Subgroups — #16019045

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	#13117109	C₂	1	2	yes	maximal, complemented
3	#13117063	C₂	1	2	yes	maximal, complemented
4	#13117036	C₂	1	4	yes	maximal
5	#13116988	C₂	1	4	yes	maximal, complemented
6	#13116635	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	—	2	1	no	—
9	C₂×#88123	—	2	1	no	—
10	C₁₃×#500	C₂²	1	1	yes	characteristic
11	C₁₃×#494	C₂²	1	1	yes	characteristic
12	C₁₃×#487	—	2	1	no	—
13	C₁₃×#467	—	2	1	no	—
14	C₁₃×#467	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	—	2	1	no	—
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88058	—	2	1	no	—
19	C₂×#88058	C₂²	1	2	yes	—
20	#11718343	—	2	1	no	—
21	#11718343	—	2	1	no	—
22	#11718343	C₂²	1	2	yes	—
23	#11718343	C₂²	1	2	yes	—
24	#11718156	C₂²	1	8	yes	—
25	C₂²⋊#6090	C₂²	1	2	yes	—
26	#11718081	C₂²	1	8	yes	—
27	C₄⋊#6090	C₂²	1	2	yes	—
28	C₄×#6090	C₂²	1	2	yes	—
29	C₂²×Dic₂₆	—	4	1	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₁₃×#125	—	4	1	no	—
37	C₁₃×C₂²⋊Q₈	—	2	2	no	—
38	C₁₃×C₂²⋊Q₈	—	4	1	no	—
39	C₄×Q₈×C₁₃	—	2	2	no	—
40	C₄×Q₈×C₁₃	—	4	1	no	—
41	C₁₃×#117	—	4	1	no	—
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄⋊Dic₁₃	—	4	1	no	—
45	#88123	—	2	2	no	—
46	#88123	—	2	2	no	—
47	#88123	—	4	1	no	—
48	#88123	—	4	1	no	—
49	C₁₃×#107	—	2	1	no	—
50	C₁₃×#107	C₂³	1	2	yes	—
51	C₁₃×#103	—	2	2	no	—
52	C₁₃×#103	—	2	2	no	—
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	#88058	—	2	2	no	—
58	#88058	—	2	2	no	—
59	#88058	—	2	2	no	—
60	#88058	C₂³	1	4	yes	—
61	C₂×#6090	C₂³	1	4	yes	—
62	#33796	—	13	1	no	maximal, Sylow
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	4	2	no	—
69	C₂×Dic₂₆	—	4	2	no	—
70	C₂×Dic₂₆	—	4	2	no	—
71	C₂×Dic₂₆	—	8	4	no	—
72	C₁₃×Q₁₆	—	2	4	no	—
73	C₁₃×Q₁₆	—	4	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₄⋊C₄	—	4	1	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₂²⋊C₄	—	4	2	no	—
87	C₄²×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	D₈	1	1	yes	characteristic
89	C₄²×C₁₃	D₈	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	C₄⋊Dic₁₃	—	4	2	no	—
96	#6090	—	2	4	no	—
97	#3170	D₁₃	1	1	yes	characteristic
98	C₂×#500	—	13	1	no	—
99	#2252	—	13	2	no	—
100	#2241	—	13	2	no	—
101	#2236	—	13	4	no	—
102	#2227	—	13	4	no	—
103	#2162	—	13	1	no	—
104	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₁₃	—	4	2	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	Q₈×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	Q₃₂	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃	—	4	4	no	—
121	C₂×Dic₁₃	—	8	4	no	—
122	Dic₂₆	—	8	4	no	—
123	Dic₂₆	—	8	4	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	#137	1	2	yes	—
127	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
128	#533	—	2	1	no	—
129	C₂×C₄⋊Q₈	—	26	1	no	—
130	#500	—	13	2	no	—
131	#500	—	13	2	no	—
132	#500	—	26	1	no	—
133	#500	—	26	1	no	—
134	#500	C₂×D₁₃	1	1	yes	characteristic
135	#494	C₂×D₁₃	1	1	yes	characteristic
136	#487	—	2	1	no	—
137	#467	—	2	1	no	—
138	#467	C₂×D₁₃	1	2	yes	—
139	C₄×Q₁₆	—	2	1	no	—
140	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
141	C₂×#107	—	13	2	no	—
142	C₂×#107	—	26	1	no	—
143	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
