Subgroups — #16789317

Order 3328.

482 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×#11718304	C₂	1	2	yes	maximal
2	#13131042	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718258	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129891	C₂	1	2	yes	maximal, complemented
5	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	#41×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×#88202	C₂²	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	#11718783	—	2	1	no	—
12	#11718783	C₂²	1	1	yes	characteristic
13	M₁₆×Dic₁₃	—	2	1	no	—
14	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
15	#11718762	—	2	1	no	—
16	#11718762	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	—	2	1	no	—
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	#11718304	C₄	1	8	yes	complemented
20	#11718258	—	2	1	no	—
21	#11718258	—	2	1	no	—
22	#11718258	—	2	1	no	—
23	#11718257	—	2	1	no	—
24	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#131	—	2	2	no	—
26	C₁₃×#131	C₈	1	4	yes	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	#88202	—	2	2	no	—
32	#88202	C₂×C₄	1	4	yes	—
33	Q₈×Dic₁₃	C₈	1	2	yes	—
34	D₄×Dic₁₃	—	2	2	no	—
35	D₄×Dic₁₃	C₈	1	2	yes	—
36	#88180	—	2	1	no	—
37	#88180	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#6099	—	2	1	no	—
41	C₂×#6099	C₂×C₄	1	1	yes	characteristic
42	C₂²×#1055	—	2	1	no	—
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	#88106	—	2	2	no	—
45	#88106	C₂×C₄	1	4	yes	complemented
46	C₂×#6092	—	2	2	no	—
47	C₂×#6092	—	2	2	no	—
48	C₂×#6092	—	2	2	no	—
49	C₂×#6092	C₂×C₄	1	4	yes	complemented
50	#88055	—	2	1	no	—
51	#88055	—	2	1	no	—
52	#88055	—	2	1	no	—
53	C₈×Dic₁₃	—	2	1	no	—
54	C₈×Dic₁₃	—	2	1	no	—
55	C₈×Dic₁₃	—	2	1	no	—
56	C₈×Dic₁₃	D₄	1	1	yes	characteristic
57	C₈×Dic₁₃	D₄	1	1	yes	characteristic
58	C₈×Dic₁₃	D₄	1	1	yes	characteristic
59	C₈×Dic₁₃	D₄	1	1	yes	characteristic
60	#12604	—	13	1	no	maximal, Sylow
61	C₁₃×#41	—	2	1	no	—
62	C₁₃×#41	—	2	1	no	—
63	C₁₃×#41	C₄²	1	2	yes	—
64	C₁₃×#41	C₂×C₈	1	2	yes	—
65	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	M₁₆	1	2	yes	—
72	C₁₃×M₁₆	—	2	1	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	C₂²⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
85	C₄×Dic₁₃	—	2	1	no	—
86	C₄×Dic₁₃	—	2	2	no	—
87	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
90	#6099	—	2	1	no	—
91	#6099	—	2	4	no	—
92	#6099	C₄²	1	2	yes	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₂×D₄	1	1	yes	characteristic
97	C₂×#1055	C₂×D₄	1	1	yes	characteristic
98	C₂×#1055	#41	1	2	yes	—
99	#6092	—	2	4	no	—
100	#6092	—	2	4	no	—
101	C₄×#131	—	13	1	no	—
102	C₂×#350	—	13	2	no	—
103	C₄×M₃₂	—	13	1	no	—
104	C₂×#346	—	13	1	no	—
105	#1379	—	13	2	no	—
106	C₂³×C₁₃	—	2	1	no	—
107	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	Q₈×C₁₃	—	2	1	no	—
109	Q₈×C₁₃	C₄×C₈	1	2	yes	—
110	D₄×C₁₃	—	2	2	no	—
111	D₄×C₁₃	—	2	2	no	—
112	D₄×C₁₃	—	2	1	no	—
113	D₄×C₁₃	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
127	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
130	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
131	#1055	—	2	2	no	—
132	#1055	C₄×D₄	1	2	yes	—
133	#1055	C₄×D₄	1	2	yes	—
134	C₂×#131	—	13	1	no	—
135	C₂×#131	#261	1	1	yes	characteristic
136	C₄×#41	—	13	1	no	—
137	C₂×M₃₂	—	13	2	no	—
138	C₂²×C₁₆	—	13	2	no	—
139	#405	—	13	1	no	—
140	#405	—	13	1	no	—
141	#405	—	26	1	no	—
142	#405	—	26	1	no	—
143	C₄×M₁₆	—	13	1	no	—
144	C₄×M₁₆	—	26	1	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	26	1	no	—
147	#350	—	13	8	no	—
148	#346	—	26	1	no	—
149	#346	—	26	1	no	—
150	#346	—	26	1	no	—
