Subgroups — #16015967

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 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
2	#13123433	C₂	1	1	yes	maximal, characteristic
3	#13123432	C₂	1	1	yes	maximal, characteristic
4	#13116624	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116623	C₂	1	2	yes	maximal, complemented
6	#13116621	C₂	1	2	yes	maximal, complemented
7	#13116618	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116617	C₂	1	2	yes	maximal, complemented
9	C₂×#11718082	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×#367	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88155	C₂²	1	1	yes	characteristic
13	#11719228	—	2	1	no	—
14	#11719209	—	2	1	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
18	#11718767	C₂²	1	1	yes	characteristic
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	1	yes	characteristic
21	#11718656	—	2	1	no	—
22	#11718625	—	2	1	no	—
23	#11718568	C₄	1	8	yes	complemented
24	C₂×#88038	C₂²	1	2	yes	—
25	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
26	#11718542	—	2	2	no	—
27	#11718541	C₂²	1	2	yes	—
28	#11718541	C₂²	1	2	yes	—
29	D₁₃×#107	C₂²	1	2	yes	—
30	D₁₃×#107	C₂²	1	2	yes	—
31	#11718522	—	2	2	no	—
32	C₁₃×#367	C₂²	1	2	yes	—
33	C₁₃×#366	C₂²	1	2	yes	—
34	#11718097	C₂²	1	2	yes	—
35	#11718096	C₂²	1	2	yes	—
36	#11718082	C₂²	1	1	yes	characteristic
37	#11718082	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	#11718082	C₂²	1	1	yes	characteristic
40	#11718081	C₂²	1	1	yes	characteristic
41	#11718081	C₂²	1	1	yes	characteristic
42	C₂×#6127	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#117	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₄×C₂²⋊D₁₃	—	4	2	no	—
47	#88180	—	2	1	no	—
48	C₂×C₄⋊Dic₁₃	—	2	1	no	—
49	C₂×#6099	D₄	1	1	yes	characteristic
50	#88155	C₂×C₄	1	4	yes	—
51	#88155	Q₈	1	4	yes	—
52	C₂×#6094	D₄	1	2	yes	—
53	C₂×C₈×D₁₃	C₂³	1	2	yes	—
54	#88147	—	4	1	no	—
55	#88146	—	2	2	no	—
56	#88146	—	2	2	no	—
57	C₄⋊C₄×D₁₃	—	2	2	no	—
58	C₄⋊C₄×D₁₃	—	2	2	no	—
59	#88141	—	4	1	no	—
60	#88135	—	4	2	no	—
61	C₄×D₅₂	—	4	1	no	—
62	C₄×Dic₂₆	—	4	1	no	—
63	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
64	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#106	—	2	1	no	—
70	#88058	—	2	1	no	—
71	#88058	C₂³	1	2	yes	—
72	#88058	C₂³	1	1	yes	characteristic
73	#88058	C₂³	1	1	yes	characteristic
74	#88057	—	2	1	no	—
75	C₂×#6090	C₂×C₄	1	4	yes	complemented
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	#88048	—	2	2	no	—
79	#88048	—	2	2	no	—
80	#88047	—	2	2	no	—
81	#88047	—	2	2	no	—
82	#88038	C₂×C₄	1	8	yes	complemented
83	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
84	#33411	—	13	1	no	maximal, Sylow
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₁₃	D₈	1	2	yes	—
88	C₂²×Dic₁₃	—	4	1	no	—
89	#6127	C₄⋊C₄	1	4	yes	—
90	#6127	Q₁₆	1	4	yes	—
91	C₂×D₅₂	D₈	1	1	yes	characteristic
92	C₂×C₄×D₁₃	—	4	2	no	—
93	C₂×C₄×D₁₃	—	4	2	no	—
94	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
95	C₂×Dic₂₆	D₈	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₂²⋊Dic₁₃	—	4	1	no	—
107	#6103	—	4	2	no	—
108	#6103	—	4	2	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄⋊Dic₁₃	—	2	2	no	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	C₄⋊Dic₁₃	—	4	1	no	—
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	C₄×Dic₁₃	—	4	1	no	—
116	C₄×Dic₁₃	—	4	2	no	—
117	C₄×Dic₁₃	—	4	2	no	—
118	#6099	C₄⋊C₄	1	4	yes	—
119	C₂×#1055	C₂×D₄	1	2	yes	—
120	#6094	C₄⋊C₄	1	8	yes	—
