Subgroups — #15778221

Order 3328.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718378	C₂	1	1	yes	maximal, characteristic
2	#13116583	C₂	1	1	yes	maximal, characteristic
3	#13116563	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718058	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112947	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1778	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	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	—	2	1	no	—
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₂×#88059	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	C₂×#88052	C₂²	1	1	yes	characteristic
17	C₂×#88045	—	2	1	no	—
18	#11718378	—	2	2	no	—
19	#11718378	C₄	1	4	yes	—
20	C₁₃×#341	—	2	1	no	—
21	#11718089	C₄	1	4	yes	complemented
22	#11718088	C₄	1	2	yes	complemented
23	#11718087	C₄	1	2	yes	complemented
24	#11718058	—	2	2	no	—
25	C₂×#6127	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
45	C₁₃×#108	C₂×C₄	1	2	yes	—
46	C₁₃×#108	C₂×C₄	1	2	yes	—
47	C₁₃×#97	—	4	1	no	—
48	C₁₃×#97	C₂×C₄	1	2	yes	—
49	C₁₃×#97	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	—	4	1	no	—
51	#88059	—	2	1	no	—
52	#88059	C₂×C₄	1	2	yes	—
53	#88052	—	2	1	no	—
54	#88052	—	2	1	no	—
55	#88052	—	2	1	no	—
56	#88052	C₂×C₄	1	2	yes	—
57	C₂×#6090	—	2	1	no	—
58	C₂×#6090	C₂×C₄	1	2	yes	—
59	#88045	—	2	4	no	—
60	#12651	—	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	—	2	2	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₁₃×M₁₆	—	4	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
88	#6099	—	2	2	no	—
89	#6099	—	4	1	no	—
90	C₂×#1055	—	4	1	no	—
91	Dic₅₂	—	2	1	no	—
92	Dic₅₂	C₄²	1	2	yes	—
93	D₁₀₄	—	2	1	no	—
94	D₁₀₄	C₄²	1	2	yes	—
95	#6095	—	2	2	no	—
96	#6095	—	2	2	no	—
97	#6094	—	4	4	no	—
98	C₈×D₁₃	—	4	4	no	—
99	#6090	—	2	2	no	—
100	#6090	—	2	2	no	—
101	C₂×#444	—	13	1	no	—
102	#2148	—	13	1	no	—
103	#2143	—	13	1	no	—
104	C₂×#347	—	13	1	no	—
105	#1778	D₁₃	1	1	yes	characteristic, complemented
106	#1388	—	13	1	no	—
107	#1385	—	13	1	no	—
108	C₂³×C₁₃	#137	1	1	yes	characteristic
109	C₂²×D₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂²⋊D₁₃	—	4	2	no	—
119	C₂²⋊D₁₃	—	4	2	no	—
120	C₂×Dic₁₃	—	4	1	no	—
121	D₅₂	—	2	2	no	—
122	D₅₂	—	4	1	no	—
123	C₄×D₁₃	—	4	2	no	—
124	C₄×D₁₃	—	4	2	no	—
125	Dic₂₆	—	2	2	no	—
126	Dic₂₆	—	4	1	no	—
127	C₈×C₁₃	—	2	2	no	—
128	C₈×C₁₃	—	4	2	no	—
129	C₈×C₁₃	—	4	2	no	—
130	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₁₃	#117	1	1	yes	characteristic
132	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
134	#1055	—	4	2	no	—
135	C₂×#135	—	13	1	no	—
136	C₂×#131	—	26	1	no	—
137	C₂×M₃₂	—	13	1	no	—
138	C₂×M₃₂	—	13	1	no	—
139	#444	—	13	4	no	—
140	#444	—	26	2	no	—
141	C₂×#108	—	13	1	no	—
142	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
143	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
144	C₂×#104	—	26	1	no	—
145	C₄×M₁₆	—	2	1	no	—
146	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
147	#362	—	13	2	no	—
148	#361	—	13	2	no	—
149	#359	—	13	4	no	—
150	#347	—	26	2	no	—
151	#341	—	2	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	#417	1	1	yes	characteristic
154	C₂²×C₁₃	#440	1	1	yes	characteristic
155	C₂²×C₁₃	#440	1	1	yes	characteristic
156	C₂×D₁₃	—	4	2	no	—
157	C₂×D₁₃	—	8	2	no	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₁₃	#418	1	1	yes	characteristic
161	C₄×C₁₃	#442	1	1	yes	characteristic
162	C₄×C₁₃	#442	1	1	yes	characteristic
163	Dic₁₃	—	4	2	no	—
164	C₂×#41	—	26	1	no	—
165	#135	—	13	2	no	—
166	#135	—	13	2	no	—
167	#135	—	26	1	no	—
