Subgroups — #16784632

Order 3328.

568 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	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718244	C₂	1	4	yes	maximal
4	#13117403	C₂	1	1	yes	maximal, characteristic
5	C₂×#88141	C₂²	1	2	yes	—
6	C₂×#88123	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	C₂²	1	2	yes	—
9	C₂×C₄×#1057	—	2	1	no	—
10	C₂×C₄×#1057	C₂²	1	2	yes	—
11	C₁₃×#384	C₄	1	1	yes	characteristic
12	C₄⋊C₄×Dic₁₃	—	2	1	no	—
13	C₄⋊C₄×Dic₁₃	—	2	1	no	—
14	#11718635	C₄	1	2	yes	—
15	#11718634	—	2	1	no	—
16	#11718634	—	2	1	no	—
17	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718290	—	2	1	no	—
19	#11718290	—	2	1	no	—
20	#11718244	—	2	8	no	—
21	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
22	C₂²×#1057	—	2	2	no	—
23	C₂²×#1057	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×#6101	—	2	2	no	—
31	C₂×C₄×Dic₁₃	—	2	2	no	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
38	#88141	—	2	4	no	—
39	#88141	—	2	4	no	—
40	#88123	—	2	2	no	—
41	#88123	C₈	1	4	yes	complemented
42	C₄⋊#1057	—	2	4	no	—
43	C₄⋊#1057	—	2	2	no	—
44	C₄⋊#1057	D₄	1	4	yes	—
45	C₄×#1057	—	2	4	no	—
46	C₄×#1057	—	4	2	no	—
47	#88071	—	2	2	no	—
48	#88071	—	2	2	no	—
49	#9673	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	2	2	no	—
55	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₂₆	—	2	4	no	—
60	C₂×Dic₂₆	—	2	4	no	—
61	C₂×Dic₂₆	—	2	2	no	—
62	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
63	C₂×#1057	—	2	4	no	—
64	C₂×#1057	—	2	4	no	—
65	C₂×#1057	—	4	2	no	—
66	C₂×#1057	—	4	2	no	—
67	C₂×#1057	—	4	2	no	—
68	C₂×#1057	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	C₂×C₈	1	2	yes	—
73	C₄²×C₁₃	M₁₆	1	2	yes	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄⋊Dic₁₃	—	2	4	no	—
76	#6101	—	4	2	no	—
77	#6101	—	4	2	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	—	2	2	no	—
80	C₄×Dic₁₃	—	2	2	no	—
81	C₄×Dic₁₃	—	4	1	no	—
82	C₄×Dic₁₃	—	4	4	no	—
83	C₄×Dic₁₃	SD₁₆	1	2	yes	—
84	C₄×Dic₁₃	Q₁₆	1	2	yes	—
85	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
86	C₄×Dic₁₃	#41	1	2	yes	—
87	C₄×C₄⋊Q₈	—	13	1	no	—
88	C₄×C₄⋊C₈	—	13	1	no	—
89	C₄×C₄⋊C₈	—	13	1	no	—
90	C₂×Q₈⋊C₈	—	13	4	no	—
91	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₃	#103	1	2	yes	—
100	C₂×C₄×C₁₃	#104	1	2	yes	—
101	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#131	1	2	yes	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	4	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	2	1	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	#103	1	2	yes	—
116	C₂×Dic₁₃	#121	1	2	yes	—
117	C₂×Dic₁₃	#124	1	1	yes	characteristic
118	C₂×Dic₁₃	#127	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×Dic₁₃	#137	1	2	yes	—
122	Dic₂₆	—	2	8	no	—
123	Dic₂₆	—	4	4	no	—
124	#1057	—	4	4	no	—
125	#1057	—	4	4	no	—
126	C₂×C₄⋊Q₈	—	13	1	no	—
127	C₂×C₄×Q₈	—	13	2	no	—
128	C₂×C₄⋊C₈	—	13	2	no	—
129	C₂×C₄⋊C₈	—	26	1	no	—
130	C₂×C₄×C₈	—	13	2	no	—
131	C₂×C₄×C₈	—	26	1	no	—
132	#391	—	13	2	no	—
133	#389	—	26	1	no	—
134	#389	—	26	1	no	—
135	#384	#261	1	1	yes	characteristic
136	C₄×C₄⋊C₄	—	26	1	no	—
137	C₄×C₄⋊C₄	—	26	1	no	—
138	C₄³	—	13	1	no	—
139	#336	—	26	1	no	—
140	#336	—	26	1	no	—
141	Q₈⋊C₈	—	26	8	no	—
142	C₂²×C₁₃	#390	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₁₃	#408	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
146	C₂²×C₁₃	#419	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
148	C₂²×C₁₃	#421	1	1	yes	characteristic
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	8	2	no	—
151	C₄×C₁₃	Q₈⋊C₈	1	4	yes	complemented
152	C₄×C₁₃	C₈×D₄	1	2	yes	—
153	C₄×C₁₃	#436	1	2	yes	—
154	Dic₁₃	—	2	1	no	—
155	Dic₁₃	—	2	1	no	—
156	Dic₁₃	—	2	2	no	—
157	Dic₁₃	—	4	4	no	—
158	Dic₁₃	—	4	1	no	—
159	Dic₁₃	#461	1	2	yes	—
160	Dic₁₃	#462	1	2	yes	—
