Subgroups — #16016104

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718801	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123546	C₂	1	1	yes	maximal, characteristic
4	#13116665	C₂	1	4	yes	maximal, complemented
5	#13116653	C₂	1	4	yes	maximal, complemented
6	#13116591	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116449	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	#11719537	C₄	1	1	yes	characteristic
12	C₂×#88155	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	#11718895	C₄	1	1	yes	characteristic
16	#11718801	—	2	4	no	—
17	#11718790	C₄	1	1	yes	characteristic
18	#11718789	C₄	1	4	yes	—
19	#11718779	C₄	1	1	yes	characteristic
20	C₈×C₂²⋊D₁₃	C₂²	1	4	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#88052	C₂²	1	1	yes	characteristic
23	C₂×#88052	C₂²	1	2	yes	—
24	C₂²×#6090	C₂²	1	2	yes	—
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718568	—	2	4	no	—
27	C₂×#88038	C₂²	1	2	yes	—
28	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
29	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
30	#11718099	C₂²	1	4	yes	—
31	#11718093	C₂²	1	4	yes	—
32	C₁₆×Dic₁₃	C₂²	1	2	yes	—
33	C₁₆×Dic₁₃	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂×#6127	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#131	—	2	2	no	—
37	C₁₃×#131	—	2	2	no	—
38	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	#88201	C₈	1	1	yes	characteristic
43	#88198	C₈	1	1	yes	characteristic
44	#88194	C₈	1	1	yes	characteristic
45	#88193	C₈	1	4	yes	—
46	#88190	C₈	1	1	yes	characteristic
47	C₄×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×#6099	C₂×C₄	1	1	yes	characteristic
50	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
51	#88155	—	2	2	no	—
52	#88155	—	2	2	no	—
53	C₂×#6094	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₁₃	C₂³	1	2	yes	—
55	C₁₃×M₃₂	—	2	2	no	—
56	C₂×C₁₃×C₁₆	—	2	2	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
59	C₂²⋊#1055	C₂×C₄	1	4	yes	—
60	#88060	C₂×C₄	1	4	yes	—
61	#88054	C₂×C₄	1	4	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂³	1	2	yes	—
64	#88052	—	2	2	no	—
65	#88052	—	2	4	no	—
66	C₂×#6090	—	2	4	no	—
67	C₂×#6090	D₄	1	4	yes	complemented
68	C₂×#6090	C₂³	1	2	yes	—
69	C₂×#6090	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	#88038	—	2	4	no	—
72	C₁₆×D₁₃	—	2	4	no	—
73	#33434	—	13	1	no	maximal, Sylow
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
82	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
83	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
84	#6127	—	2	2	no	—
85	#6127	—	2	2	no	—
86	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
89	C₁₃×M₁₆	—	2	2	no	—
90	C₁₃×M₁₆	—	2	4	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃	#41	1	2	yes	—
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
99	#6103	C₂×C₈	1	4	yes	—
100	#6101	C₂×C₈	1	4	yes	—
101	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
104	#6099	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	C₂²×C₄	1	2	yes	—
107	C₂×#1055	C₂²×C₄	1	2	yes	—
108	#6094	—	2	4	no	—
109	C₈×D₁₃	—	2	2	no	—
110	C₈×D₁₃	—	2	2	no	—
111	C₁₃×C₁₆	—	2	2	no	—
112	C₁₃×C₁₆	—	2	2	no	—
113	#6090	—	2	2	no	—
114	#6090	C₂×D₄	1	2	yes	—
115	#6090	C₂×D₄	1	2	yes	—
116	C₂×#504	—	13	1	no	—
117	C₂×#504	—	13	1	no	—
118	#2931	—	13	1	no	—
119	#2169	—	13	4	no	—
120	D₄×C₁₆	—	13	4	no	—
121	#2150	—	13	1	no	—
122	#2113	—	13	1	no	—
123	#2113	D₁₃	1	1	yes	characteristic, complemented
124	C₂×C₄×C₁₆	—	13	1	no	—
125	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
126	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
128	Q₈×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	2	4	no	—
130	D₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#131	1	2	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₁₃	—	2	4	no	—
139	C₂²⋊D₁₃	—	2	8	no	—
140	C₂×Dic₁₃	—	2	4	no	—
141	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
142	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
143	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
144	D₅₂	—	2	2	no	—
145	C₄×D₁₃	—	2	2	no	—
146	C₄×D₁₃	—	2	2	no	—
147	Dic₂₆	—	2	2	no	—
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	C₄×D₄	1	2	yes	—
150	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
152	#1055	—	2	2	no	—
153	#1055	—	2	2	no	—
154	C₂×#131	—	13	1	no	—
155	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
156	#532	—	13	1	no	—
157	#504	—	26	4	no	—
158	#504	—	26	4	no	—
159	C₂×M₃₂	—	13	1	no	—
160	C₂×M₃₂	—	13	2	no	—
161	C₂×M₃₂	—	13	2	no	—
162	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
163	C₂²×C₁₆	—	13	1	no	—
164	C₂²×C₁₆	—	13	2	no	—
165	C₂²×C₁₆	—	13	2	no	—
166	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
