Subgroups — #16017665

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123754	C₂	1	1	yes	maximal, characteristic
2	#13117009	C₂	1	4	yes	maximal, complemented
3	#13117004	C₂	1	4	yes	maximal, complemented
4	#13116608	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718079	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116602	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116571	C₂	1	1	yes	maximal, characteristic
8	C₂×#11718080	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#544	—	2	1	no	—
11	#11719450	—	2	1	no	—
12	C₂×#88158	C₂²	1	1	yes	characteristic
13	C₁₃×#501	—	2	2	no	—
14	C₁₃×#499	C₂²	1	2	yes	complemented
15	C₁₃×#498	C₂²	1	2	yes	complemented
16	C₁₃×#428	—	2	1	no	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
19	C₁₃×#405	C₂²	1	1	yes	characteristic
20	C₂×#88059	C₂²	1	1	yes	characteristic
21	C₂×#88052	C₂²	1	1	yes	characteristic
22	C₂×#88052	C₂²	1	1	yes	characteristic
23	#11718666	—	2	1	no	—
24	#11718378	—	2	2	no	—
25	#11718361	C₂²	1	4	yes	—
26	#11718080	C₂²	1	2	yes	—
27	#11718080	C₂²	1	4	yes	—
28	#11718080	C₂²	1	2	yes	—
29	#11718079	C₂²	1	2	yes	complemented
30	#11718079	C₂²	1	4	yes	complemented
31	#11718079	C₂²	1	2	yes	complemented
32	#11718055	C₂²	1	4	yes	complemented
33	C₂×#6127	—	2	1	no	—
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	4	1	no	—
38	C₁₃×#125	—	2	2	no	—
39	C₁₃×#125	—	4	1	no	—
40	C₁₃×C₂²⋊Q₈	—	4	2	no	—
41	C₁₃×#117	—	2	1	no	—
42	C₁₃×#117	D₄	1	1	yes	characteristic
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×#6099	—	2	1	no	—
45	#88161	—	4	2	no	—
46	M₁₆×D₁₃	—	4	2	no	—
47	C₂×Dic₅₂	Q₈	1	1	yes	characteristic, complemented
48	#88158	—	2	1	no	—
49	#88158	—	2	1	no	—
50	#88158	Q₈	1	2	yes	complemented
51	#88158	C₂³	1	2	yes	—
52	C₂×D₁₀₄	Q₈	1	1	yes	characteristic, complemented
53	C₂×#6095	—	2	1	no	—
54	#88155	—	2	4	no	—
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	—	2	2	no	—
57	C₁₃×#106	—	2	1	no	—
58	C₁₃×#106	—	2	2	no	—
59	C₁₃×#106	C₂³	1	2	yes	—
60	C₁₃×#106	C₂³	1	1	yes	characteristic
61	C₁₃×#106	C₂³	1	1	yes	characteristic
62	C₁₃×#97	D₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂³	1	2	yes	—
64	#88059	—	2	1	no	—
65	#88059	C₂³	1	2	yes	—
66	#88052	—	2	1	no	—
67	#88052	—	2	1	no	—
68	#88052	C₂³	1	2	yes	—
69	#88052	C₂³	1	2	yes	—
70	C₂×#6090	—	2	1	no	—
71	C₂×#6090	—	2	1	no	—
72	#88045	—	2	4	no	—
73	#33638	—	13	1	no	maximal, Sylow
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₁₃	—	4	1	no	—
78	#6127	—	2	2	no	—
79	#6127	—	2	2	no	—
80	#6127	—	4	1	no	—
81	#6127	—	4	1	no	—
82	C₂×D₅₂	—	2	1	no	—
83	C₂×C₄×D₁₃	—	4	1	no	—
84	C₂×Dic₂₆	—	2	1	no	—
85	C₁₃×M₁₆	SD₁₆	1	2	yes	—
86	C₁₃×M₁₆	SD₁₆	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	2	1	no	—
103	C₁₃×C₂²⋊C₄	—	4	1	no	—
104	C₄²×C₁₃	—	4	1	no	—
105	C₄²×C₁₃	C₂×D₄	1	2	yes	—
106	#6099	—	2	2	no	—
107	#6099	—	4	1	no	—
108	C₂×#1055	—	4	1	no	—
109	Dic₅₂	—	2	1	no	—
110	Dic₅₂	C₂×Q₈	1	2	yes	—
111	D₁₀₄	—	2	1	no	—
112	D₁₀₄	C₂×Q₈	1	2	yes	—
113	#6095	—	2	2	no	—
114	#6095	—	2	2	no	—
115	#6094	—	4	4	no	—
116	C₈×D₁₃	—	4	4	no	—
117	#6090	—	2	2	no	—
118	#6090	—	2	2	no	—
119	#3163	D₁₃	1	1	yes	characteristic, complemented
