Subgroups — #15778157

Order 3328.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718361	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718089	C₂	1	1	yes	maximal, characteristic
3	#13116541	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112931	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112929	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88158	C₂²	1	1	yes	characteristic
9	C₂×#88155	—	2	1	no	—
10	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
11	C₁₃×#427	C₂²	1	1	yes	characteristic
12	C₁₃×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88059	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	C₂×#88045	—	2	1	no	—
18	#11718361	—	2	2	no	—
19	#11718361	C₄	1	4	yes	complemented
20	C₁₃×#339	—	2	1	no	—
21	#11718089	C₄	1	4	yes	complemented
22	#11718084	C₄	1	2	yes	complemented
23	#11718083	C₄	1	2	yes	complemented
24	C₄×#6090	—	2	2	no	—
25	C₂×#6127	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#117	—	2	1	no	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×#6099	—	2	1	no	—
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	#88155	—	2	4	no	—
40	#88155	—	4	2	no	—
41	C₂×#6094	—	4	1	no	—
42	C₂×C₈×D₁₃	—	4	1	no	—
43	C₁₃×#108	—	2	1	no	—
44	C₁₃×#108	C₂×C₄	1	2	yes	—
45	C₁₃×#107	C₂×C₄	1	2	yes	—
46	C₁₃×#106	—	2	1	no	—
47	C₁₃×#97	—	4	1	no	—
48	C₄×C₈×C₁₃	—	4	1	no	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	D₄	1	2	yes	—
51	#88059	—	2	1	no	—
52	#88059	C₂×C₄	1	2	yes	—
53	C₂×#6090	—	2	1	no	—
54	C₂×#6090	—	2	1	no	—
55	C₂×#6090	—	2	1	no	—
56	C₂×#6090	—	2	1	no	—
57	C₂×#6090	C₂×C₄	1	2	yes	—
58	C₂×#6090	C₂×C₄	1	2	yes	—
59	#88045	—	2	4	no	—
60	#12641	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	4	1	no	—
64	#6127	—	2	2	no	—
65	#6127	—	2	2	no	—
66	#6127	—	4	1	no	—
67	#6127	—	4	1	no	—
68	C₂×D₅₂	—	2	1	no	—
69	C₂×C₄×D₁₃	—	4	1	no	—
70	C₂×Dic₂₆	—	2	1	no	—
71	C₁₃×M₁₆	—	2	2	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
77	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	1	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	4	1	no	—
88	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
89	#6099	—	2	2	no	—
90	#6099	—	4	1	no	—
91	C₂×#1055	—	4	1	no	—
92	Dic₅₂	—	2	1	no	—
93	Dic₅₂	C₄²	1	2	yes	—
94	D₁₀₄	—	2	1	no	—
95	D₁₀₄	C₄²	1	2	yes	—
96	#6095	—	2	2	no	—
97	#6095	—	2	2	no	—
98	#6094	—	4	4	no	—
99	C₈×D₁₃	—	4	4	no	—
100	#6090	—	2	2	no	—
101	#6090	—	2	2	no	—
102	C₂×#443	—	13	1	no	—
103	C₂×#359	—	13	1	no	—
104	#2138	—	13	1	no	—
105	C₂×C₄×C₁₆	—	13	1	no	—
106	#1772	D₁₃	1	1	yes	characteristic, complemented
107	#1382	—	13	1	no	—
108	#1381	—	13	1	no	—
109	C₂³×C₁₃	#135	1	1	yes	characteristic
110	C₂²×D₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
120	C₂²⋊D₁₃	—	4	2	no	—
121	C₂²⋊D₁₃	—	4	2	no	—
122	C₂×Dic₁₃	—	4	1	no	—
123	D₅₂	—	2	2	no	—
124	D₅₂	—	4	1	no	—
125	C₄×D₁₃	—	4	2	no	—
126	C₄×D₁₃	—	4	2	no	—
127	Dic₂₆	—	2	2	no	—
128	Dic₂₆	—	4	1	no	—
129	C₈×C₁₃	—	4	2	no	—
130	C₈×C₁₃	#102	1	2	yes	—
131	C₈×C₁₃	#102	1	2	yes	—
132	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₁₃	#117	1	1	yes	characteristic
134	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
135	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
136	#1055	—	4	2	no	—
137	C₂×#135	—	13	1	no	—
138	C₂×#131	—	26	1	no	—
139	C₂²×C₁₆	—	13	1	no	—
140	C₂²×C₁₆	—	13	1	no	—
141	#443	—	13	4	no	—
142	#443	—	26	2	no	—
143	C₂×#108	—	13	1	no	—
144	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
145	#427	C₂×D₁₃	1	1	yes	characteristic
146	C₂×#104	—	26	1	no	—
147	#405	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	#359	—	13	4	no	—
150	#358	—	13	2	no	—
151	#357	—	13	2	no	—
152	C₄×C₁₆	—	26	2	no	—
153	#339	—	2	1	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	#416	1	1	yes	characteristic
156	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
157	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
158	C₂×D₁₃	—	4	2	no	—
159	C₂×D₁₃	—	8	2	no	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
