Subgroups — #16798270

Order 3328.

1126 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719092	C₂	1	4	yes	maximal
2	#13131285	C₂	1	2	yes	maximal
3	#13130773	C₂	1	4	yes	maximal, complemented
4	#13130772	C₂	1	2	yes	maximal, complemented
5	#13130771	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	C₂²	1	4	yes	—
8	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	#11719136	—	2	4	no	—
11	#11719092	—	2	16	no	—
12	C₂×#88102	C₂²	1	8	yes	—
13	C₂×#88101	C₂²	1	8	yes	—
14	#11719060	C₂²	1	4	yes	—
15	#11719059	C₂²	1	4	yes	—
16	C₄×C₄⋊#261	—	2	4	no	—
17	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
19	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
20	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic, complemented
21	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic, complemented
22	C₄×C₈×D₁₃	C₄	1	2	yes	complemented
23	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	2	yes	—
26	C₂²×C₄×D₁₃	D₄	1	2	yes	—
27	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊#261	—	2	4	no	—
29	#88237	—	2	16	no	—
30	C₂×C₄⋊#261	—	2	8	no	—
31	C₂×C₄⋊#261	C₂³	1	8	yes	—
32	C₂×C₄×#261	—	2	4	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
37	C₂²×#1055	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	8	yes	—
41	C₂×C₈×D₁₃	Q₈	1	8	yes	—
42	C₂×C₈×D₁₃	C₂³	1	2	yes	—
43	C₂×C₈×D₁₃	C₂³	1	2	yes	—
44	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₁₃	D₄	1	2	yes	—
47	C₄²×D₁₃	Q₈	1	2	yes	—
48	C₄²×D₁₃	C₂³	1	1	yes	characteristic
49	C₄²×D₁₃	C₂³	1	1	yes	characteristic
50	(C₂×#261)⋊C₄	—	2	8	no	—
51	#88102	—	2	16	no	—
52	#88101	—	2	16	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
60	C₄×#1055	C₂×C₄	1	1	yes	characteristic
61	C₄×#1055	C₂×C₄	1	1	yes	characteristic
62	C₄×#1055	C₂×C₄	1	2	yes	—
63	#20797	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×#261	—	2	8	no	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
69	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	—	2	8	no	—
71	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
76	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
79	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊#261	—	4	8	no	—
82	C₄⋊#261	—	2	16	no	—
83	C₄⋊#261	—	4	16	no	—
84	C₄×#261	—	4	8	no	—
85	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	complemented
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
92	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
93	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
94	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
97	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
98	C₂×#1055	C₄⋊C₄	1	8	yes	complemented
99	C₂×#1055	C₂²×C₄	1	2	yes	—
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₈×D₁₃	C₄⋊C₄	1	8	yes	complemented
102	C₈×D₁₃	C₄⋊C₄	1	8	yes	complemented
103	C₈×D₁₃	C₂×D₄	1	4	yes	—
104	C₈×D₁₃	C₂×D₄	1	4	yes	—
105	C₈×D₁₃	C₂×Q₈	1	4	yes	—
106	C₈×D₁₃	C₂×Q₈	1	4	yes	—
107	C₂×#427	—	13	4	no	—
108	C₂²×C₄×C₈	—	13	1	no	—
109	#2285	—	13	2	no	—
110	#1846	—	13	4	no	—
111	#1845	—	13	2	no	—
112	#1844	—	13	2	no	—
113	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	4	no	—
115	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×D₁₃	#127	1	2	yes	—
117	C₂²×D₁₃	C₄⋊Q₈	1	2	yes	—
118	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×#261	—	4	8	no	—
120	C₂×#261	—	4	16	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
132	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
133	C₄×D₁₃	—	2	4	no	—
