Subgroups — #16016446

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116745	C₂	1	2	yes	maximal, complemented
2	#13116668	C₂	1	2	yes	maximal, complemented
3	#13116662	C₂	1	4	yes	maximal, complemented
4	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116635	C₂	1	2	yes	maximal, complemented
6	C₄×#88052	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
13	C₈×D₄×C₁₃	C₄	1	2	yes	—
14	C₈×D₄×C₁₃	C₂²	1	2	yes	—
15	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#431	C₄	1	2	yes	—
17	C₁₃×#431	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
20	C₂×#88052	C₂²	1	4	yes	—
21	C₂²×#6090	C₂²	1	2	yes	—
22	C₂²⋊#6090	C₂²	1	2	yes	—
23	C₂²⋊#6090	C₂²	1	4	yes	—
24	C₄⋊#6090	C₂²	1	4	yes	—
25	C₄⋊#6090	C₂²	1	2	yes	—
26	#11718053	C₂²	1	4	yes	—
27	C₄×#6090	C₂²	1	1	yes	characteristic
28	C₄×#6090	C₂²	1	2	yes	—
29	C₄×#6090	C₂²	1	1	yes	characteristic
30	C₄×#6090	C₂²	1	2	yes	—
31	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₁₃	C₈	1	2	yes	—
39	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
40	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#117	C₈	1	2	yes	—
43	C₁₃×#117	C₈	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
51	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
52	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	#88052	—	2	8	no	—
61	C₂×#6090	—	2	4	no	—
62	C₂×#6090	C₂³	1	2	yes	—
63	C₂×#6090	C₂³	1	4	yes	—
64	C₂×#6090	C₂³	1	2	yes	—
65	#33479	—	13	1	no	maximal, Sylow
66	C₁₃×#41	—	2	4	no	—
67	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
91	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
92	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
93	C₄²×C₁₃	C₂×C₈	1	2	yes	—
94	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×C₈	1	2	yes	—
96	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	#6090	—	2	2	no	—
101	#6090	—	2	4	no	—
102	#6090	#41	1	4	yes	complemented
103	C₈×#41	D₁₃	1	1	yes	characteristic
104	#2183	—	13	2	no	—
105	#2169	—	13	2	no	—
106	#2168	—	13	4	no	—
107	#2162	—	13	2	no	—
108	#2162	—	13	1	no	—
109	C₄×M₃₂	—	13	2	no	—
110	C₂×C₄×C₁₆	—	13	1	no	—
111	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
112	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₁₃	—	2	2	no	—
114	D₄×C₁₃	—	2	4	no	—
115	D₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
124	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	2	no	—
137	C₈×C₁₃	#131	1	2	yes	—
138	C₈×C₁₃	C₂×#41	1	2	yes	—
139	C₄×#41	Dic₁₃	1	1	yes	characteristic
140	C₂×M₃₂	—	13	4	no	—
141	C₂²×C₁₆	—	13	2	no	—
142	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
143	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
144	C₈×D₄	Dic₁₃	1	1	yes	characteristic
145	C₈×D₄	Dic₁₃	1	2	yes	—
146	C₈×D₄	C₂×D₁₃	1	2	yes	—
147	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
148	#431	Dic₁₃	1	2	yes	—
149	#431	Dic₁₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
152	C₄⋊C₁₆	—	13	4	no	—
153	C₄⋊C₁₆	—	13	2	no	—
154	C₂²⋊C₁₆	—	13	2	no	—
155	C₂²⋊C₁₆	—	13	4	no	—
156	#346	—	13	4	no	—
157	C₄×C₁₆	—	13	2	no	—
158	C₄×C₁₆	—	13	1	no	—
159	C₄×C₁₆	—	13	1	no	—
160	C₄×C₁₆	—	13	2	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
167	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
174	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
175	C₄×C₁₃	#504	1	2	yes	—
176	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
177	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
178	C₂×#41	#1055	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
181	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
183	C₄×Q₈	#1055	1	1	yes	characteristic
184	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
185	C₄×D₄	#1055	1	2	yes	—
186	C₄×D₄	#1055	1	1	yes	characteristic
187	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
188	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
189	#117	#1055	1	2	yes	—
190	#117	#1055	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
192	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
193	M₃₂	—	26	8	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	13	2	no	—
196	C₂×C₁₆	—	13	4	no	—
197	C₂×C₁₆	—	26	4	no	—
198	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
199	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
201	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
202	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
203	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
204	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
205	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
206	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
207	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
208	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
209	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₃	1	2	yes	—
211	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	—	4	2	no	—
214	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
215	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
217	#41	—	2	4	no	—
218	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
219	C₂×D₄	C₂×#1055	1	2	yes	—
220	C₂×D₄	C₂×#1055	1	1	yes	characteristic
221	C₂²×C₄	C₂×#1055	1	2	yes	—
222	C₂²×C₄	C₂×#1055	1	2	yes	—
223	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
224	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
233	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
235	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
237	C₄⋊C₄	C₂×#1055	1	2	yes	—
238	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
239	C₄⋊C₄	C₂×#1055	1	2	yes	—
240	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
241	C₂²⋊C₄	C₂×#1055	1	2	yes	—
242	C₂²⋊C₄	C₂×#1055	1	2	yes	—
243	C₂²⋊C₄	C₂×#1055	1	2	yes	—
244	C₄²	C₂×#1055	1	2	yes	—
245	C₄²	C₂×#1055	1	1	yes	characteristic
246	C₄²	C₂×#1055	1	2	yes	—
247	C₄²	C₂×#1055	1	1	yes	characteristic
248	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
249	C₄²	C₂²×Dic₁₃	1	2	yes	—
250	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
251	C₁₆	—	13	4	no	—
252	C₁₆	—	26	2	no	—
253	C₁₆	—	26	4	no	—
254	C₁₃	#33479	1	1	yes	characteristic, complemented, Sylow
255	C₂³	C₂²×#1055	1	2	yes	—
256	C₂³	C₂²×#1055	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	#88052	1	4	yes	complemented
268	C₂×C₄	C₂²×#1055	1	2	yes	—
269	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
270	C₂×C₄	C₂²×#1055	1	2	yes	—
271	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
272	C₂×C₄	C₂²×#1055	1	2	yes	—
273	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
274	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
275	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	#88202	1	2	yes	—
282	C₈	#41×D₁₃	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#88052	1	2	yes	—
289	C₂²	C₂³×#1055	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#88052	1	2	yes	—
296	C₄	C₂×#88052	1	2	yes	—
297	C₄	#11718801	1	2	yes	—
298	C₄	C₂×#88202	1	1	yes	characteristic
299	C₄	#41×Dic₁₃	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#41×#1055	1	1	yes	characteristic
303	C₂	C₂²×#88052	1	1	yes	characteristic
304	C₂	C₂×#11718801	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₈×C₁₃×#41	C₂
prime radical	C₂×C₁₃×#41	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	#33479
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #13116745	C₂	semidirect
2. #13116668	C₂	semidirect
3. #13116662	C₂	semidirect
4. #13116635	C₂	semidirect
5. #13116635	C₂	semidirect
6. C₄×#88052	C₂	semidirect
7. C₂×C₄×#6090	C₂	semidirect
102. #6090	#41	semidirect
123. C₂×C₄×C₁₃	M₃₂	semidirect
254. C₁₃	#33479	semidirect
267. C₂×C₄	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116745	2
#13116668	2
#13116662	2
#13116635	2
C₄×#88052	2
C₂×C₄×#6090	2
C₈×C₁₃×#41	2
#33479	13