120	#2957	—	13	1	no	—
121	#2231	—	13	4	no	—
122	#2230	—	13	4	no	—
123	#2155	—	13	1	no	—
124	C₂×#365	—	13	1	no	—
125	#2153	—	13	1	no	—
126	#2145	—	13	1	no	—
127	C₂×#360	—	13	1	no	—
128	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂²×D₁₃	—	4	1	no	—
130	Q₈×C₁₃	—	16	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	16	1	no	—
136	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₁₃	—	4	2	no	—
143	C₂²⋊D₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	1	no	—
145	D₅₂	—	2	2	no	—
146	D₅₂	—	4	1	no	—
147	C₄×D₁₃	—	4	2	no	—
148	C₄×D₁₃	—	4	2	no	—
149	Dic₂₆	—	2	2	no	—
150	Dic₂₆	—	4	1	no	—
151	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
152	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
154	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	#1055	—	4	2	no	—
157	C₂×#135	—	13	1	no	—
158	#568	—	26	1	no	—
159	#544	—	2	1	no	—
160	C₂×M₃₂	—	13	1	no	—
161	C₂×M₃₂	—	13	1	no	—
162	#501	—	2	2	no	—
163	#499	C₂×D₁₃	1	2	yes	complemented
164	#498	C₂×D₁₃	1	2	yes	complemented
165	#444	—	26	2	no	—
166	#443	—	13	4	no	—
167	C₂×#108	—	13	1	no	—
168	#428	—	2	1	no	—
169	#427	C₂×D₁₃	1	1	yes	characteristic
170	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
171	#421	—	26	1	no	—
172	#405	C₂×D₁₃	1	1	yes	characteristic
173	#365	—	13	4	no	—
174	#365	—	13	2	no	—
175	#365	—	13	2	no	—
176	#364	—	13	4	no	—
177	#360	—	13	4	no	—
178	#360	—	13	2	no	—
179	#360	—	13	2	no	—
180	C₂²×C₁₃	—	4	1	no	—
181	C₂²×C₁₃	#476	1	1	yes	characteristic
182	C₂²×C₁₃	#476	1	1	yes	characteristic
183	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×D₁₃	—	4	2	no	—
185	C₂×D₁₃	—	16	1	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	8	1	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	1	no	—
190	C₄×C₁₃	#476	1	1	yes	characteristic
191	C₄×C₁₃	#476	1	1	yes	characteristic
192	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
193	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
194	Dic₁₃	—	4	2	no	—
195	C₂×#41	—	26	1	no	—
196	#135	—	13	2	no	—
197	#135	—	13	2	no	—
198	#135	—	26	1	no	—
199	#135	—	26	1	no	—
200	C₂×Q₁₆	—	13	1	no	—
201	C₂×SD₁₆	—	26	1	no	—
202	C₂×D₈	—	13	1	no	—
203	#131	—	26	4	no	—
204	#131	—	52	2	no	—
205	C₂×M₁₆	—	26	1	no	—
206	C₂×M₁₆	—	52	2	no	—
207	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	4	1	no	—
211	#125	—	2	2	no	—
212	#125	—	4	1	no	—
213	C₂²⋊Q₈	—	4	2	no	—
214	#117	—	2	1	no	—
215	#117	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	2	1	no	—
217	M₃₂	—	13	2	no	—
218	M₃₂	—	13	2	no	—
219	M₃₂	—	26	1	no	—
220	M₃₂	—	26	1	no	—
221	C₂×C₁₆	—	26	1	no	—
222	C₂×C₁₆	—	26	1	no	—
223	#108	—	13	2	no	—
224	#108	—	26	1	no	—
225	#107	—	2	1	no	—
226	#107	—	2	2	no	—
227	#106	—	2	1	no	—
228	#106	—	2	2	no	—
229	#106	C₂²×D₁₃	1	2	yes	—
230	#106	C₂²×D₁₃	1	1	yes	characteristic
231	#106	C₂²×D₁₃	1	1	yes	characteristic
232	#104	—	26	4	no	—
233	#97	C₂²⋊D₁₃	1	2	yes	—
234	C₄×C₈	C₂²×D₁₃	1	2	yes	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
239	D₁₃	—	8	2	no	—
240	#41	—	26	2	no	—
241	#41	—	26	2	no	—
242	#41	—	52	1	no	—
243	#41	—	52	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	26	1	no	—
