Subgroups — #15834912

Order 3328.

572 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	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13123937	C₂	1	2	yes	maximal
3	#13113734	C₂	1	2	yes	maximal, complemented
4	#13113709	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₅₂	C₂	1	2	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic
7	#13113258	C₂	1	2	yes	maximal
8	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718777	C₄	1	2	yes	—
14	#11718777	C₄	1	2	yes	—
15	#11718586	C₂²	1	2	yes	—
16	#11718492	—	2	2	no	—
17	#11718491	—	2	2	no	—
18	#11718381	—	2	4	no	—
19	#11718360	C₄	1	1	yes	characteristic
20	C₄×Dic₅₂	C₂²	1	2	yes	—
21	C₄×Dic₅₂	C₂²	1	2	yes	—
22	#11718343	C₄	1	1	yes	characteristic
23	#11718342	C₄	1	2	yes	—
24	#11718340	C₂²	1	4	yes	—
25	C₈×Dic₂₆	C₂²	1	4	yes	—
26	C₁₃×#322	C₂²	1	2	yes	—
27	C₈²×C₁₃	C₂²	1	2	yes	—
28	#11718050	C₂²	1	1	yes	characteristic
29	#11718050	C₂²	1	1	yes	characteristic
30	#11718049	C₂²	1	2	yes	—
31	#11718048	C₂²	1	4	yes	—
32	#11718048	C₂²	1	4	yes	—
33	C₂×C₁₃×M₁₆	—	2	1	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	#88180	—	2	2	no	—
38	#88178	—	2	2	no	—
39	#88178	C₂×C₄	1	4	yes	—
40	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
41	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
42	#88124	—	2	2	no	—
43	#88123	C₂×C₄	1	2	yes	—
44	#88123	C₂×C₄	1	2	yes	—
45	C₄×Dic₂₆	—	2	2	no	—
46	C₄×Dic₂₆	C₂³	1	4	yes	—
47	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
50	C₁₃×#97	—	2	1	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂²⋊#1055	—	2	2	no	—
58	#88058	C₂×C₄	1	2	yes	—
59	#88058	C₂×C₄	1	2	yes	—
60	#88056	C₂×C₄	1	4	yes	—
61	#88056	C₂×C₄	1	4	yes	—
62	#88055	—	2	4	no	—
63	#88054	—	2	4	no	—
64	#88054	—	2	4	no	—
65	C₈×Dic₁₃	—	2	4	no	—
66	C₄⋊#1055	C₂³	1	2	yes	—
67	C₄⋊#1055	C₂³	1	2	yes	—
68	C₄×#1055	—	2	2	no	—
69	#18086	—	13	1	no	maximal, Sylow
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	#41	1	1	yes	characteristic
88	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	2	2	no	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
94	#6101	—	2	4	no	—
95	#6101	—	2	4	no	—
96	C₄×Dic₁₃	—	2	4	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	2	2	no	—
99	Dic₅₂	—	2	4	no	—
100	Dic₅₂	—	4	2	no	—
101	#3170	—	13	1	no	—
102	#2986	—	13	2	no	—
103	#1591	—	13	2	no	—
104	#1587	—	13	4	no	—
105	C₈×Q₁₆	—	13	2	no	—
106	#1564	—	13	1	no	—
107	#1485	—	13	2	no	—
108	C₈×M₁₆	D₁₃	1	1	yes	characteristic
109	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	4	no	—
122	C₂×Dic₁₃	—	2	4	no	—
123	Dic₂₆	—	2	4	no	—
124	Dic₂₆	—	2	4	no	—
125	Dic₂₆	—	4	4	no	—
126	C₈×C₁₃	—	2	4	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	—	2	2	no	—
132	C₈×C₁₃	#131	1	2	yes	—
133	#1055	—	4	2	no	—
134	#1055	—	4	2	no	—
135	#533	—	13	2	no	—
136	#500	—	13	1	no	—
137	#494	—	13	1	no	—
138	#487	—	13	2	no	—
139	#467	—	13	2	no	—
140	#467	—	13	2	no	—
141	#446	—	13	4	no	—
142	#446	—	26	2	no	—
143	C₈×Q₈	—	13	4	no	—
144	C₈×Q₈	—	26	2	no	—
145	C₄×Q₁₆	—	13	2	no	—
146	C₄×Q₁₆	—	13	2	no	—
147	#433	—	26	4	no	—
148	#431	—	13	2	no	—
149	#431	C₂×D₁₃	1	1	yes	characteristic
150	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
152	#335	—	13	1	no	—
153	#335	—	13	1	no	—
154	#334	—	13	2	no	—
155	Q₈⋊C₈	—	13	4	no	—