161	C₂²×Q₈	—	13	2	no	—
162	C₂²×C₈	—	26	2	no	—
163	C₂²×C₈	—	26	2	no	—
164	C₄⋊Q₈	—	13	4	no	—
165	C₄⋊Q₈	—	26	2	no	—
166	C₄×Q₈	—	26	4	no	—
167	C₄×Q₈	—	26	4	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	26	1	no	—
171	C₂×C₄⋊C₄	—	26	1	no	—
172	C₂×C₄⋊C₄	—	26	2	no	—
173	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
174	C₂×C₄²	—	13	1	no	—
175	C₂×C₄²	—	13	1	no	—
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	13	1	no	—
178	C₂×C₄²	—	13	1	no	—
179	C₂×C₄²	—	13	1	no	—
180	C₂×C₄²	—	26	2	no	—
181	C₂×C₄²	C₂×#261	1	1	yes	characteristic
182	C₄⋊C₈	—	13	4	no	—
183	C₄⋊C₈	—	26	2	no	—
184	C₄⋊C₈	—	26	4	no	—
185	C₄×C₈	—	26	4	no	—
186	C₄×C₈	—	52	2	no	—
187	#95	—	26	2	no	—
188	#95	—	26	2	no	—
189	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
190	C₂×C₁₃	#1479	1	1	yes	characteristic
191	C₂×C₁₃	#1481	1	1	yes	characteristic
192	C₂×C₁₃	#1485	1	1	yes	characteristic
193	C₂×C₁₃	#1954	1	1	yes	characteristic
194	C₂×C₁₃	#1980	1	1	yes	characteristic
195	C₂×C₁₃	#1981	1	1	yes	characteristic
196	C₂×Q₈	—	13	4	no	—
197	C₂×Q₈	—	26	2	no	—
198	C₂×Q₈	—	26	4	no	—
199	C₂×Q₈	—	26	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	26	2	no	—
206	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
208	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	—	52	2	no	—
212	C₂×C₈	—	52	2	no	—
213	C₂×C₈	—	52	2	no	—
214	C₂×C₈	—	52	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	26	4	no	—
219	C₄⋊C₄	—	26	4	no	—
220	C₄⋊C₄	—	52	2	no	—
221	C₄⋊C₄	—	52	2	no	—
222	C₄²	—	13	2	no	—
223	C₄²	—	13	2	no	—
224	C₄²	—	13	2	no	—
225	C₄²	—	13	2	no	—
226	C₄²	—	26	2	no	—
227	C₄²	—	26	2	no	—
228	C₄²	—	26	2	no	—
229	C₄²	—	52	1	no	—
230	C₄²	—	52	4	no	—
231	C₄²	D₁₃⋊C₈	1	2	yes	—
232	C₄²	#6118	1	2	yes	—
233	C₁₃	#9673	1	1	yes	characteristic, complemented, Sylow
234	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
235	Q₈	—	26	8	no	—
236	Q₈	—	52	4	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	1	no	—
248	C₂×C₄	—	13	2	no	—
249	C₂×C₄	—	26	1	no	—
250	C₂×C₄	—	26	1	no	—
251	C₂×C₄	—	26	1	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	4	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	26	1	no	—
257	C₂×C₄	—	52	2	no	—
258	C₂×C₄	—	52	2	no	—
259	C₂×C₄	#88111	1	2	yes	—
260	C₂×C₄	#88111	1	2	yes	—
261	C₂×C₄	#88116	1	2	yes	—
262	C₂×C₄	#88117	1	2	yes	—
263	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
264	C₂×C₄	C₂×#6118	1	1	yes	characteristic
265	C₂×C₄	D₄×#261	1	2	yes	—
266	C₂×C₄	#88240	1	2	yes	—
267	C₈	—	52	4	no	—
268	C₈	—	52	4	no	—
269	C₂²	C₂×#88111	1	1	yes	characteristic
270	C₂²	C₂×#88116	1	1	yes	characteristic
271	C₂²	#11719140	1	1	yes	characteristic
272	C₂²	#11719151	1	1	yes	characteristic
273	C₂²	C₂×#88117	1	1	yes	characteristic
274	C₂²	#11719153	1	1	yes	characteristic
275	C₂²	#11719159	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	13	2	no	—
279	C₄	—	13	2	no	—
280	C₄	—	26	2	no	—
281	C₄	—	26	1	no	—
282	C₄	—	26	1	no	—
283	C₄	—	52	1	no	—
284	C₄	—	52	4	no	—
285	C₄	#11718244	1	4	yes	complemented
286	C₄	#11719073	1	2	yes	—
287	C₄	#11719076	1	2	yes	—
288	C₂	C₂×#11718244	1	1	yes	characteristic
289	C₂	#13130140	1	1	yes	characteristic
290	C₂	#13130146	1	1	yes	characteristic
291	C₂	#13130160	1	1	yes	characteristic
292	C₂	#13130958	1	1	yes	characteristic
293	C₂	#13131000	1	1	yes	characteristic
294	C₂	#13131002	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#384	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#384	#261
hypocenter	C₁₃	#9673
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
41. #88123	C₈	semidirect
151. C₄×C₁₃	Q₈⋊C₈	semidirect
233. C₁₃	#9673	semidirect
285. C₄	#11718244	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1057	2
C₂×#11718244	2
#13117403	2
#9673	13