Subgroups — #16789949

Order 3328.

566 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
3	#13129874	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	4	yes	—
9	C₂×C₄×#1055	—	2	4	no	—
10	C₂×C₄×#1055	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂²⋊#6092	—	2	4	no	—
15	C₄⋊#6092	—	2	4	no	—
16	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	—	2	2	no	—
20	C₂×C₄²×C₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	—	2	2	no	—
27	C₂²×#1055	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	C₂²×#1055	—	2	2	no	—
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	D₄	1	2	yes	—
34	C₂²×#1055	Q₈	1	2	yes	—
35	C₂²×#1055	C₂³	1	1	yes	characteristic
36	C₂×#6092	—	2	4	no	—
37	C₂×#6092	—	2	4	no	—
38	C₂×#6092	—	2	4	no	—
39	C₄×#1055	—	2	2	no	—
40	C₄×#1055	—	2	2	no	—
41	C₄×#1055	—	2	2	no	—
42	C₄×#1055	D₄	1	4	yes	—
43	#12783	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
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	C₁₆	1	4	yes	complemented
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	—	2	1	no	—
67	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₃	M₁₆	1	2	yes	—
69	C₂×#1055	—	2	8	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	—	2	1	no	—
75	C₂×#1055	—	2	1	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	2	4	no	—
78	C₂×#1055	C₂²⋊C₄	1	2	yes	—
79	C₂×#1055	C₄⋊C₄	1	2	yes	—
80	C₂×#1055	M₁₆	1	2	yes	—
81	C₂×#1055	C₂×D₄	1	1	yes	characteristic
82	C₂×#1055	C₂×D₄	1	2	yes	—
83	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
84	C₂×#1055	#41	1	2	yes	—
85	#6092	—	4	4	no	—
86	C₂²×C₄×C₈	—	13	1	no	—
87	C₂×C₄⋊C₁₆	—	13	2	no	—
88	C₂×C₂²⋊C₁₆	—	13	2	no	—
89	#1374	—	13	2	no	—
90	C₂³×C₁₃	—	2	1	no	—
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	2	1	no	—
94	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
95	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
96	C₂³×C₁₃	M₃₂	1	2	yes	—
97	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	8	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
118	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
125	#1055	—	2	2	no	—
126	#1055	—	2	2	no	—
127	#1055	—	2	2	no	—
128	#1055	#121	1	2	yes	—
129	#1055	C₂²⋊Q₈	1	2	yes	—
130	C₂³×C₈	—	13	1	no	—
131	C₂³×C₈	—	13	1	no	—
132	C₂²×C₄²	#261	1	1	yes	characteristic
133	C₂²×C₁₆	—	13	4	no	—
134	C₂×C₄×C₈	—	13	1	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	—	26	4	no	—
139	C₄⋊C₁₆	—	26	4	no	—
140	C₂²⋊C₁₆	—	26	4	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	C₄⋊C₁₆	1	4	yes	complemented
148	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₂²×C₁₃	#431	1	1	yes	characteristic
151	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	C₂²⋊C₁₆	1	4	yes	complemented
160	C₄×C₁₃	#380	1	1	yes	characteristic
161	C₄×C₁₃	#407	1	1	yes	characteristic
162	C₄×C₁₃	#423	1	1	yes	characteristic
163	C₄×C₁₃	#432	1	1	yes	characteristic
164	C₂³×C₄	#1057	1	1	yes	characteristic
165	C₂³×C₄	#1057	1	1	yes	characteristic
166	C₂³×C₄	C₂×#261	1	1	yes	characteristic
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	—	13	2	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	13	1	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	—	26	2	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	—	26	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	#1057	1	1	yes	characteristic
181	C₂×C₄²	#1057	1	1	yes	characteristic
182	C₂×C₄²	C₂×#261	1	1	yes	characteristic
183	C₂×C₄²	C₂×#261	1	1	yes	characteristic
184	C₂×C₁₆	—	26	4	no	—