156	Q₈⋊C₈	—	13	4	no	—
157	#322	C₂×D₁₃	1	2	yes	—
158	C₈²	C₂×D₁₃	1	2	yes	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	#443	1	2	yes	—
167	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
168	C₄×C₁₃	#568	1	1	yes	characteristic
169	C₄×C₁₃	#578	1	1	yes	characteristic
170	C₄×C₁₃	#578	1	1	yes	characteristic
171	Dic₁₃	—	4	4	no	—
172	Dic₁₃	—	4	4	no	—
173	C₂×Q₁₆	—	13	2	no	—
174	C₂×Q₁₆	—	13	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	13	2	no	—
179	C₄⋊Q₈	—	13	2	no	—
180	#125	—	26	2	no	—
181	C₂²⋊Q₈	—	13	4	no	—
182	C₂²⋊Q₈	—	26	2	no	—
183	C₄×Q₈	—	13	4	no	—
184	C₄×Q₈	—	26	2	no	—
185	#117	—	26	2	no	—
186	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
187	#107	—	13	2	no	—
188	#107	—	13	2	no	—
189	C₄⋊C₈	—	13	2	no	—
190	C₄⋊C₈	—	13	2	no	—
191	C₄⋊C₈	—	26	4	no	—
192	C₄⋊C₈	—	26	4	no	—
193	C₄⋊C₈	D₅₂	1	1	yes	characteristic
194	C₄⋊C₈	D₅₂	1	1	yes	characteristic
195	#103	—	13	4	no	—
196	#103	—	13	4	no	—
197	C₂²⋊C₈	—	26	2	no	—
198	C₂²⋊C₈	D₅₂	1	2	yes	—
199	#97	—	2	1	no	—
200	#97	—	26	4	no	—
201	C₄×C₈	—	26	4	no	—
202	C₄×C₈	—	26	2	no	—
203	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
204	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₁₃	1	2	yes	—
206	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₁₃	1	2	yes	—
209	C₂×C₁₃	—	4	1	no	—
210	C₂×C₁₃	#2935	1	1	yes	characteristic
211	C₂×C₁₃	#2950	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
213	C₂×Q₈	—	13	4	no	—
214	C₂×Q₈	—	26	2	no	—
215	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
217	C₂²×C₄	#6127	1	1	yes	characteristic
218	Q₁₆	—	26	4	no	—
219	Q₁₆	—	52	2	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₅₂	1	2	yes	—
231	C₂×C₈	C₂×D₅₂	1	2	yes	—
232	C₄⋊C₄	—	13	4	no	—
233	C₄⋊C₄	—	26	2	no	—
234	C₄⋊C₄	—	26	2	no	—
235	C₄⋊C₄	—	26	4	no	—
236	C₄⋊C₄	—	26	2	no	—
237	C₄⋊C₄	—	26	4	no	—
238	C₂²⋊C₄	—	26	4	no	—
239	C₄²	—	26	4	no	—
240	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₄²	C₂×D₅₂	1	1	yes	characteristic
242	C₄²	#6127	1	1	yes	characteristic
243	C₄²	C₂³×D₁₃	1	1	yes	characteristic
244	C₁₃	#18086	1	1	yes	characteristic, complemented, Sylow
245	C₂³	C₄×D₅₂	1	1	yes	characteristic
246	Q₈	—	26	4	no	—
247	Q₈	—	26	4	no	—
248	Q₈	—	52	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
257	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
258	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
261	C₂×C₄	C₂×#6127	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	52	2	no	—
269	C₈	—	52	2	no	—
270	C₈	#88161	1	2	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	52	4	no	—
279	C₄	—	52	4	no	—
280	C₄	#11718361	1	2	yes	—
281	C₄	C₂×#88161	1	1	yes	characteristic
282	C₄	#11719450	1	1	yes	characteristic
283	C₄	#11719458	1	1	yes	characteristic
284	C₄	#11719458	1	1	yes	characteristic
285	C₂	—	4	1	no	—
286	C₂	#13123582	1	1	yes	characteristic
287	C₂	#13123705	1	1	yes	characteristic
288	C₂	C₂×#11718361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#18086
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13113734	C₂	semidirect
4. #13113709	C₂	semidirect
5. C₈×Dic₅₂	C₂	semidirect
244. C₁₃	#18086	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13123937	2
#13113734	2
#13113709	2
C₈×Dic₅₂	2
#13113584	2
#13113258	2
C₈×C₁₃×M₁₆	2
#18086	13