Subgroups — #16784445

Order 3328.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130768	C₂	1	1	yes	maximal, characteristic
2	#13130622	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718255	C₂	1	2	yes	maximal
4	#13129666	C₂	1	2	yes	maximal
5	#13114684	C₂	1	1	yes	maximal, characteristic
6	C₂×#88110	—	2	1	no	—
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	—	2	1	no	—
9	C₂×C₄⋊#1057	C₂²	1	2	yes	—
10	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
12	C₂×#88054	C₄	1	2	yes	—
13	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718290	—	2	1	no	—
16	#11718255	—	2	4	no	—
17	C₁₃×#336	C₄	1	1	yes	characteristic
18	#11718116	C₄	1	1	yes	characteristic
19	C₂²×#1057	—	2	1	no	—
20	C₂²×#1057	—	2	1	no	—
21	C₂²×#1057	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	#88110	—	4	2	no	—
32	C₄⋊#1057	—	2	2	no	—
33	C₄⋊#1057	—	2	1	no	—
34	C₄⋊#1057	—	2	1	no	—
35	C₄⋊#1057	—	2	2	no	—
36	C₄⋊#1057	—	4	2	no	—
37	C₄⋊#1057	—	4	2	no	—
38	C₄⋊#1057	D₄	1	2	yes	—
39	C₄⋊#1057	Q₈	1	2	yes	—
40	#88054	—	2	4	no	—
41	C₈×Dic₁₃	—	2	2	no	—
42	C₈×Dic₁₃	—	2	2	no	—
43	#9605	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×#1057	—	2	2	no	—
52	C₂×#1057	—	2	2	no	—
53	C₂×#1057	—	4	1	no	—
54	C₂×#1057	—	4	1	no	—
55	C₂×#1057	—	4	1	no	—
56	C₂×#1057	—	4	1	no	—
57	C₂×#1057	—	4	2	no	—
58	C₂×#1057	—	4	2	no	—
59	C₂×#1057	—	4	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	2	2	no	—
66	C₄×Dic₁₃	—	2	1	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	—	4	1	no	—
69	C₄×Dic₁₃	SD₁₆	1	2	yes	—
70	C₄×Dic₁₃	SD₁₆	1	2	yes	—
71	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₄×Dic₁₃	#41	1	1	yes	characteristic
74	C₄×Dic₁₃	#41	1	1	yes	characteristic
75	C₂×#1055	—	2	2	no	—
76	C₂×#1055	—	2	2	no	—
77	C₂×#1055	M₁₆	1	4	yes	—
78	#1842	—	13	1	no	—
79	#1767	—	13	1	no	—
80	C₄×C₄⋊C₈	—	13	1	no	—
81	C₂×#334	—	13	2	no	—
82	#1277	—	13	2	no	—
83	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	#106	1	2	yes	—
89	C₂×C₄×C₁₃	#108	1	2	yes	—
90	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₁₃	#131	1	2	yes	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	1	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	#106	1	2	yes	—
102	C₂×Dic₁₃	#121	1	1	yes	characteristic
103	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×Dic₁₃	#125	1	1	yes	characteristic
105	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×Dic₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×Dic₁₃	#137	1	1	yes	characteristic
110	#1057	—	4	2	no	—
111	#1057	—	4	2	no	—
112	#1057	—	8	2	no	—
113	C₈×C₁₃	—	4	2	no	—
114	#1055	—	2	4	no	—
115	C₂×C₄⋊C₈	—	13	1	no	—
116	C₂×C₄⋊C₈	—	13	1	no	—
117	C₂×C₄⋊C₈	—	13	2	no	—
118	C₂×C₄⋊C₈	—	13	2	no	—
119	C₂×C₄⋊C₈	—	26	1	no	—
120	C₂×C₄⋊C₈	—	26	1	no	—
121	C₂×#97	—	26	1	no	—
122	C₂×C₄×C₈	—	13	2	no	—
123	C₄³	—	13	1	no	—
124	#336	—	13	1	no	—
125	#336	—	26	1	no	—
126	#336	#261	1	1	yes	characteristic
127	#334	—	26	4	no	—
128	C₂²×C₁₃	#389	1	1	yes	characteristic
129	C₂²×C₁₃	#423	1	1	yes	characteristic
130	C₂²×C₁₃	#424	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
132	C₂²×C₁₃	#428	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
134	C₂²×C₁₃	#430	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	#435	1	2	yes	—
138	C₄×C₁₃	#446	1	2	yes	—
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	2	1	no	—
141	Dic₁₃	—	2	1	no	—
142	Dic₁₃	—	4	1	no	—
143	Dic₁₃	#460	1	1	yes	characteristic
144	Dic₁₃	#461	1	1	yes	characteristic
145	Dic₁₃	#475	1	1	yes	characteristic
