Subgroups — #16017671

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	#13125368	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123753	C₂	1	1	yes	maximal, characteristic
3	#13117010	C₂	1	4	yes	maximal, complemented
4	#13117005	C₂	1	4	yes	maximal, complemented
5	#13116608	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718105	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116601	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718158	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2145	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
11	#11719524	—	2	1	no	—
12	#11719224	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#427	C₂²	1	1	yes	characteristic
15	#11718897	—	2	1	no	—
16	#11718858	—	2	2	no	—
17	#11718857	C₂²	1	4	yes	—
18	C₂×#88059	C₂²	1	1	yes	characteristic
19	C₂×#88057	C₂²	1	1	yes	characteristic
20	#11718762	C₂²	1	1	yes	characteristic
21	C₂×#88052	C₂²	1	1	yes	characteristic
22	#11718656	—	2	1	no	—
23	#11718537	—	2	2	no	—
24	#11718519	C₂²	1	2	yes	complemented
25	#11718516	C₂²	1	2	yes	complemented
26	C₁₃×#360	C₂²	1	2	yes	—
27	C₁₃×#360	C₂²	1	2	yes	—
28	#11718158	C₂²	1	4	yes	—
29	#11718105	C₂²	1	4	yes	complemented
30	#11718103	C₂²	1	4	yes	complemented
31	#11718079	C₂²	1	2	yes	complemented
32	#11718079	C₂²	1	2	yes	complemented
33	C₂×C₁₃×#41	—	2	1	no	—
34	C₁₃×#135	—	2	1	no	—
35	C₁₃×#135	—	2	1	no	—
36	C₁₃×#135	Q₈	1	2	yes	complemented
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×C₁₃×SD₁₆	—	2	1	no	—
40	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	—	2	1	no	—
43	#88202	—	2	4	no	—
44	#88202	—	4	2	no	—
45	#88180	D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×#6099	—	2	1	no	—
48	C₂×#6099	—	4	2	no	—
49	C₂×#6099	D₄	1	1	yes	characteristic
50	#88144	—	2	2	no	—
51	#88142	—	2	2	no	—
52	#88132	—	4	2	no	—
53	#88124	—	4	1	no	—
54	#88123	—	4	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₁₃×M₃₂	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	—	2	1	no	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#106	C₂³	1	1	yes	characteristic
60	C₁₃×#106	C₂³	1	1	yes	characteristic
61	#88077	—	2	4	no	—
62	#88059	—	2	1	no	—
63	#88059	C₂³	1	2	yes	—
64	#88057	—	2	1	no	—
65	#88057	C₂³	1	2	yes	—
66	#88055	D₄	1	2	yes	—
67	C₈×Dic₁₃	C₂³	1	2	yes	—
68	#88052	—	2	1	no	—
69	#88052	C₂³	1	2	yes	—
70	C₂×#6090	—	2	1	no	—
71	#88048	—	2	2	no	—
72	#88047	—	2	2	no	—
73	#33638	—	13	1	no	maximal, Sylow
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₁₃×#41	—	4	1	no	—
77	C₁₃×#41	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	2	1	no	—
79	C₂×D₄×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₁₃	—	4	1	no	—
84	C₂×Dic₂₆	—	4	2	no	—
85	C₁₃×Q₁₆	—	2	1	no	—
86	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
87	C₁₃×SD₁₆	—	2	2	no	—
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×D₈	—	2	1	no	—
90	C₁₃×D₈	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	1	no	—
100	C₄²×C₁₃	—	4	1	no	—
101	C₂²⋊Dic₁₃	—	2	1	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	4	2	no	—
105	C₄⋊Dic₁₃	—	4	1	no	—
106	#6101	—	4	2	no	—
107	#6101	—	4	2	no	—
108	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
109	#6099	—	2	2	no	—
110	#6099	—	4	1	no	—
111	#6099	—	4	4	no	—
112	#6099	SD₁₆	1	2	yes	—
113	#6099	SD₁₆	1	2	yes	—
114	C₂×#1055	—	4	1	no	—
115	C₂×#1055	—	4	4	no	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	C₁₃×C₁₆	—	2	2	no	—
118	#6090	—	2	2	no	—
119	#3163	—	13	1	no	—
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	D₁₃	1	1	yes	characteristic
127	C₂×#360	—	13	1	no	—
128	C₂³×C₁₃	—	4	1	no	—
129	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Q₈×C₁₃	—	2	2	no	—
131	Q₈×C₁₃	—	4	1	no	—
132	D₄×C₁₃	—	2	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	16	1	no	—
150	Dic₂₆	—	16	2	no	—
151	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
152	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	#1055	—	4	2	no	—
156	#1055	C₂×SD₁₆	1	4	yes	—
157	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
158	#568	—	26	1	no	—
159	#544	—	26	1	no	—
160	C₂×M₃₂	—	13	1	no	—
161	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
162	#501	—	26	2	no	—
163	#499	—	13	2	no	—
164	#498	—	13	2	no	—
165	#444	—	26	2	no	—
166	#443	—	13	4	no	—
167	C₂×#108	—	13	1	no	—
168	#428	—	26	1	no	—
169	#427	C₂×D₁₃	1	1	yes	characteristic
170	C₂×#106	—	13	1	no	—
171	#421	—	26	1	no	—
172	#405	—	13	1	no	—
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	C₂×D₁₃	1	2	yes	—
179	#360	C₂×D₁₃	1	2	yes	—