121	C₈×D₁₃	C₂²×C₄	1	4	yes	—
122	C₈×D₁₃	C₂×Q₈	1	4	yes	—
123	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
124	#6090	C₂²×C₄	1	2	yes	—
125	#6090	C₂²×C₄	1	2	yes	—
126	C₂×#504	—	13	1	no	—
127	#2913	—	13	1	no	—
128	#2913	—	13	1	no	—
129	#2159	—	13	1	no	—
130	#2159	—	13	2	no	—
131	#2159	—	13	2	no	—
132	#2159	D₁₃	1	1	yes	characteristic, complemented
133	#2158	—	13	1	no	—
134	#2158	—	13	2	no	—
135	C₂×#367	—	13	1	no	—
136	C₂×#367	—	13	2	no	—
137	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂²×D₁₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	16	1	no	—
143	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₁₃	#107	1	8	yes	—
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	—	4	2	no	—
151	C₂×Dic₁₃	—	16	1	no	—
152	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
153	D₅₂	#107	1	4	yes	—
154	C₄×D₁₃	—	16	2	no	—
155	C₄×D₁₃	—	16	2	no	—
156	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
157	C₄×D₁₃	C₂×Q₁₆	1	4	yes	—
158	Dic₂₆	#107	1	4	yes	—
159	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
160	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
161	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
162	#1055	C₂×C₄⋊C₄	1	4	yes	—
163	C₂×#131	—	13	1	no	—
164	#520	—	26	1	no	—
165	#520	—	26	1	no	—
166	#504	—	13	8	no	—
167	C₂×M₃₂	—	13	2	no	—
168	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
169	C₂²×C₁₆	—	13	1	no	—
170	C₂²×C₁₆	—	13	2	no	—
171	#428	—	26	2	no	—
172	#428	—	26	2	no	—
173	#428	—	26	1	no	—
174	#428	—	26	1	no	—
175	#427	—	13	1	no	—
176	#427	—	13	2	no	—
177	#427	—	13	2	no	—
178	#427	C₂×D₁₃	1	1	yes	characteristic
179	C₂×#107	—	13	1	no	—
180	C₂×#107	—	13	2	no	—
181	C₂×#107	—	13	2	no	—
182	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
183	#367	—	13	2	no	—
184	#367	—	13	1	no	—
185	#367	—	13	2	no	—
186	#367	—	13	2	no	—
187	#367	—	13	1	no	—
188	#367	C₂×D₁₃	1	2	yes	—
189	#366	—	13	2	no	—
190	#366	—	13	1	no	—
191	#366	—	13	1	no	—
192	#366	C₂×D₁₃	1	2	yes	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	2	1	no	—
195	C₂²×C₁₃	C₂×#107	1	2	yes	—
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₁₃	—	2	2	no	—
198	C₂×D₁₃	—	2	2	no	—
199	C₂×D₁₃	C₂×#107	1	4	yes	—
200	C₄×C₁₃	—	8	1	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	C₂×#107	1	2	yes	—
204	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
205	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
206	Dic₁₃	—	8	1	no	—
207	Dic₁₃	—	8	2	no	—
208	Dic₁₃	—	8	1	no	—
209	Dic₁₃	C₂×#107	1	4	yes	—
210	C₂×#41	—	13	1	no	—
211	#131	—	13	4	no	—
212	#131	—	13	4	no	—
213	C₂×M₁₆	—	13	1	no	—
214	C₂×M₁₆	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄×Q₈	—	52	1	no	—
218	C₄×Q₈	—	52	1	no	—
219	C₄×D₄	—	52	1	no	—
220	C₄×D₄	—	52	2	no	—
221	C₄×D₄	—	52	1	no	—
222	C₄×D₄	—	52	2	no	—
223	#117	—	2	1	no	—
224	#117	—	26	2	no	—
225	#117	—	26	2	no	—
226	#117	—	26	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	26	2	no	—
229	C₂×C₄⋊C₄	—	26	2	no	—
230	C₂×C₄⋊C₄	—	26	1	no	—
231	M₃₂	—	13	8	no	—
232	M₃₂	C₄×D₁₃	1	4	yes	complemented
233	C₂×C₁₆	—	13	4	no	—
234	C₂×C₁₆	—	13	1	no	—
235	C₂×C₁₆	—	13	1	no	—
236	C₂×C₁₆	—	13	8	no	—
237	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
238	#107	—	2	1	no	—
239	#107	—	13	1	no	—
240	#107	—	13	2	no	—
241	#107	—	13	1	no	—
242	#107	—	26	2	no	—
243	#107	—	26	2	no	—
244	#107	—	26	1	no	—
245	#107	C₂²×D₁₃	1	1	yes	characteristic
246	#107	C₂²×D₁₃	1	1	yes	characteristic
247	#107	C₂²×D₁₃	1	2	yes	—
248	#106	—	2	1	no	—