246	C₂×D₄	—	26	1	no	—
247	C₂×D₄	—	52	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	52	1	no	—
250	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	Q₁₆	—	13	2	no	—
252	Q₁₆	—	26	1	no	—
253	SD₁₆	—	26	2	no	—
254	SD₁₆	—	26	2	no	—
255	D₈	—	13	2	no	—
256	D₈	—	26	1	no	—
257	M₁₆	—	26	2	no	—
258	M₁₆	—	52	1	no	—
259	M₁₆	—	52	4	no	—
260	M₁₆	#6106	1	2	yes	—
261	M₁₆	#6106	1	2	yes	—
262	C₂×C₈	—	52	1	no	—
263	C₂×C₈	—	52	4	no	—
264	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
265	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
266	C₂×C₈	#6131	1	1	yes	characteristic
267	C₂×C₈	#6131	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
283	C₁₆	—	26	2	no	—
284	C₁₆	—	26	2	no	—
285	C₁₃	#33638	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	52	1	no	—
287	C₂³	#88186	1	1	yes	characteristic
288	Q₈	—	16	2	no	—
289	Q₈	—	26	2	no	—
290	Q₈	—	52	1	no	—
291	D₄	—	26	2	no	—
292	D₄	—	52	2	no	—
293	D₄	—	52	2	no	—
294	D₄	—	52	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	52	1	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	#88186	1	1	yes	characteristic
304	C₂×C₄	C₂×#6106	1	1	yes	characteristic
305	C₂×C₄	C₂×#6106	1	1	yes	characteristic
306	C₂×C₄	#88200	1	1	yes	characteristic
307	C₂×C₄	#88200	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₈	—	52	2	no	—
310	C₈	C₂×#6106	1	4	yes	—
311	C₈	#88200	1	1	yes	characteristic
312	C₈	#88200	1	1	yes	characteristic
313	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
314	C₈	C₂×#6131	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	52	2	no	—
317	C₂²	—	208	1	no	—
318	C₂²	#11718745	1	1	yes	characteristic
319	C₂²	#11718745	1	1	yes	characteristic
320	C₂²	C₂×#88186	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	52	2	no	—
326	C₄	#11718745	1	1	yes	characteristic
327	C₄	#11718745	1	1	yes	characteristic
328	C₄	C₂²×#6106	1	1	yes	characteristic
329	C₄	C₂×#88200	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	104	2	no	—
334	C₂	C₂×#11718745	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718745
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3163	C₂
socle	C₂×C₁₃	C₂×#476
prime radical	C₂×#88158	C₂²
hypercenter	#3163	D₁₃
hypocenter	C₁₃	#33638
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #3163

Complements

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

N	H ≅ G/N	type
2. #13117009	C₂	semidirect
3. #13117004	C₂	semidirect
4. #13116608	C₂	semidirect
5. C₂×#11718079	C₂	semidirect
6. #13116602	C₂	semidirect
9. C₁₃×#3163	C₂	semidirect
14. C₁₃×#499	C₂²	semidirect
15. C₁₃×#498	C₂²	semidirect
29. #11718079	C₂²	semidirect
30. #11718079	C₂²	semidirect
31. #11718079	C₂²	semidirect
32. #11718055	C₂²	semidirect
47. C₂×Dic₅₂	Q₈	semidirect
50. #88158	Q₈	semidirect
52. C₂×D₁₀₄	Q₈	semidirect
119. #3163	D₁₃	semidirect
163. #499	C₂×D₁₃	semidirect
164. #498	C₂×D₁₃	semidirect
285. C₁₃	#33638	semidirect

Maximal subgroups

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

subgroup	index
#13123754	2
#13117009	2
#13117004	2
#13116608	2
C₂×#11718079	2
#13116602	2
#13116571	2
C₂×#11718080	2
C₁₃×#3163	2
#33638	13