂³	#88152	1	2	yes	—
260	C₂³	#88200	1	1	yes	characteristic
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	52	4	no	—
263	C₂×C₄	—	52	8	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
266	C₂×C₄	#88126	1	1	yes	characteristic
267	C₂×C₄	C₄×D₅₂	1	2	yes	—
268	C₂×C₄	#88131	1	1	yes	characteristic
269	C₂×C₄	#88131	1	1	yes	characteristic
270	C₂×C₄	#88131	1	2	yes	—
271	C₂×C₄	#88133	1	2	yes	—
272	C₂×C₄	#88135	1	2	yes	—
273	C₂×C₄	#88139	1	2	yes	—
274	C₂×C₄	#88140	1	2	yes	—
275	C₂×C₄	#88141	1	2	yes	—
276	C₂×C₄	#88143	1	2	yes	—
277	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
278	C₂×C₄	#88151	1	1	yes	characteristic
279	C₂×C₄	C₂×#6103	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
281	C₂×C₄	#88189	1	2	yes	—
282	C₂×C₄	#88192	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	4	4	no	—
285	C₈	—	26	2	no	—
286	C₈	—	26	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#11718309	1	1	yes	characteristic
293	C₂²	#11718312	1	1	yes	characteristic
294	C₂²	#11718331	1	2	yes	—
295	C₂²	#11718639	1	2	yes	—
296	C₂²	#11718648	1	1	yes	characteristic
297	C₄	—	52	4	no	—
298	C₄	—	104	4	no	—
299	C₄	#11718310	1	1	yes	characteristic
300	C₄	#11718311	1	1	yes	characteristic
301	C₄	#11718314	1	2	yes	—
302	C₄	C₄×#6103	1	1	yes	characteristic
303	C₄	#11718604	1	1	yes	characteristic
304	C₄	#11718605	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	#13113060	1	1	yes	characteristic
310	C₂	#13114592	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#338	C₂
socle	C₂²×C₁₃	#376
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#338	D₁₃
hypocenter	C₁₃	#10800
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
13. C₂×#88055	C₄	semidirect
14. C₂×C₈×Dic₁₃	C₄	semidirect
18. #11718142	C₄	semidirect
254. C₁₃	#10800	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718783	2
C₂×#11718149	2
#13114534	2
C₂×C₁₃×#338	2
#10800	13