164	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
165	C₄×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
166	Dic₁₃	—	4	2	no	—
167	C₂×#41	—	26	1	no	—
168	#135	—	13	2	no	—
169	#135	—	13	2	no	—
170	#135	—	26	1	no	—
171	#135	—	26	1	no	—
172	C₂×Q₁₆	—	13	1	no	—
173	C₂×SD₁₆	—	26	1	no	—
174	C₂×D₈	—	13	1	no	—
175	#131	—	26	4	no	—
176	#131	—	52	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	26	1	no	—
179	C₂×M₁₆	—	52	1	no	—
180	C₂²×C₈	—	52	1	no	—
181	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂×C₁₆	—	13	2	no	—
186	C₂×C₁₆	—	13	2	no	—
187	C₂×C₁₆	—	26	1	no	—
188	C₂×C₁₆	—	26	1	no	—
189	C₂×C₁₆	—	26	1	no	—
190	C₂×C₁₆	—	26	1	no	—
191	#108	—	2	1	no	—
192	#108	—	13	2	no	—
193	#108	—	26	1	no	—
194	#108	C₄×D₁₃	1	2	yes	—
195	#107	C₄×D₁₃	1	2	yes	—
196	#106	—	2	1	no	—
197	#104	—	26	4	no	—
198	#97	—	4	1	no	—
199	C₄×C₈	—	4	1	no	—
200	C₄×C₈	C₄×D₁₃	1	2	yes	—
201	C₄×C₈	D₅₂	1	2	yes	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
205	C₂×C₁₃	#2178	1	2	yes	—
206	D₁₃	—	8	2	no	—
207	#41	—	26	2	no	—
208	#41	—	26	2	no	—
209	#41	—	52	1	no	—
210	#41	—	52	1	no	—
211	C₂×Q₈	—	26	1	no	—
212	C₂×D₄	—	26	1	no	—
213	C₂×D₄	—	52	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	52	1	no	—
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	Q₁₆	—	13	2	no	—
218	Q₁₆	—	26	1	no	—
219	SD₁₆	—	26	2	no	—
220	SD₁₆	—	26	2	no	—
221	D₈	—	13	2	no	—
222	D₈	—	26	1	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	26	2	no	—
226	M₁₆	—	52	1	no	—
227	M₁₆	—	52	4	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	52	1	no	—
230	C₂×C₈	—	52	4	no	—
231	C₂×C₈	#6103	1	2	yes	—
232	C₂×C₈	#6104	1	1	yes	characteristic
233	C₂×C₈	#6104	1	1	yes	characteristic
234	C₂×C₈	#6106	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
239	C₂×C₈	#6127	1	1	yes	characteristic
240	C₂×C₈	#6127	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	4	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	#6103	1	2	yes	—
246	C₁₆	—	26	2	no	—
247	C₁₆	—	26	2	no	—
248	C₁₃	#12641	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	52	1	no	—
250	C₂³	#88204	1	1	yes	characteristic
251	Q₈	—	26	2	no	—
252	Q₈	—	52	1	no	—
253	D₄	—	26	2	no	—
254	D₄	—	52	2	no	—
255	D₄	—	52	2	no	—
256	D₄	—	52	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	52	1	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	—	52	2	no	—
263	C₂×C₄	C₂×#6103	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#6104	1	1	yes	characteristic
267	C₂×C₄	C₂×#6106	1	1	yes	characteristic
268	C₂×C₄	#88204	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	52	2	no	—
271	C₈	#88049	1	2	yes	—
272	C₈	#88049	1	2	yes	—
273	C₈	C₄²×D₁₃	1	1	yes	characteristic
274	C₈	#88126	1	1	yes	characteristic
275	C₈	C₄×D₅₂	1	1	yes	characteristic
276	C₈	C₄×D₅₂	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	52	2	no	—
279	C₂²	—	104	2	no	—
280	C₂²	#11718663	1	1	yes	characteristic
281	C₂²	C₄×#6104	1	1	yes	characteristic
282	C₂²	C₄×#6106	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	8	2	no	—
285	C₄	—	52	2	no	—
286	C₄	C₄×#6103	1	1	yes	characteristic
287	C₄	C₂×#88049	1	1	yes	characteristic
288	C₄	C₄×#6104	1	1	yes	characteristic
289	C₄	C₄×#6106	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	104	2	no	—
293	C₂	C₄×#88049	1	1	yes	characteristic
294	C₂	#13116715	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#6104
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1772	C₂
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1772	D₁₃
hypocenter	C₁₃	#12641
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×#6090	C₂	semidirect
5. #13112931	C₂	semidirect
6. #13112929	C₂	semidirect
7. C₁₃×#1772	C₂	semidirect
19. #11718361	C₄	semidirect
21. #11718089	C₄	semidirect
22. #11718084	C₄	semidirect
23. #11718083	C₄	semidirect
106. #1772	D₁₃	semidirect
248. C₁₃	#12641	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718361	2
C₂×#11718089	2
#13116541	2
C₂×C₄×#6090	2
#13112931	2
#13112929	2
C₁₃×#1772	2
#12641	13