Subgroups — #16789093

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88106	C₂	1	1	yes	maximal, characteristic
3	#13129891	C₂	1	4	yes	maximal, complemented
4	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂×#88180	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	—	2	2	no	—
9	C₂×#88106	—	2	1	no	—
10	C₂×#88106	—	2	2	no	—
11	C₂×#88106	—	2	1	no	—
12	C₂×#88106	—	2	1	no	—
13	C₂×#88106	C₄	1	4	yes	complemented
14	C₂×#88106	C₄	1	4	yes	complemented
15	C₂×#88106	C₂²	1	2	yes	—
16	C₂²×#6092	—	2	1	no	—
17	C₂²×#6092	C₂²	1	2	yes	—
18	#11718783	—	2	1	no	—
19	#11718783	—	2	1	no	—
20	#11718783	C₄	1	2	yes	—
21	#11718783	C₂²	1	2	yes	—
22	C₂×#88055	—	2	1	no	—
23	C₂×C₈×Dic₁₃	—	2	1	no	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	C₈	1	4	yes	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
31	#88180	—	2	1	no	—
32	#88180	—	2	1	no	—
33	#88180	C₈	1	2	yes	—
34	#88180	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
36	C₂×C₄×Dic₁₃	—	2	1	no	—
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	2	yes	—
39	C₂²×#1055	C₂×C₄	1	2	yes	—
40	C₂²×#1055	D₄	1	2	yes	—
41	C₂²×#1055	D₄	1	1	yes	characteristic
42	C₂²×#1055	D₄	1	1	yes	characteristic
43	C₂²×#1055	D₄	1	1	yes	characteristic
44	C₂²×#1055	D₄	1	1	yes	characteristic
45	C₂²×#1055	Q₈	1	2	yes	—
46	C₂²×#1055	C₂³	1	1	yes	characteristic
47	#88106	—	2	4	no	—
48	#88106	—	2	4	no	—
49	#88106	—	2	4	no	—
50	#88106	—	2	2	no	—
51	#88106	C₂×C₄	1	4	yes	complemented
52	C₂×#6092	—	2	4	no	—
53	C₂×#6092	—	2	2	no	—
54	C₂×#6092	—	2	2	no	—
55	C₂×#6092	—	2	2	no	—
56	C₂×#6092	C₂×C₄	1	4	yes	complemented
57	#88055	—	2	2	no	—
58	#88055	—	2	2	no	—
59	C₈×Dic₁₃	—	2	2	no	—
60	C₈×Dic₁₃	—	2	2	no	—
61	#12554	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	2	2	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	2	2	no	—
76	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂²⋊Dic₁₃	—	2	2	no	—
80	C₂²⋊Dic₁₃	—	2	2	no	—
81	C₄⋊Dic₁₃	—	2	2	no	—
82	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
83	C₄×Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	—	2	2	no	—
85	C₂×#1055	C₄²	1	2	yes	—
86	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#1055	C₂²⋊C₄	1	2	yes	—
89	C₂×#1055	C₂²⋊C₄	1	2	yes	—
90	C₂×#1055	C₂²⋊C₄	1	2	yes	—
91	C₂×#1055	C₂²⋊C₄	1	2	yes	—
92	C₂×#1055	C₄⋊C₄	1	2	yes	—
93	C₂×#1055	C₄⋊C₄	1	2	yes	—
94	C₂×#1055	C₄⋊C₄	1	2	yes	—
95	C₂×#1055	C₄⋊C₄	1	2	yes	—
96	C₂×#1055	C₄⋊C₄	1	2	yes	—
97	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
98	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
99	C₂×#1055	C₂×D₄	1	1	yes	characteristic
100	C₂×#1055	C₂×D₄	1	1	yes	characteristic
101	C₂×#1055	C₂×D₄	1	1	yes	characteristic
102	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
103	#6092	—	2	4	no	—
104	#6092	—	2	4	no	—
105	C₂²×M₃₂	—	13	1	no	—
106	C₂²×M₃₂	—	13	1	no	—
107	C₂×#405	—	13	1	no	—
108	#1379	—	13	4	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	#97	1	2	yes	—
111	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
112	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
113	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
114	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#95	1	2	yes	—
116	C₂×C₄×C₁₃	#95	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
119	C₂×C₄×C₁₃	#97	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	4	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₈×C₁₃	—	2	4	no	—
136	#1055	#95	1	2	yes	—
137	#1055	#95	1	2	yes	—
138	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
141	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×M₁₆	#261	1	1	yes	characteristic
143	C₂³×C₈	—	13	1	no	—
144	C₂×#117	—	13	1	no	—
145	C₂×M₃₂	—	13	4	no	—
146	C₂×M₃₂	—	13	2	no	—
147	C₂×M₃₂	—	13	4	no	—
148	C₂×M₃₂	—	26	2	no	—
149	C₂×M₃₂	—	26	1	no	—
150	C₂×M₃₂	—	26	1	no	—
151	C₂×M₃₂	—	26	2	no	—
152	C₂×M₃₂	—	26	1	no	—
153	C₂²×C₁₆	—	13	2	no	—
154	C₂²×C₁₆	—	26	1	no	—
155	#405	—	13	2	no	—
156	#405	—	13	2	no	—
157	#405	—	26	1	no	—
158	#405	—	26	1	no	—
159	C₂×#97	—	26	1	no	—
160	C₂×C₄×C₈	—	26	1	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	4	2	no	—
165	C₂²×C₁₃	—	4	2	no	—