167	#436	—	13	4	no	—
168	C₈×D₄	—	13	4	no	—
169	#423	—	13	1	no	—
170	#408	—	13	1	no	—
171	#408	—	13	1	no	—
172	C₂×C₄×C₈	—	13	1	no	—
173	C₄⋊C₁₆	—	13	4	no	—
174	C₂²⋊C₁₆	—	13	4	no	—
175	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	—
176	C₄×C₁₆	—	13	2	no	—
177	C₄×C₁₆	—	13	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	4	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	C₈×D₄	1	2	yes	—
182	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
183	C₂×D₁₃	—	2	4	no	—
184	C₂×D₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	C₈×D₄	1	2	yes	—
187	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	2	no	—
191	Dic₁₃	#504	1	4	yes	complemented
192	C₂×#41	—	13	1	no	—
193	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
194	#131	—	2	2	no	—
195	#131	—	2	2	no	—
196	#131	—	26	2	no	—
197	#131	—	26	2	no	—
198	C₂×M₁₆	—	13	1	no	—
199	C₂×M₁₆	—	13	2	no	—
200	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
201	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
205	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	13	1	no	—
207	#127	—	13	1	no	—
208	#124	—	13	1	no	—
209	#124	—	13	1	no	—
210	#121	—	13	4	no	—
211	C₄×D₄	—	13	4	no	—
212	C₂×C₄²	—	13	1	no	—
213	M₃₂	—	2	2	no	—
214	M₃₂	—	26	4	no	—
215	M₃₂	—	26	4	no	—
216	M₃₂	—	26	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	13	4	no	—
219	C₂×C₁₆	—	13	1	no	—
220	C₂×C₁₆	—	13	1	no	—
221	C₂×C₁₆	—	13	2	no	—
222	C₂×C₁₆	—	26	4	no	—
223	C₂×C₁₆	—	26	4	no	—
224	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
225	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
226	C₄⋊C₈	—	13	4	no	—
227	C₂²⋊C₈	—	13	4	no	—
228	C₂²⋊C₈	—	13	4	no	—
229	C₄×C₈	—	13	2	no	—
230	C₄×C₈	—	13	2	no	—
231	C₂×C₁₃	—	2	2	no	—
232	C₂×C₁₃	—	4	2	no	—
233	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
234	C₂×C₁₃	C₂×#504	1	2	yes	—
235	D₁₃	—	4	2	no	—
236	#41	—	2	2	no	—
237	#41	—	2	2	no	—
238	#41	—	26	2	no	—
239	#41	—	26	2	no	—
240	C₂×Q₈	—	13	1	no	—
241	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
242	C₂×D₄	—	13	4	no	—
243	C₂×D₄	—	13	2	no	—
244	C₂×D₄	—	13	1	no	—
245	C₂×D₄	C₈×D₁₃	1	2	yes	—
246	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	4	no	—
253	M₁₆	—	26	2	no	—
254	M₁₆	—	26	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	2	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
263	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₈	D₄×D₁₃	1	2	yes	—
265	C₂×C₈	#6129	1	2	yes	—
266	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
267	C₄⋊C₄	—	13	4	no	—
268	C₂²⋊C₄	—	13	4	no	—
269	C₂²⋊C₄	—	13	4	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	13	1	no	—
272	C₄²	—	13	2	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	13	2	no	—
276	C₁₆	—	13	2	no	—
277	C₁₆	—	26	2	no	—
278	C₁₃	#33434	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	13	2	no	—
280	C₂³	C₂×C₈×D₁₃	1	2	yes	—
281	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	26	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	26	8	no	—
287	D₄	—	26	4	no	—
288	D₄	—	26	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	26	4	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	#88135	1	2	yes	—
298	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
300	C₂×C₄	#88161	1	2	yes	—
301	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	26	2	no	—
304	C₈	—	26	2	no	—
305	C₈	#88135	1	2	yes	—
306	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
307	C₈	C₂×#6129	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	26	2	no	—
312	C₂²	—	26	4	no	—
313	C₂²	#11718386	1	2	yes	—
314	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	13	4	no	—
317	C₄	—	26	2	no	—
318	C₄	—	26	2	no	—
319	C₄	#11718386	1	2	yes	—
320	C₄	C₂×#88135	1	1	yes	characteristic
321	C₄	C₂×#88161	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	52	2	no	—
325	C₂	C₂×#11718386	1	1	yes	characteristic
326	C₂	D₁₃×#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2113	C₂
prime radical	#11719537	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2113	D₁₃
hypocenter	C₁₃	#33434
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. C₂×#11718568	C₂	semidirect
2. C₂×#11718801	C₂	semidirect
4. #13116665	C₂	semidirect
5. #13116653	C₂	semidirect
6. #13116591	C₂	semidirect
7. #13116449	C₂	semidirect
8. C₂×C₁₆×Dic₁₃	C₂	semidirect
9. C₁₃×#2113	C₂	semidirect
67. C₂×#6090	D₄	semidirect
123. #2113	D₁₃	semidirect
191. Dic₁₃	#504	semidirect
278. C₁₃	#33434	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718568	2
C₂×#11718801	2
#13123546	2
#13116665	2
#13116653	2
#13116591	2
#13116449	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2113	2
#33434	13