Subgroups — #15749482

Order 3328.

448 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114816	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	—	2	1	no	—
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊#1055	—	2	1	no	—
7	C₂×C₂²⋊#1055	C₄	1	2	yes	—
8	C₂×C₄⋊#1055	C₄	1	2	yes	—
9	C₂×C₄×#1055	—	2	1	no	—
10	C₁₃×#336	—	2	1	no	—
11	C₁₃×#336	C₄	1	2	yes	—
12	#11718116	—	2	1	no	—
13	#11718116	—	2	1	no	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	4	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	4	2	no	—
20	C₂²×C₈×C₁₃	—	4	1	no	—
21	C₂²×C₈×C₁₃	—	4	1	no	—
22	C₂×C₄²×C₁₃	—	4	2	no	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×#1055	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	C₂²⋊#1055	—	2	2	no	—
30	C₂²⋊#1055	—	4	1	no	—
31	C₂²⋊#1055	—	4	1	no	—
32	C₂²⋊#1055	C₈	1	4	yes	complemented
33	C₄⋊#1055	—	4	2	no	—
34	C₄×#1055	—	4	2	no	—
35	#7029	—	13	1	no	maximal, Sylow
36	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	4	1	no	—
42	C₂²×C₄×C₁₃	—	4	1	no	—
43	C₂²×C₄×C₁₃	—	4	1	no	—
44	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₃	—	2	4	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	—	4	4	no	—
59	C₂×C₈×C₁₃	—	4	4	no	—
60	C₂×C₈×C₁₃	—	4	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	4	1	no	—
64	C₄²×C₁₃	—	4	1	no	—
65	C₂×#1055	—	2	4	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	4	2	no	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	—	4	2	no	—
70	C₂×#1055	—	4	2	no	—
71	#1797	—	13	1	no	—
72	C₄×C₂²⋊C₈	—	13	1	no	—
73	C₂×#336	D₁₃	1	1	yes	characteristic
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	#95	1	1	yes	characteristic
79	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂³×C₁₃	#106	1	1	yes	characteristic
81	C₂³×C₁₃	#107	1	1	yes	characteristic
82	C₂³×C₁₃	#108	1	2	yes	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₃	#99	1	2	yes	—
105	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#104	1	2	yes	—
110	C₂×C₄×C₁₃	#108	1	2	yes	—
111	C₈×C₁₃	—	4	4	no	—
112	#1055	—	4	4	no	—
113	#1055	—	8	2	no	—
114	C₂³×C₈	—	2	1	no	—
115	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	13	2	no	—
117	C₂×C₂²⋊C₈	—	13	2	no	—
118	C₂×C₂²⋊C₈	—	26	1	no	—
119	C₂×C₄×C₈	—	26	1	no	—
120	#336	—	2	1	no	—
121	#336	—	26	1	no	—
122	#336	—	26	1	no	—
123	#336	Dic₁₃	1	2	yes	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	#334	1	2	yes	—
133	C₂²×C₁₃	#335	1	2	yes	—
134	C₂²×C₁₃	#336	1	1	yes	characteristic
135	C₂²×C₁₃	#339	1	1	yes	characteristic
136	C₂²×C₁₃	#340	1	1	yes	characteristic
137	C₂²×C₁₃	#341	1	1	yes	characteristic
138	C₂²×C₁₃	#341	1	1	yes	characteristic
139	C₂²×C₁₃	#342	1	1	yes	characteristic
140	C₂²×C₁₃	#344	1	1	yes	characteristic
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	4	1	no	—
148	C₄×C₁₃	#327	1	2	yes	—
149	C₄×C₁₃	#328	1	2	yes	—
150	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
151	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
152	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	2	no	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	—	4	1	no	—
158	C₂²×C₈	—	26	2	no	—
159	C₂²×C₈	—	26	2	no	—
160	C₂×C₄²	—	4	2	no	—
161	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
