Subgroups — #15755775

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114816	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112671	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#431	—	2	2	no	—
8	C₁₃×#431	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	#11718116	—	2	2	no	—
17	#11718050	—	2	2	no	—
18	#11718049	—	2	2	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₃	—	2	1	no	—
25	C₂×C₄²×C₁₃	—	2	1	no	—
26	C₂×C₄²×C₁₃	—	2	1	no	—
27	C₂×C₄²×C₁₃	—	2	1	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×#1055	—	2	2	no	—
33	C₂²×#1055	—	2	2	no	—
34	C₁₃×C₄⋊C₈	—	2	4	no	—
35	C₁₃×C₂²⋊C₈	—	2	4	no	—
36	C₄×C₈×C₁₃	—	2	1	no	—
37	C₄×C₈×C₁₃	—	2	1	no	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	C₂²⋊#1055	—	4	4	no	—
41	C₄⋊#1055	—	2	2	no	—
42	C₄⋊#1055	—	2	2	no	—
43	C₄⋊#1055	—	2	2	no	—
44	C₄⋊#1055	—	2	2	no	—
45	#8600	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃	#41	1	1	yes	characteristic
77	C₄²×C₁₃	#41	1	1	yes	characteristic
78	C₂×#1055	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	C₂×#1055	—	4	2	no	—
82	C₂×#1055	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#431	D₁₃	1	1	yes	characteristic
85	#1797	—	13	2	no	—
86	C₂×#335	—	13	1	no	—
87	C₂×#334	—	13	1	no	—
88	#1277	—	13	2	no	—
89	C₂³×C₁₃	—	2	1	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	—	2	2	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	#108	1	2	yes	—
94	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	#108	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#131	1	4	yes	—
118	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₃	#137	1	2	yes	—
120	C₈×C₁₃	—	2	4	no	—
121	C₈×C₁₃	—	4	2	no	—
122	#1055	—	8	2	no	—
123	#1055	—	8	2	no	—
124	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
125	#431	—	2	2	no	—
126	#431	—	2	2	no	—
127	C₂×C₄⋊C₈	—	13	2	no	—
128	C₂×C₄⋊C₈	—	13	2	no	—
129	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
130	C₂×C₂²⋊C₈	—	26	2	no	—
131	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
134	#336	—	26	2	no	—
135	#335	—	26	2	no	—
136	#334	—	26	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	4	1	no	—
140	C₂²×C₁₃	—	4	1	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	#380	1	1	yes	characteristic
143	C₂²×C₁₃	#424	1	2	yes	—
144	C₂²×C₁₃	#428	1	2	yes	—
145	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	4	1	no	—
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	#408	1	2	yes	—
153	C₄×C₁₃	#433	1	2	yes	—
154	C₄×C₁₃	#468	1	2	yes	—
155	C₄×C₁₃	#469	1	1	yes	characteristic
156	C₄×C₁₃	#470	1	1	yes	characteristic
157	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
158	C₂³×C₄	D₅₂	1	1	yes	characteristic
159	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂²×C₈	—	26	2	no	—
161	C₂²×C₈	—	26	2	no	—
162	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
163	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
169	C₂×C₄²	D₅₂	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	—	26	2	no	—
174	C₄⋊C₈	—	26	2	no	—
175	C₄⋊C₈	—	26	2	no	—
176	C₄⋊C₈	—	26	2	no	—
177	C₂²⋊C₈	—	2	4	no	—
178	C₂²⋊C₈	—	52	4	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
182	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
183	C₂×C₁₃	—	4	2	no	—
184	C₂×C₁₃	#1573	1	2	yes	—
185	C₂×C₁₃	#1574	1	2	yes	—
186	C₂×C₁₃	#1798	1	1	yes	characteristic
187	C₂×C₁₃	#1809	1	1	yes	characteristic
188	C₂×C₁₃	#1810	1	1	yes	characteristic
189	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	52	2	no	—
211	C₂×C₈	—	52	2	no	—
212	C₂×C₈	—	52	2	no	—
213	C₂×C₈	—	52	2	no	—
214	C₂×C₈	—	52	2	no	—
215	C₂×C₈	—	52	2	no	—
216	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	#6127	1	1	yes	characteristic
224	C₄²	#6127	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
226	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₁₃	#8600	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	#88059	1	2	yes	—
233	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#88059	1	2	yes	—
250	C₂×C₄	#88162	1	2	yes	—
251	C₂×C₄	#88163	1	2	yes	—
252	C₂×C₄	#88178	1	1	yes	characteristic
253	C₂×C₄	#88178	1	1	yes	characteristic
254	C₂×C₄	#88180	1	1	yes	characteristic
255	C₂×C₄	#88184	1	1	yes	characteristic
256	C₂×C₄	#88184	1	1	yes	characteristic
257	C₂×C₄	#88202	1	4	yes	—
258	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	104	2	no	—
262	C₈	—	104	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#11718660	1	2	yes	—
269	C₂²	C₂×#88059	1	1	yes	characteristic
270	C₂²	C₂×#88059	1	1	yes	characteristic
271	C₂²	#11718782	1	2	yes	—
272	C₂²	#11718873	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	#11718661	1	2	yes	—
279	C₄	#11718795	1	2	yes	—
280	C₄	#11718796	1	1	yes	characteristic
281	C₄	#11718798	1	1	yes	characteristic
282	C₄	#11718895	1	2	yes	—
283	C₂	—	4	2	no	—
284	C₂	#13113621	1	2	yes	—
285	C₂	#13113625	1	2	yes	—
286	C₂	#13114820	1	1	yes	characteristic
287	C₂	#13114869	1	1	yes	characteristic
288	C₂	#13114874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#8600
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. C₂×#11718050	C₂	semidirect
3. C₂×#11718049	C₂	semidirect
4. #13112671	C₂	semidirect
227. C₁₃	#8600	semidirect

Maximal subgroups

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

subgroup	index
#13114816	2
C₂×#11718050	2
C₂×#11718049	2
#13112671	2
C₂×C₁₃×#431	2
#8600	13