249	#106	—	26	2	no	—
250	#106	—	26	2	no	—
251	#106	—	26	1	no	—
252	C₂×C₁₃	—	2	2	no	—
253	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
254	C₂×C₁₃	#3416	1	1	yes	characteristic
255	C₂×C₁₃	#3416	1	1	yes	characteristic
256	D₁₃	—	2	4	no	—
257	#41	—	13	4	no	—
258	#41	—	13	4	no	—
259	C₂×Q₈	—	13	1	no	—
260	C₂×D₄	—	13	2	no	—
261	C₂×D₄	—	13	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	13	2	no	—
264	C₂²×C₄	—	52	2	no	—
265	C₂²×C₄	—	52	2	no	—
266	C₂²×C₄	—	52	1	no	—
267	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
268	M₁₆	—	13	4	no	—
269	M₁₆	—	13	8	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	13	4	no	—
272	C₂×C₈	—	13	4	no	—
273	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
274	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
275	C₂×C₈	Q₈×D₁₃	1	2	yes	—
276	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	26	1	no	—
282	C₄⋊C₄	—	26	2	no	—
283	C₄⋊C₄	—	26	1	no	—
284	C₄⋊C₄	—	52	2	no	—
285	C₄⋊C₄	—	52	2	no	—
286	C₄⋊C₄	—	52	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	52	2	no	—
289	C₂²⋊C₄	—	52	2	no	—
290	C₂²⋊C₄	—	52	1	no	—
291	C₄²	—	4	1	no	—
292	C₄²	—	52	2	no	—
293	C₄²	—	52	2	no	—
294	C₄²	—	52	1	no	—
295	C₁₆	—	13	2	no	—
296	C₁₆	—	13	2	no	—
297	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
298	C₁₃	#33411	1	1	yes	characteristic, complemented, Sylow
299	C₂³	—	13	2	no	—
300	C₂³	D₈×D₁₃	1	1	yes	characteristic
301	Q₈	—	13	4	no	—
302	D₄	—	13	8	no	—
303	D₄	—	13	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	4	no	—
309	C₂×C₄	—	13	4	no	—
310	C₂×C₄	—	16	1	no	—
311	C₂×C₄	—	52	1	no	—
312	C₂×C₄	—	52	2	no	—
313	C₂×C₄	—	52	1	no	—
314	C₂×C₄	—	208	2	no	—
315	C₂×C₄	—	208	2	no	—
316	C₂×C₄	—	208	1	no	—
317	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
318	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
319	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
321	C₈	—	13	4	no	—
322	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
323	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
324	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	13	4	no	—
328	C₂²	—	26	2	no	—
329	C₂²	—	26	2	no	—
330	C₂²	D₁₃×#107	1	2	yes	—
331	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	13	4	no	—
336	C₄	—	104	1	no	—
337	C₄	—	104	2	no	—
338	C₄	—	104	1	no	—
339	C₄	D₁₃×#107	1	2	yes	—
340	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
341	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	26	4	no	—
344	C₂	C₂×D₁₃×#107	1	1	yes	characteristic
345	C₂	#13127981	1	1	yes	characteristic
346	C₂	#13127982	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2159	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#6127	D₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2159	D₁₃
hypocenter	C₁₃	#33411
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
4. #13116624	C₂	semidirect
5. #13116623	C₂	semidirect
6. #13116621	C₂	semidirect
7. #13116618	C₂	semidirect
8. #13116617	C₂	semidirect
9. C₂×#11718082	C₂	semidirect
10. D₁₃×#367	C₂	semidirect
11. C₁₃×#2159	C₂	semidirect
23. #11718568	C₄	semidirect
63. C₁₃×M₃₂	C₂×C₄	semidirect
75. C₂×#6090	C₂×C₄	semidirect
82. #88038	C₂×C₄	semidirect
83. C₁₆×D₁₃	C₂×C₄	semidirect
132. #2159	D₁₃	semidirect
232. M₃₂	C₄×D₁₃	semidirect
298. C₁₃	#33411	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
#13123433	2
#13123432	2
#13116624	2
#13116623	2
#13116621	2
#13116618	2
#13116617	2
C₂×#11718082	2
D₁₃×#367	2
C₁₃×#2159	2
#33411	13