168	#135	—	26	1	no	—
169	C₂×Q₁₆	—	13	1	no	—
170	C₂×SD₁₆	—	26	1	no	—
171	C₂×D₈	—	13	1	no	—
172	#131	—	26	4	no	—
173	#131	—	52	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	26	1	no	—
177	C₂×M₁₆	—	52	1	no	—
178	C₂²×C₈	—	52	1	no	—
179	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
182	M₃₂	—	13	2	no	—
183	M₃₂	—	26	1	no	—
184	M₃₂	—	26	1	no	—
185	M₃₂	—	26	1	no	—
186	C₂×C₁₆	—	13	2	no	—
187	C₂×C₁₆	—	26	1	no	—
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	—	13	2	no	—
191	#108	—	26	1	no	—
192	#108	C₄×D₁₃	1	2	yes	—
193	#108	C₄×D₁₃	1	2	yes	—
194	#104	—	26	4	no	—
195	#97	—	4	1	no	—
196	#97	C₄×D₁₃	1	2	yes	—
197	#97	D₅₂	1	2	yes	—
198	C₄×C₈	—	4	1	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	#1762	1	1	yes	characteristic
203	D₁₃	—	8	2	no	—
204	#41	—	26	2	no	—
205	#41	—	26	2	no	—
206	#41	—	52	1	no	—
207	#41	—	52	1	no	—
208	C₂×Q₈	—	26	1	no	—
209	C₂×D₄	—	26	1	no	—
210	C₂×D₄	—	52	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	52	1	no	—
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	Q₁₆	—	13	2	no	—
215	Q₁₆	—	26	1	no	—
216	SD₁₆	—	26	2	no	—
217	SD₁₆	—	26	2	no	—
218	D₈	—	13	2	no	—
219	D₈	—	26	1	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	26	2	no	—
225	M₁₆	—	52	4	no	—
226	M₁₆	—	52	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	52	4	no	—
232	C₂×C₈	—	52	1	no	—
233	C₂×C₈	#6103	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
238	C₂×C₈	#6127	1	1	yes	characteristic
239	C₂×C₈	#6127	1	1	yes	characteristic
240	C₄²	—	2	1	no	—
241	C₄²	#6103	1	2	yes	—
242	C₁₆	—	26	2	no	—
243	C₁₆	—	26	2	no	—
244	C₁₃	#12651	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	52	1	no	—
246	C₂³	#88196	1	1	yes	characteristic
247	Q₈	—	26	2	no	—
248	Q₈	—	52	1	no	—
249	D₄	—	26	2	no	—
250	D₄	—	52	2	no	—
251	D₄	—	52	2	no	—
252	D₄	—	52	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	52	1	no	—
256	C₂×C₄	—	52	2	no	—
257	C₂×C₄	—	52	2	no	—
258	C₂×C₄	C₂×#6103	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂×C₄	#88186	1	1	yes	characteristic
262	C₂×C₄	#88203	1	1	yes	characteristic
263	C₂×C₄	#88203	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	52	2	no	—
268	C₈	C₄²×D₁₃	1	1	yes	characteristic
269	C₈	#88126	1	1	yes	characteristic
270	C₈	C₄×D₅₂	1	1	yes	characteristic
271	C₈	C₄×D₅₂	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	52	2	no	—
274	C₂²	—	104	2	no	—
275	C₂²	#11718627	1	1	yes	characteristic
276	C₂²	#11718687	1	1	yes	characteristic
277	C₂²	#11718687	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	52	2	no	—
280	C₄	C₄×#6103	1	1	yes	characteristic
281	C₄	#11718662	1	1	yes	characteristic
282	C₄	#11718675	1	1	yes	characteristic
283	C₄	#11718675	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	104	2	no	—
288	C₂	#13114664	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718675
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1778	C₂
socle	C₂×C₁₃	#1762
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1778	D₁₃
hypocenter	C₁₃	#12651
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1778

Complements

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

N	H ≅ G/N	type
4. C₂×#11718058	C₂	semidirect
5. #13112947	C₂	semidirect
6. #13112939	C₂	semidirect
7. C₁₃×#1778	C₂	semidirect
21. #11718089	C₄	semidirect
22. #11718088	C₄	semidirect
23. #11718087	C₄	semidirect
105. #1778	D₁₃	semidirect
244. C₁₃	#12651	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718378	2
#13116583	2
#13116563	2
C₂×#11718058	2
#13112947	2
#13112939	2
C₁₃×#1778	2
#12651	13