Subgroups — #15778147

Order 3328.

442 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123607	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718102	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718089	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718094	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112931	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88158	C₂²	1	1	yes	characteristic
9	#11719227	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₁₃×#427	C₂²	1	1	yes	characteristic
13	M₁₆×Dic₁₃	—	2	1	no	—
14	C₂×#88059	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	#11718663	—	2	1	no	—
18	#11718534	C₄	1	2	yes	—
19	#11718533	C₄	1	2	yes	—
20	#11718513	—	2	2	no	—
21	#11718150	—	2	1	no	—
22	#11718102	C₄	1	4	yes	complemented
23	#11718094	—	2	2	no	—
24	#11718089	C₄	1	4	yes	complemented
25	C₂×#6127	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#117	—	2	1	no	—
29	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×#6099	—	2	1	no	—
31	C₂²×#1055	D₄	1	1	yes	characteristic
32	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
33	#88158	—	2	1	no	—
34	#88158	—	2	1	no	—
35	#88158	C₂×C₄	1	2	yes	—
36	#88158	C₂×C₄	1	2	yes	—
37	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
38	C₂×#6095	—	2	1	no	—
39	#88147	—	2	2	no	—
40	#88141	—	2	2	no	—
41	#88135	—	4	2	no	—
42	#88126	—	4	1	no	—
43	C₄²×D₁₃	—	4	1	no	—
44	C₂×C₁₃×C₁₆	—	2	1	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
47	C₁₃×#108	—	2	1	no	—
48	C₁₃×#108	C₂×C₄	1	2	yes	complemented
49	C₁₃×#107	C₂×C₄	1	2	yes	—
50	C₁₃×#106	—	2	1	no	—
51	#88059	—	2	1	no	—
52	#88059	C₂×C₄	1	2	yes	complemented
53	#88055	—	4	1	no	—
54	C₈×Dic₁₃	—	4	1	no	—
55	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	D₄	1	2	yes	—
57	C₂×#6090	—	2	1	no	—
58	C₂×#6090	—	2	1	no	—
59	C₂×#6090	C₂×C₄	1	2	yes	complemented
60	#88050	—	2	2	no	—
61	#88049	—	2	2	no	—
62	#12640	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	—	4	1	no	—
65	C₂²×Dic₁₃	SD₁₆	1	2	yes	—
66	#6127	—	2	2	no	—
67	#6127	—	2	2	no	—
68	#6127	—	4	1	no	—
69	#6127	—	4	1	no	—
70	C₂×D₅₂	—	2	1	no	—
71	C₂×C₄×D₁₃	—	4	1	no	—
72	C₂×C₄×D₁₃	—	4	2	no	—
73	C₂×Dic₂₆	—	2	1	no	—
74	C₁₃×M₁₆	—	2	2	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₂²⋊C₄	—	4	1	no	—
85	C₄²×C₁₃	—	4	1	no	—
86	#6103	—	4	2	no	—
87	#6101	—	4	2	no	—
88	C₄×Dic₁₃	—	4	2	no	—
89	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
90	C₄×Dic₁₃	D₈	1	1	yes	characteristic
91	C₄×Dic₁₃	D₈	1	1	yes	characteristic
92	#6099	—	2	2	no	—
93	#6099	—	4	1	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	—	4	1	no	—
97	C₂×#1055	C₂²⋊C₄	1	2	yes	—
98	Dic₅₂	—	2	1	no	—
99	Dic₅₂	C₄²	1	2	yes	—
100	D₁₀₄	—	2	1	no	—
101	D₁₀₄	C₄²	1	2	yes	—
102	#6095	—	2	2	no	—
103	#6095	—	2	2	no	—
104	C₁₃×C₁₆	—	2	2	no	—
105	#6090	—	2	2	no	—
106	C₄×#135	—	13	1	no	—
107	C₂×#359	—	13	1	no	—
108	C₂×#359	—	13	1	no	—
109	C₂×#346	—	13	1	no	—
110	C₄×#108	—	13	1	no	—
111	#1382	—	13	1	no	—
112	#1382	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂²×D₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
122	C₂²⋊D₁₃	—	4	2	no	—
123	C₂²⋊D₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	1	no	—
126	C₂×Dic₁₃	#102	1	2	yes	—
127	C₂×Dic₁₃	#102	1	2	yes	—
128	D₅₂	—	2	2	no	—
129	D₅₂	—	4	1	no	—
130	C₄×D₁₃	—	4	2	no	—
131	C₄×D₁₃	—	4	2	no	—
132	C₄×D₁₃	—	8	4	no	—
133	Dic₂₆	—	2	2	no	—
134	Dic₂₆	—	4	1	no	—
135	C₈×C₁₃	—	4	2	no	—
136	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
137	C₈×C₁₃	#117	1	1	yes	characteristic
138	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
140	#1055	—	2	2	no	—
141	#1055	—	4	2	no	—
142	C₂×#135	—	13	1	no	—
143	C₄×#41	—	26	1	no	—
144	C₂²×C₁₆	—	13	1	no	—
145	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
146	C₄×Q₁₆	—	13	2	no	—
147	C₄×SD₁₆	—	26	2	no	—
148	C₄×D₈	—	13	2	no	—
149	C₂×#108	—	13	1	no	—
150	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
151	#427	C₂×D₁₃	1	1	yes	characteristic
152	#416	—	26	1	no	—
153	C₄×M₁₆	—	26	1	no	—
154	C₂×C₄×C₈	—	13	1	no	—
155	#359	—	13	4	no	—
156	#359	—	13	4	no	—
157	#346	—	26	2	no	—
158	#341	—	26	1	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
161	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
162	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
163	C₂×D₁₃	—	4	2	no	—
164	C₂×D₁₃	—	8	2	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₁₃	#416	1	1	yes	characteristic
168	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