162	C₂×C₄²	D₅₂	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₄⋊C₈	—	52	2	no	—
166	C₂²⋊C₈	—	13	4	no	—
167	C₂²⋊C₈	—	26	2	no	—
168	C₂²⋊C₈	—	52	1	no	—
169	C₂²⋊C₈	—	52	1	no	—
170	C₄×C₈	—	52	2	no	—
171	C₂×C₁₃	—	2	2	no	—
172	C₂×C₁₃	—	2	2	no	—
173	C₂×C₁₃	#1277	1	1	yes	characteristic
174	C₂×C₁₃	#1278	1	1	yes	characteristic
175	C₂×C₁₃	#1282	1	1	yes	characteristic
176	C₂×C₁₃	#1294	1	1	yes	characteristic
177	C₂×C₁₃	#1298	1	1	yes	characteristic
178	C₂×C₁₃	#1301	1	1	yes	characteristic
179	C₂×C₁₃	#1302	1	1	yes	characteristic
180	C₂⁴	#6101	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	C₈×D₁₃	1	2	yes	—
189	C₂²×C₄	#6094	1	2	yes	—
190	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
191	C₂²×C₄	#6101	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
193	C₂²×C₄	#6103	1	1	yes	characteristic
194	C₂²×C₄	#6103	1	1	yes	characteristic
195	C₂²×C₄	#6104	1	1	yes	characteristic
196	C₂²×C₄	#6105	1	1	yes	characteristic
197	C₂²×C₄	#6106	1	1	yes	characteristic
198	C₂²×C₄	#6107	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	26	4	no	—
206	C₂×C₈	—	52	2	no	—
207	C₂×C₈	—	52	2	no	—
208	C₂×C₈	—	52	2	no	—
209	C₂×C₈	—	52	2	no	—
210	C₂×C₈	—	52	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₁₃	#7029	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	#88047	1	1	yes	characteristic
221	C₂³	#88048	1	1	yes	characteristic
222	C₂³	#88054	1	2	yes	—
223	C₂³	#88062	1	2	yes	—
224	C₂³	#88071	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	#88045	1	2	yes	—
244	C₂×C₄	#88046	1	2	yes	—
245	C₂×C₄	#88049	1	1	yes	characteristic
246	C₂×C₄	#88050	1	1	yes	characteristic
247	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
248	C₂×C₄	#88055	1	1	yes	characteristic
249	C₂×C₄	#88059	1	2	yes	—
250	C₂×C₄	#88060	1	2	yes	—
251	C₂×C₄	#88072	1	1	yes	characteristic
252	C₂×C₄	#88075	1	1	yes	characteristic
253	C₈	—	4	4	no	—
254	C₈	—	52	4	no	—
255	C₈	—	104	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	#11718071	1	2	yes	—
265	C₂²	#11718072	1	2	yes	—
266	C₂²	#11718115	1	1	yes	characteristic
267	C₂²	#11718119	1	1	yes	characteristic
268	C₂²	#11718121	1	1	yes	characteristic
269	C₂²	#11718142	1	1	yes	characteristic
270	C₂²	#11718144	1	1	yes	characteristic
271	C₂²	#11718150	1	1	yes	characteristic
272	C₂²	#11718151	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	#11718043	1	2	yes	—
281	C₄	#11718044	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	#13112670	1	1	yes	characteristic
285	C₂	#13112673	1	1	yes	characteristic
286	C₂	#13112682	1	1	yes	characteristic
287	C₂	#13112715	1	1	yes	characteristic
288	C₂	#13112724	1	1	yes	characteristic
289	C₂	#13112733	1	1	yes	characteristic
290	C₂	#13112735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6103
commutator	C₂×C₄×C₁₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₃
Fitting	C₂×C₁₃×#336	C₂
socle	C₂³×C₁₃	#95
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₁₃	C₂²
hypercenter	C₂×#336	D₁₃
hypocenter	C₁₃	#7029
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
32. C₂²⋊#1055	C₈	semidirect
215. C₁₃	#7029	semidirect

Maximal subgroups

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

subgroup	index
#13114816	2
C₄×C₂²⋊#1055	2
C₂×C₁₃×#336	2
#7029	13