Subgroups — #16781869

Order 3328.

462 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	#13130383	C₂	1	2	yes	maximal, complemented
2	#13130067	C₂	1	2	yes	maximal
3	#13129662	C₂	1	2	yes	maximal, complemented
4	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×#88180	C₂²	1	1	yes	characteristic
7	C₂²×#6099	C₄	1	1	yes	characteristic
8	#11719119	C₂²	1	4	yes	—
9	#11718762	—	2	2	no	—
10	#11718762	—	2	2	no	—
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×#88055	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718305	—	2	2	no	—
16	#11718285	—	2	2	no	—
17	#11718254	—	2	2	no	—
18	#11718253	—	2	2	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₈	1	4	yes	complemented
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₃	D₄	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₃	Q₈	1	1	yes	characteristic
28	#88180	—	2	2	no	—
29	#88180	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	2	no	—
36	C₂×#6099	—	2	2	no	—
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊#1057	—	2	4	no	—
40	C₄⋊#1057	—	2	4	no	—
41	C₄×#1057	—	2	4	no	—
42	#88055	—	2	2	no	—
43	#88055	—	2	2	no	—
44	C₈×Dic₁₃	—	2	1	no	—
45	C₈×Dic₁₃	—	2	1	no	—
46	C₈×Dic₁₃	D₄	1	2	yes	—
47	C₈×Dic₁₃	D₄	1	2	yes	—
48	#8601	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×#1057	—	4	4	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
72	C₂²⋊Dic₁₃	—	2	2	no	—
73	C₂²⋊Dic₁₃	—	2	2	no	—
74	C₄⋊Dic₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	C₄×Dic₁₃	—	2	1	no	—
77	C₄×Dic₁₃	—	2	1	no	—
78	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
79	C₄×Dic₁₃	#41	1	2	yes	—
80	#6099	—	2	8	no	—
81	C₂×#1055	—	2	1	no	—
82	C₂×#1055	—	2	1	no	—
83	C₂×#1055	—	2	1	no	—
84	C₂×#1055	—	2	1	no	—
85	C₂×#1055	C₂²⋊C₄	1	2	yes	—
86	C₂×#1055	M₁₆	1	2	yes	—
87	C₂×#405	—	13	1	no	—
88	#1564	—	13	2	no	—
89	#1458	—	13	2	no	—
90	#1275	—	13	2	no	—
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	4	1	no	—
94	C₂³×C₁₃	—	4	1	no	—
95	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	4	1	no	—
113	C₂×Dic₁₃	—	4	1	no	—
114	C₂×Dic₁₃	—	4	1	no	—
115	C₂×Dic₁₃	—	4	1	no	—
116	C₂×Dic₁₃	#121	1	2	yes	—
117	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
118	#1057	—	8	4	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
121	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
122	#1055	—	2	2	no	—
123	#1055	—	2	2	no	—
124	C₂²×M₁₆	—	13	1	no	—
125	C₂³×C₈	#261	1	1	yes	characteristic
126	C₂×#117	—	13	1	no	—
127	#431	—	13	4	no	—
128	#405	—	26	2	no	—
129	#405	—	26	2	no	—
130	C₂×#97	—	13	1	no	—
131	C₂×#97	—	13	1	no	—
132	C₂×C₄×C₈	—	13	1	no	—
133	C₂×C₄×C₈	—	13	1	no	—
134	#335	—	26	2	no	—
135	#334	—	26	2	no	—
136	#324	—	26	2	no	—
137	#323	—	26	2	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	1	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	#380	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₂²×C₁₃	#423	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₁₃	#407	1	1	yes	characteristic
151	C₄×C₁₃	#431	1	1	yes	characteristic
152	C₄×C₁₃	#432	1	1	yes	characteristic
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	—	2	2	no	—
155	Dic₁₃	—	4	1	no	—
156	Dic₁₃	—	4	1	no	—
157	C₂³×C₄	C₂×#261	1	1	yes	characteristic