180	C₂²×C₁₃	—	4	1	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	16	1	no	—
183	C₂²×C₁₃	#476	1	1	yes	characteristic
184	C₂²×C₁₃	#476	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	8	1	no	—
188	C₄×C₁₃	—	8	1	no	—
189	C₄×C₁₃	#476	1	1	yes	characteristic
190	C₄×C₁₃	#476	1	1	yes	characteristic
191	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	—	8	2	no	—
195	C₂×#41	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	—	2	1	no	—
198	#135	Dic₂₆	1	2	yes	complemented
199	#135	C₂²×D₁₃	1	2	yes	—
200	C₂×Q₁₆	Dic₂₆	1	1	yes	characteristic, complemented
201	C₂×SD₁₆	—	2	1	no	—
202	C₂×D₈	Dic₂₆	1	1	yes	characteristic, complemented
203	#131	—	26	4	no	—
204	#131	—	52	2	no	—
205	C₂×M₁₆	—	13	1	no	—
206	C₂×M₁₆	—	26	1	no	—
207	C₂×M₁₆	—	52	2	no	—
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊Q₈	—	26	2	no	—
210	C₄⋊Q₈	—	52	1	no	—
211	#125	—	26	2	no	—
212	#125	—	52	1	no	—
213	C₂²⋊Q₈	—	52	2	no	—
214	#117	—	2	1	no	—
215	#117	—	13	1	no	—
216	C₂×C₄⋊C₄	—	26	1	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	13	2	no	—
219	M₃₂	—	26	1	no	—
220	M₃₂	C₂²×D₁₃	1	2	yes	—
221	C₂×C₁₆	—	2	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	—	26	2	no	—
227	#106	—	13	2	no	—
228	#106	—	26	2	no	—
229	#106	—	26	1	no	—
230	#106	C₂²×D₁₃	1	1	yes	characteristic
231	#106	C₂²×D₁₃	1	1	yes	characteristic
232	#104	—	26	4	no	—
233	#97	—	13	2	no	—
234	C₄×C₈	—	13	2	no	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	8	2	no	—
239	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	52	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	52	1	no	—
250	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	C₂×Dic₂₆	1	2	yes	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	C₂×Dic₂₆	1	2	yes	—
257	M₁₆	—	13	2	no	—
258	M₁₆	—	13	2	no	—
259	M₁₆	—	26	2	no	—
260	M₁₆	—	52	4	no	—
261	M₁₆	—	52	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	52	4	no	—
264	C₂×C₈	—	52	1	no	—
265	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
266	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
267	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
268	C₂×C₈	#6129	1	1	yes	characteristic
269	C₂×C₈	#6129	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	26	1	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	52	2	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₄⋊C₄	—	52	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	26	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	13	2	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	26	2	no	—
285	C₁₃	#33638	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	4	1	no	—
287	C₂³	#88165	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	208	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	208	1	no	—
303	C₂×C₄	#88132	1	1	yes	characteristic
304	C₂×C₄	#88132	1	1	yes	characteristic
305	C₂×C₄	#88165	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₈	—	13	4	no	—
310	C₈	—	52	2	no	—
311	C₈	#88132	1	1	yes	characteristic
312	C₈	#88132	1	1	yes	characteristic
313	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
314	C₈	C₂×#6129	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#11718422	1	1	yes	characteristic
319	C₂²	#11718422	1	1	yes	characteristic
320	C₂²	C₂×#88165	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	52	2	no	—
325	C₄	—	104	2	no	—
326	C₄	#11718422	1	1	yes	characteristic
327	C₄	#11718422	1	1	yes	characteristic
328	C₄	C₂×#88132	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#11718422	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718422
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2145	C₂
socle	C₂×C₁₃	C₂×#476
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	#2145	D₁₃
hypocenter	C₁₃	#33638
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
1. #13125368	C₂	semidirect
3. #13117010	C₂	semidirect
4. #13117005	C₂	semidirect
5. #13116608	C₂	semidirect
6. C₂×#11718105	C₂	semidirect
7. #13116601	C₂	semidirect
24. #11718519	C₂²	semidirect
25. #11718516	C₂²	semidirect
29. #11718105	C₂²	semidirect
30. #11718103	C₂²	semidirect
31. #11718079	C₂²	semidirect
32. #11718079	C₂²	semidirect
36. C₁₃×#135	Q₈	semidirect
38. C₂×C₁₃×Q₁₆	Q₈	semidirect
40. C₂×C₁₃×D₈	Q₈	semidirect
198. #135	Dic₂₆	semidirect
200. C₂×Q₁₆	Dic₂₆	semidirect
202. C₂×D₈	Dic₂₆	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
#13125368	2
#13123753	2
#13117010	2
#13117005	2
#13116608	2
C₂×#11718105	2
#13116601	2
C₂×#11718158	2
C₁₃×#2145	2
#33638	13