144	#366	—	13	8	no	—
145	C₄⋊C₁₆	—	13	2	no	—
146	#358	—	13	8	no	—
147	C₂²⋊C₁₆	—	13	2	no	—
148	C₄×C₁₆	—	13	2	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
152	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	8	2	no	—
156	C₄×C₁₃	—	8	2	no	—
157	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
158	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
159	C₄×C₁₃	#509	1	2	yes	—
160	C₄×C₁₃	#536	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
162	Dic₁₃	—	8	4	no	—
163	C₂²×Q₈	—	52	1	no	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	26	2	no	—
171	C₄⋊Q₈	—	26	2	no	—
172	C₄⋊Q₈	—	52	1	no	—
173	C₄⋊Q₈	—	52	1	no	—
174	#125	—	4	1	no	—
175	C₂²⋊Q₈	—	2	2	no	—
176	C₂²⋊Q₈	—	4	1	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×Q₈	—	4	1	no	—
179	#117	—	4	1	no	—
180	C₂×C₄⋊C₄	—	26	2	no	—
181	C₂×C₄⋊C₄	—	52	1	no	—
182	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂×C₁₆	—	13	4	no	—
184	#107	—	2	1	no	—
185	#107	—	13	4	no	—
186	#107	—	26	2	no	—
187	#107	—	26	2	no	—
188	#107	—	26	2	no	—
189	#107	C₂²×D₁₃	1	2	yes	—
190	#103	—	2	2	no	—
191	#103	—	2	2	no	—
192	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	#3226	1	1	yes	characteristic
198	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
199	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	52	2	no	—
203	C₂×Q₈	—	52	2	no	—
204	C₂×Q₈	—	104	4	no	—
205	C₂²×C₄	—	52	2	no	—
206	C₂²×C₄	#6104	1	1	yes	characteristic
207	C₂²×C₄	#6104	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	26	4	no	—
224	C₄⋊C₄	—	52	2	no	—
225	C₄⋊C₄	—	52	2	no	—
226	C₄⋊C₄	—	52	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₄²	—	4	2	no	—
229	C₄²	#6104	1	1	yes	characteristic
230	C₄²	#6104	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	C₁₆	—	26	4	no	—
234	C₁₃	#33796	1	1	yes	characteristic, complemented, Sylow
235	C₂³	C₂×#6104	1	1	yes	characteristic
236	Q₈	—	4	2	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	8	2	no	—
239	Q₈	—	104	4	no	—
240	Q₈	—	104	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	52	4	no	—
247	C₂×C₄	—	104	4	no	—
248	C₂×C₄	#88069	1	4	yes	—
249	C₂×C₄	C₂×#6104	1	1	yes	characteristic
250	C₂×C₄	C₂×#6104	1	1	yes	characteristic
251	C₂×C₄	C₂×#6104	1	1	yes	characteristic
252	C₂×C₄	C₂×#6104	1	1	yes	characteristic
253	C₂×C₄	C₂×#6104	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	#88205	1	2	yes	—
258	C₈	#88257	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₂×#88069	1	2	yes	—
262	C₂²	C₂²×#6104	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	104	4	no	—
268	C₄	C₂×#88069	1	2	yes	—
269	C₄	C₂×#88069	1	2	yes	—
270	C₄	#11718854	1	2	yes	—
271	C₄	C₂×#88205	1	1	yes	characteristic
272	C₄	#11719530	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	#13125954	1	1	yes	characteristic
275	C₂	C₂²×#88069	1	1	yes	characteristic
276	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#33796
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13117109	C₂	semidirect
3. #13117063	C₂	semidirect
5. #13116988	C₂	semidirect
234. C₁₃	#33796	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718343	2
#13117109	2
#13117063	2
#13117036	2
#13116988	2
#13116635	2
C₁₃×#3170	2
#33796	13