151	C₄×C₁₆	—	26	1	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₁₃	#431	1	1	yes	characteristic
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
161	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₁₃	#431	1	1	yes	characteristic
163	Dic₁₃	—	2	2	no	—
164	Dic₁₃	—	2	2	no	—
165	Dic₁₃	—	4	2	no	—
166	C₂×#41	C₂×#261	1	1	yes	characteristic
167	#131	—	2	2	no	—
168	#131	—	13	4	no	—
169	#131	—	26	2	no	—
170	#131	#1057	1	4	yes	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	13	1	no	—
173	C₂×M₁₆	—	26	1	no	—
174	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	26	1	no	—
178	C₂²×C₈	C₂×#261	1	1	yes	characteristic
179	C₄×Q₈	—	13	2	no	—
180	C₄×D₄	—	13	2	no	—
181	C₄×D₄	—	26	2	no	—
182	#117	—	13	1	no	—
183	#117	—	26	1	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	—	26	1	no	—
186	M₃₂	—	13	4	no	—
187	M₃₂	—	26	2	no	—
188	C₂×C₁₆	—	13	4	no	—
189	C₂×C₁₆	—	26	2	no	—
190	C₂×C₁₆	—	26	2	no	—
191	C₂×C₁₆	—	26	2	no	—
192	#97	—	26	1	no	—
193	#97	—	26	1	no	—
194	#97	—	26	1	no	—
195	C₄×C₈	—	13	1	no	—
196	C₄×C₈	—	13	1	no	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	26	1	no	—
200	C₄×C₈	—	26	1	no	—
201	C₄×C₈	—	26	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	4	2	no	—
205	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
206	C₂×C₁₃	#2117	1	2	yes	—
207	#41	—	2	1	no	—
208	#41	—	2	1	no	—
209	#41	C₄×#261	1	2	yes	—
210	#41	C₂×#1057	1	2	yes	—
211	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	13	2	no	—
216	C₂²×C₄	—	26	2	no	—
217	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	13	2	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	—	26	1	no	—
223	M₁₆	C₂×#1057	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	C₂×#1057	1	2	yes	—
233	C₂×C₈	#6122	1	2	yes	—
234	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	26	2	no	—
238	C₂²⋊C₄	—	26	2	no	—
239	C₂²⋊C₄	—	26	1	no	—
240	C₄²	—	13	2	no	—
241	C₄²	—	13	1	no	—
242	C₄²	—	13	1	no	—
243	C₄²	—	26	2	no	—
244	C₄²	—	26	1	no	—
245	C₁₆	—	26	4	no	—
246	C₁₆	—	26	4	no	—
247	C₁₃	#12604	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	2	1	no	—
249	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#1057	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#1057	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	52	2	no	—
263	C₂×C₄	—	52	4	no	—
264	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
266	C₂×C₄	#88237	1	1	yes	characteristic
267	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
268	C₂×C₄	C₂×#6122	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	13	2	no	—
272	C₈	—	13	2	no	—
273	C₈	—	26	2	no	—
274	C₈	C₂²⋊#1057	1	2	yes	—
275	C₈	C₂²⋊#1057	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	#11719119	1	1	yes	characteristic
280	C₂²	C₄×#6122	1	1	yes	characteristic
281	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	26	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	52	2	no	—
286	C₄	C₂×C₄×#1057	1	1	yes	characteristic
287	C₄	#11719125	1	1	yes	characteristic
288	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
289	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#1057	1	1	yes	characteristic
294	C₂	#13131079	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#12604
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #13131042	C₂	semidirect
3. C₂×#11718258	C₂	semidirect
4. #13129891	C₂	semidirect
19. #11718304	C₄	semidirect
45. #88106	C₂×C₄	semidirect
49. C₂×#6092	C₂×C₄	semidirect
247. C₁₃	#12604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718304	2
#13131042	2
C₂×#11718258	2
#13129891	2
#131×Dic₁₃	2
#12604	13