146	Dic₁₃	#476	1	1	yes	characteristic
147	C₂²×C₈	—	13	2	no	—
148	C₂²×C₈	—	26	1	no	—
149	C₂²×C₈	—	26	1	no	—
150	C₂²×C₈	—	26	2	no	—
151	C₂²×C₈	C₂×#261	1	2	yes	—
152	C₂×C₄²	—	13	1	no	—
153	C₂×C₄²	—	13	1	no	—
154	C₂×C₄²	—	13	1	no	—
155	C₂×C₄²	—	13	1	no	—
156	C₂×C₄²	—	13	1	no	—
157	C₂×C₄²	—	13	1	no	—
158	C₂×C₄²	C₂×#261	1	1	yes	characteristic
159	C₄⋊C₈	—	13	2	no	—
160	C₄⋊C₈	—	13	2	no	—
161	C₄⋊C₈	—	26	1	no	—
162	C₄⋊C₈	—	26	1	no	—
163	C₄⋊C₈	—	26	2	no	—
164	C₄⋊C₈	—	26	2	no	—
165	C₄⋊C₈	—	26	4	no	—
166	C₄⋊C₈	—	52	2	no	—
167	C₄⋊C₈	—	52	2	no	—
168	#97	—	52	2	no	—
169	C₄×C₈	—	26	2	no	—
170	C₄×C₈	—	26	2	no	—
171	C₂×C₁₃	#1566	1	1	yes	characteristic
172	C₂×C₁₃	#1569	1	1	yes	characteristic
173	C₂×C₁₃	#1570	1	1	yes	characteristic
174	C₂×C₁₃	#1573	1	1	yes	characteristic
175	C₂×C₁₃	#1940	1	1	yes	characteristic
176	C₂×C₁₃	#1948	1	1	yes	characteristic
177	C₂×C₁₃	#1950	1	1	yes	characteristic
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
184	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	13	4	no	—
189	C₂×C₈	—	26	2	no	—
190	C₂×C₈	—	26	2	no	—
191	C₂×C₈	—	26	2	no	—
192	C₂×C₈	—	26	2	no	—
193	C₂×C₈	—	52	1	no	—
194	C₂×C₈	—	52	1	no	—
195	C₂×C₈	—	52	1	no	—
196	C₂×C₈	—	52	1	no	—
197	C₂×C₈	—	52	2	no	—
198	C₂×C₈	—	52	2	no	—
199	C₂×C₈	—	52	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	13	2	no	—
202	C₄²	—	13	2	no	—
203	C₄²	—	13	1	no	—
204	C₄²	—	13	1	no	—
205	C₄²	—	13	1	no	—
206	C₄²	—	13	1	no	—
207	C₄²	—	26	1	no	—
208	C₄²	—	26	1	no	—
209	C₄²	—	26	2	no	—
210	C₄²	—	26	2	no	—
211	C₄²	—	52	1	no	—
212	C₁₃	#9605	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	13	1	no	—
219	C₂×C₄	—	13	1	no	—
220	C₂×C₄	—	13	1	no	—
221	C₂×C₄	—	13	1	no	—
222	C₂×C₄	—	13	1	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	1	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	26	1	no	—
228	C₂×C₄	—	26	1	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	26	1	no	—
231	C₂×C₄	—	26	1	no	—
232	C₂×C₄	—	26	1	no	—
233	C₂×C₄	—	26	1	no	—
234	C₂×C₄	#88101	1	2	yes	—
235	C₂×C₄	#88104	1	2	yes	—
236	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
237	C₂×C₄	#88238	1	2	yes	—
238	C₂×C₄	D₄×#261	1	1	yes	characteristic
239	C₂×C₄	Q₈×#261	1	1	yes	characteristic
240	C₈	—	4	2	no	—
241	C₈	—	26	4	no	—
242	C₈	—	52	2	no	—
243	C₈	—	52	2	no	—
244	C₈	—	104	2	no	—
245	C₂²	#11719053	1	1	yes	characteristic
246	C₂²	#11719067	1	1	yes	characteristic
247	C₂²	#11719083	1	1	yes	characteristic
248	C₂²	C₂×#88101	1	1	yes	characteristic
249	C₂²	C₂×#88104	1	1	yes	characteristic
250	C₂²	#11719098	1	1	yes	characteristic
251	C₂²	#11719100	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	13	1	no	—
255	C₄	—	13	1	no	—
256	C₄	—	13	1	no	—
257	C₄	—	13	1	no	—
258	C₄	—	26	2	no	—
259	C₄	—	26	1	no	—
260	C₄	—	26	1	no	—
261	C₄	—	52	1	no	—
262	C₄	#11719065	1	2	yes	—
263	C₄	#11719078	1	2	yes	—
264	C₂	#13130392	1	1	yes	characteristic
265	C₂	#13130402	1	1	yes	characteristic
266	C₂	#13130403	1	1	yes	characteristic
267	C₂	#13130411	1	1	yes	characteristic
268	C₂	#13130937	1	1	yes	characteristic
269	C₂	#13130948	1	1	yes	characteristic
270	C₂	#13130950	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#336	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#336	#261
hypocenter	C₁₃	#9605
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
212. C₁₃	#9605	semidirect

Maximal subgroups

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

subgroup	index
#13130768	2
#13130622	2
C₂×#11718255	2
#13129666	2
#13114684	2
#9605	13