169	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
170	Dic₁₃	—	2	2	no	—
171	Dic₁₃	—	2	2	no	—
172	Dic₁₃	—	4	2	no	—
173	C₂×#41	—	26	1	no	—
174	#135	—	13	2	no	—
175	#135	—	13	2	no	—
176	#135	—	26	1	no	—
177	#135	—	26	1	no	—
178	C₂×Q₁₆	—	13	1	no	—
179	C₂×SD₁₆	—	26	1	no	—
180	C₂×D₈	—	13	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	26	1	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	C₄×Q₈	—	26	2	no	—
186	C₄×D₄	—	26	2	no	—
187	C₄×D₄	—	52	2	no	—
188	#117	—	2	1	no	—
189	#117	—	52	1	no	—
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	—	52	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	26	1	no	—
196	C₂×C₁₆	—	26	1	no	—
197	C₂×C₁₆	C₄×D₁₃	1	2	yes	complemented
198	#108	—	2	1	no	—
199	#108	—	13	2	no	—
200	#108	—	26	1	no	—
201	#108	C₄×D₁₃	1	2	yes	complemented
202	#107	C₄×D₁₃	1	2	yes	—
203	#106	—	2	1	no	—
204	#103	—	26	2	no	—
205	#102	—	26	2	no	—
206	#97	—	52	1	no	—
207	C₄×C₈	—	13	2	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	52	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	2	1	no	—
212	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
213	C₂×C₁₃	#2143	1	2	yes	—
214	D₁₃	—	8	2	no	—
215	#41	—	26	2	no	—
216	#41	—	26	2	no	—
217	#41	—	52	1	no	—
218	#41	—	52	1	no	—
219	C₂×Q₈	—	26	1	no	—
220	C₂×D₄	—	26	1	no	—
221	C₂×D₄	—	52	1	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	52	2	no	—
224	C₂²×C₄	—	52	1	no	—
225	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
226	Q₁₆	—	13	2	no	—
227	Q₁₆	—	26	1	no	—
228	SD₁₆	—	26	2	no	—
229	SD₁₆	—	26	2	no	—
230	D₈	—	13	2	no	—
231	D₈	—	26	1	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	4	1	no	—
234	M₁₆	—	26	2	no	—
235	M₁₆	—	52	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	26	1	no	—
239	C₂×C₈	—	26	1	no	—
240	C₂×C₈	—	52	1	no	—
241	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
244	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
245	C₂×C₈	#6129	1	1	yes	characteristic
246	C₂×C₈	#6131	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	52	2	no	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	52	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	13	2	no	—
253	C₄²	—	13	1	no	—
254	C₄²	—	13	1	no	—
255	C₄²	—	52	2	no	—
256	C₁₆	—	2	2	no	—
257	C₁₆	—	26	2	no	—
258	C₁₃	#12640	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	52	1	no	—
260	C₂³	SD₁₆×D₁₃	1	1	yes	characteristic
261	Q₈	—	26	2	no	—
262	Q₈	—	52	1	no	—
263	D₄	—	26	2	no	—
264	D₄	—	52	2	no	—
265	D₄	—	52	2	no	—
266	D₄	—	52	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	52	1	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	52	2	no	—
274	C₂×C₄	—	104	4	no	—
275	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	#88135	1	1	yes	characteristic
277	C₂×C₄	#88147	1	1	yes	characteristic
278	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
279	C₂×C₄	#88170	1	1	yes	characteristic
280	C₂×C₄	#88173	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	26	2	no	—
283	C₈	—	52	2	no	—
284	C₈	C₄²×D₁₃	1	1	yes	characteristic
285	C₈	#88135	1	1	yes	characteristic
286	C₈	#88146	1	1	yes	characteristic
287	C₈	#88147	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	52	2	no	—
290	C₂²	—	104	2	no	—
291	C₂²	D₁₃×#102	1	1	yes	characteristic
292	C₂²	#11718449	1	1	yes	characteristic
293	C₂²	#11718513	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	26	2	no	—
296	C₄	—	26	2	no	—
297	C₄	—	52	2	no	—
298	C₄	#11718327	1	1	yes	characteristic
299	C₄	#11718417	1	1	yes	characteristic
300	C₄	#11718465	1	1	yes	characteristic
301	C₄	#11718533	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	104	2	no	—
305	C₂	#13114650	1	1	yes	characteristic
306	C₂	#13116562	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1382	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1382	D₁₃
hypocenter	C₁₃	#12640
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₂×#11718094	C₂	semidirect
5. #108×Dic₁₃	C₂	semidirect
6. #13112931	C₂	semidirect
7. C₁₃×#1382	C₂	semidirect
22. #11718102	C₄	semidirect
24. #11718089	C₄	semidirect
46. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
48. C₁₃×#108	C₂×C₄	semidirect
52. #88059	C₂×C₄	semidirect
59. C₂×#6090	C₂×C₄	semidirect
112. #1382	D₁₃	semidirect
197. C₂×C₁₆	C₄×D₁₃	semidirect
201. #108	C₄×D₁₃	semidirect
258. C₁₃	#12640	semidirect

Maximal subgroups

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

subgroup	index
#13123607	2
C₂×#11718102	2
C₂×#11718089	2
C₂×#11718094	2
#108×Dic₁₃	2
#13112931	2
C₁₃×#1382	2
#12640	13