185	C₂×C₁₆	—	26	4	no	—
186	C₂×C₁₆	—	26	4	no	—
187	C₄×C₈	—	13	4	no	—
188	C₄×C₈	—	26	2	no	—
189	C₄×C₈	—	26	2	no	—
190	C₄×C₈	—	26	2	no	—
191	C₂×C₁₃	—	2	4	no	—
192	C₂×C₁₃	#1797	1	1	yes	characteristic
193	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
194	C₂×C₁₃	#2112	1	1	yes	characteristic
195	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
196	C₂×C₁₃	#2150	1	1	yes	characteristic
197	C₂×C₁₃	#2162	1	1	yes	characteristic
198	C₂×C₁₃	#2163	1	1	yes	characteristic
199	C₂⁴	C₂×#1057	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	#6092	1	4	yes	complemented
210	C₂²×C₄	#6118	1	2	yes	—
211	C₂²×C₄	C₄⋊#261	1	2	yes	—
212	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
213	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
214	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
215	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
216	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
217	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
218	C₂×C₈	—	13	1	no	—
219	C₂×C₈	—	13	1	no	—
220	C₂×C₈	—	13	2	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	4	no	—
227	C₂×C₈	—	26	8	no	—
228	C₂×C₈	—	26	1	no	—
229	C₂×C₈	—	26	1	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	1	no	—
233	C₂×C₈	—	26	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	#6122	1	2	yes	—
239	C₄²	C₂²⋊#261	1	2	yes	—
240	C₁₆	—	52	4	no	—
241	C₁₃	#12783	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₂×#6092	1	2	yes	—
247	C₂³	#88106	1	2	yes	—
248	C₂³	C₄⋊#1057	1	2	yes	—
249	C₂³	C₂²×#1057	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	C₂×#6092	1	2	yes	—
265	C₂×C₄	C₂×#6092	1	2	yes	—
266	C₂×C₄	#88106	1	2	yes	—
267	C₂×C₄	#88106	1	2	yes	—
268	C₂×C₄	C₄⋊#1057	1	2	yes	—
269	C₂×C₄	C₂²⋊#1057	1	2	yes	—
270	C₂×C₄	C₂²⋊#1057	1	2	yes	—
271	C₂×C₄	C₂×#6118	1	1	yes	characteristic
272	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
274	C₂×C₄	#88237	1	1	yes	characteristic
275	C₂×C₄	C₂×#6122	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
277	C₈	—	13	2	no	—
278	C₈	—	13	2	no	—
279	C₈	—	26	2	no	—
280	C₈	—	26	2	no	—
281	C₈	—	26	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₄⋊#6092	1	4	yes	complemented
289	C₂²	C₂²×#6092	1	1	yes	characteristic
290	C₂²	C₂×#88106	1	1	yes	characteristic
291	C₂²	C₂×#88106	1	1	yes	characteristic
292	C₂²	C₂×#88106	1	1	yes	characteristic
293	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
294	C₂²	#11719119	1	1	yes	characteristic
295	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	C₂²⋊#6092	1	4	yes	complemented
301	C₄	#11719127	1	1	yes	characteristic
302	C₄	#11719129	1	1	yes	characteristic
303	C₄	#11719130	1	1	yes	characteristic
304	C₄	#11719136	1	1	yes	characteristic
305	C₂	—	2	4	no	—
306	C₂	#13130690	1	1	yes	characteristic
307	C₂	C₂×C₂²⋊#6092	1	1	yes	characteristic
308	C₂	#13131063	1	1	yes	characteristic
309	C₂	C₂×C₄⋊#6092	1	1	yes	characteristic
310	C₂	#13131168	1	1	yes	characteristic
311	C₂	#13131193	1	1	yes	characteristic
312	C₂	#13131197	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	#12783
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#6092	C₂	semidirect
3. #13129874	C₂	semidirect
54. C₂²×C₄×C₁₃	C₁₆	semidirect
147. C₂²×C₁₃	C₄⋊C₁₆	semidirect
159. C₄×C₁₃	C₂²⋊C₁₆	semidirect
209. C₂²×C₄	#6092	semidirect
241. C₁₃	#12783	semidirect
288. C₂²	C₄⋊#6092	semidirect
300. C₄	C₂²⋊#6092	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×C₂²⋊#6092	2
#13129874	2
C₂²×C₄×#1055	2
#12783	13