158	C₂×M₁₆	—	26	2	no	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	#1057	1	4	yes	complemented
167	C₂²×C₈	C₂×#261	1	1	yes	characteristic
168	C₂²×C₈	C₂×#261	1	1	yes	characteristic
169	#117	—	26	2	no	—
170	#117	—	26	2	no	—
171	C₂×C₄⋊C₄	—	13	1	no	—
172	C₂×C₄⋊C₄	—	13	1	no	—
173	C₂×C₂²⋊C₄	—	13	1	no	—
174	C₂×C₂²⋊C₄	—	13	1	no	—
175	C₂×C₄²	—	13	1	no	—
176	C₂×C₄²	—	13	1	no	—
177	C₄⋊C₈	—	26	4	no	—
178	C₂²⋊C₈	—	26	4	no	—
179	#97	—	26	2	no	—
180	#97	—	26	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	26	4	no	—
184	C₄×C₈	—	26	1	no	—
185	C₄×C₈	—	26	1	no	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	#1797	1	1	yes	characteristic
190	C₂×C₁₃	#1812	1	1	yes	characteristic
191	C₂×C₁₃	#1813	1	1	yes	characteristic
192	C₂⁴	C₄⋊#261	1	1	yes	characteristic
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	#6118	1	2	yes	—
202	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
203	C₂²×C₄	C₂×#1057	1	2	yes	—
204	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
205	M₁₆	—	26	8	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	26	1	no	—
213	C₂×C₈	—	26	1	no	—
214	C₂×C₈	—	26	1	no	—
215	C₂×C₈	—	26	1	no	—
216	C₂×C₈	—	52	4	no	—
217	C₂×C₈	C₂×#1057	1	2	yes	—
218	C₂×C₈	C₂×#1057	1	2	yes	—
219	C₂×C₈	#6122	1	2	yes	—
220	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
222	C₄⋊C₄	—	26	2	no	—
223	C₄⋊C₄	—	26	2	no	—
224	C₂²⋊C₄	—	26	2	no	—
225	C₂²⋊C₄	—	26	2	no	—
226	C₄²	—	13	2	no	—
227	C₄²	—	13	2	no	—
228	C₄²	—	26	1	no	—
229	C₄²	—	26	1	no	—
230	C₁₃	#8601	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	—	4	1	no	—
235	C₂³	C₄⋊#1057	1	2	yes	—
236	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	26	1	no	—
246	C₂×C₄	—	26	1	no	—
247	C₂×C₄	—	52	1	no	—
248	C₂×C₄	—	52	1	no	—
249	C₂×C₄	—	52	1	no	—
250	C₂×C₄	—	52	1	no	—
251	C₂×C₄	C₄⋊#1057	1	2	yes	—
252	C₂×C₄	C₂×#6118	1	1	yes	characteristic
253	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
254	C₂×C₄	#88237	1	1	yes	characteristic
255	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
256	C₂×C₄	C₂×#6122	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	26	2	no	—
260	C₈	—	26	2	no	—
261	C₈	—	104	4	no	—
262	C₈	C₂²⋊#1057	1	2	yes	—
263	C₈	C₂²⋊#1057	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
272	C₂²	#11719129	1	1	yes	characteristic
273	C₂²	#11719136	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	26	2	no	—
276	C₄	—	26	2	no	—
277	C₄	—	52	1	no	—
278	C₄	—	52	1	no	—
279	C₄	#11719119	1	1	yes	characteristic
280	C₄	#11719127	1	1	yes	characteristic
281	C₄	#11719130	1	1	yes	characteristic
282	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	4	2	no	—
286	C₂	#13130690	1	1	yes	characteristic
287	C₂	#13130718	1	1	yes	characteristic
288	C₂	#13130720	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88237
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	#8601
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. #13130383	C₂	semidirect
3. #13129662	C₂	semidirect
23. C₂²×C₈×C₁₃	C₈	semidirect
166. C₂²×C₈	#1057	semidirect
230. C₁₃	#8601	semidirect

Maximal subgroups

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

subgroup	index
#13130383	2
#13130067	2
#13129662	2
C₂×#11718762	2
#8601	13