166	C₂²×C₁₃	#336	1	2	yes	—
167	C₂²×C₁₃	#336	1	2	yes	—
168	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₄×C₁₃	#336	1	2	yes	—
172	C₄×C₁₃	#336	1	2	yes	—
173	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
175	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
177	Dic₁₃	—	4	4	no	—
178	C₂³×C₄	C₂×#261	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	#1057	1	4	yes	—
182	C₂×M₁₆	C₂×#261	1	2	yes	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	1	no	—
189	C₂²×C₈	—	13	1	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	1	no	—
192	C₂²×C₈	—	13	1	no	—
193	C₂²×C₈	—	13	1	no	—
194	#117	—	13	2	no	—
195	#117	—	13	2	no	—
196	#117	—	26	1	no	—
197	#117	—	26	1	no	—
198	C₂×C₄⋊C₄	—	13	2	no	—
199	C₂×C₂²⋊C₄	—	26	1	no	—
200	C₂×C₄²	—	26	1	no	—
201	M₃₂	—	13	4	no	—
202	M₃₂	—	26	4	no	—
203	M₃₂	—	26	4	no	—
204	M₃₂	—	26	4	no	—
205	M₃₂	—	26	2	no	—
206	C₂×C₁₆	—	13	4	no	—
207	C₂×C₁₆	—	26	2	no	—
208	C₂×C₁₆	—	26	2	no	—
209	C₂×C₁₆	—	26	4	no	—
210	C₂×C₁₆	—	26	2	no	—
211	#97	—	26	2	no	—
212	#97	—	26	2	no	—
213	C₄×C₈	—	26	2	no	—
214	C₄×C₈	—	26	2	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
220	C₂×C₁₃	#2117	1	2	yes	—
221	C₂⁴	#6118	1	1	yes	characteristic
222	C₂²×C₄	—	26	2	no	—
223	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
224	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
225	C₂²×C₄	#6118	1	1	yes	characteristic
226	C₂²×C₄	C₄×#261	1	2	yes	—
227	C₂²×C₄	C₄⋊#261	1	2	yes	—
228	C₂²×C₄	C₂×#1057	1	2	yes	—
229	C₂²×C₄	#6122	1	2	yes	—
230	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
232	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	C₂×#1057	1	4	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	13	2	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	13	2	no	—
244	C₂×C₈	—	13	2	no	—
245	C₂×C₈	—	13	2	no	—
246	C₂×C₈	—	13	1	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	1	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	1	no	—
256	C₄⋊C₄	—	13	4	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₂²⋊C₄	—	26	2	no	—
259	C₂²⋊C₄	—	26	2	no	—
260	C₄²	—	26	2	no	—
261	C₄²	—	26	2	no	—
262	C₁₆	—	26	4	no	—
263	C₁₆	—	26	4	no	—
264	C₁₃	#12554	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	4	2	no	—
266	C₂³	C₄⋊#1057	1	2	yes	—
267	C₂³	#88110	1	2	yes	—
268	C₂³	#88111	1	1	yes	characteristic
269	C₂³	#88111	1	1	yes	characteristic
270	C₂³	C₂×#6118	1	1	yes	characteristic
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	52	4	no	—
273	C₂×C₄	C₄×#1057	1	2	yes	—
274	C₂×C₄	C₄×#1057	1	2	yes	—
275	C₂×C₄	C₄⋊#1057	1	2	yes	—
276	C₂×C₄	#88110	1	2	yes	—
277	C₂×C₄	#88111	1	1	yes	characteristic
278	C₂×C₄	#88111	1	1	yes	characteristic
279	C₂×C₄	#88112	1	2	yes	—
280	C₂×C₄	#88112	1	2	yes	—
281	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
282	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
283	C₂×C₄	C₂²⋊#1057	1	2	yes	—
284	C₂×C₄	C₂²⋊#1057	1	2	yes	—
285	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
288	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
289	C₂×C₄	C₂×#6122	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	13	1	no	—
293	C₈	—	13	1	no	—
294	C₈	—	13	2	no	—
295	C₈	—	13	2	no	—
296	C₈	—	13	1	no	—
297	C₈	—	13	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#11718290	1	2	yes	—
304	C₂²	#11718290	1	2	yes	—
305	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
306	C₂²	C₂×#88110	1	1	yes	characteristic
307	C₂²	C₂×#88111	1	1	yes	characteristic
308	C₄	—	52	4	no	—
309	C₄	#11718290	1	2	yes	—
310	C₄	#11718290	1	2	yes	—
311	C₄	C₂×C₄×#1057	1	1	yes	characteristic
312	C₄	C₂×#88112	1	1	yes	characteristic
313	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
314	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	C₂×#11718290	1	1	yes	characteristic
320	C₂	#13131080	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	#12554
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13129891	C₂	semidirect
13. C₂×#88106	C₄	semidirect
14. C₂×#88106	C₄	semidirect
51. #88106	C₂×C₄	semidirect
56. C₂×#6092	C₂×C₄	semidirect
264. C₁₃	#12554	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13129891	2
C₂×#11718783	2
#12554	13