134	C₄×D₁₃	—	2	4	no	—
135	C₄×D₁₃	—	2	4	no	—
136	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₄×D₁₃	#127	1	2	yes	—
138	C₄×D₁₃	C₄⋊Q₈	1	4	yes	—
139	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
140	C₄×D₁₃	#135	1	8	yes	—
141	C₄×D₁₃	C₂²×D₄	1	2	yes	—
142	C₄×D₁₃	C₂²×Q₈	1	2	yes	—
143	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
144	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
145	#1055	C₂×C₄⋊C₄	1	4	yes	—
146	#1055	C₂×C₄⋊C₄	1	4	yes	—
147	C₂³×C₈	—	13	2	no	—
148	C₂×#117	—	13	4	no	—
149	C₂²×C₄²	—	13	1	no	—
150	#427	—	26	16	no	—
151	C₂×#107	—	13	8	no	—
152	C₂×#106	—	13	8	no	—
153	C₂×C₄×C₈	—	13	2	no	—
154	C₂×C₄×C₈	—	13	1	no	—
155	C₂×C₄×C₈	—	13	1	no	—
156	C₂×C₄×C₈	—	13	1	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	—	13	2	no	—
159	C₂×C₄×C₈	—	13	1	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
162	#384	—	13	4	no	—
163	#382	—	13	4	no	—
164	#380	—	26	4	no	—
165	C₄×C₄⋊C₄	—	26	4	no	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	#384	1	2	yes	—
168	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂×D₁₃	—	2	2	no	—
170	C₂×D₁₃	—	2	4	no	—
171	C₂×D₁₃	—	2	4	no	—
172	C₂×D₁₃	#384	1	2	yes	—
173	C₂×D₁₃	C₂×#127	1	1	yes	characteristic
174	C₂×D₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×D₁₃	C₂×#135	1	4	yes	—
176	#261	—	8	8	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	#384	1	2	yes	—
179	C₄×C₁₃	#427	1	4	yes	complemented
180	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
181	Dic₁₃	—	2	2	no	—
182	Dic₁₃	#384	1	2	yes	—
183	Dic₁₃	#427	1	4	yes	complemented
184	Dic₁₃	C₂×C₄⋊Q₈	1	2	yes	—
185	C₂³×C₄	—	13	1	no	—
186	C₂³×C₄	—	13	2	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	8	no	—
189	C₂²×C₈	—	13	8	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₂²×C₈	C₂×#261	1	2	yes	—
195	#117	—	26	16	no	—
196	C₂×C₄⋊C₄	—	13	8	no	—
197	C₂×C₄⋊C₄	—	26	8	no	—
198	C₂×C₂²⋊C₄	—	26	4	no	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	—	13	1	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	—	13	1	no	—
204	C₂×C₄²	—	13	1	no	—
205	C₂×C₄²	—	13	1	no	—
206	C₂×C₄²	—	13	1	no	—
207	C₂×C₄²	—	26	4	no	—
208	C₂×C₄²	C₂×#261	1	1	yes	characteristic
209	#107	—	26	16	no	—
210	#106	—	26	16	no	—
211	C₄×C₈	—	13	2	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	—	13	2	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	—	13	1	no	—
216	C₄×C₈	—	13	1	no	—
217	C₄×C₈	—	13	2	no	—
218	C₄×C₈	C₂×#261	1	1	yes	characteristic
219	C₄×C₈	C₂×#261	1	1	yes	characteristic
220	C₄×C₈	C₂×#261	1	2	yes	—
221	#95	—	26	8	no	—
222	C₂×C₁₃	—	2	2	no	—
223	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
224	C₂×C₁₃	C₂×#427	1	2	yes	—
225	D₁₃	—	2	2	no	—
226	D₁₃	#3147	1	2	yes	—
227	D₁₃	#3160	1	2	yes	—
228	C₂⁴	—	13	1	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	13	4	no	—
232	C₂²×C₄	—	13	4	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	—	13	2	no	—
241	C₂²×C₄	—	26	8	no	—
242	C₂²×C₄	—	26	8	no	—
243	C₂²×C₄	C₄⋊#261	1	2	yes	—
244	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
245	C₂×C₈	—	13	8	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	4	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	—	13	8	no	—
251	C₂×C₈	—	13	8	no	—
252	C₂×C₈	—	13	4	no	—
253	C₂×C₈	—	13	4	no	—
254	C₂×C₈	C₄⋊#261	1	8	yes	complemented
255	C₂×C₈	C₂²×#261	1	2	yes	—
256	C₂×C₈	C₂²×#261	1	2	yes	—
257	C₄⋊C₄	—	26	16	no	—
258	C₄⋊C₄	—	52	16	no	—
259	C₂²⋊C₄	—	52	8	no	—
260	C₄²	—	13	2	no	—
261	C₄²	—	13	2	no	—
262	C₄²	—	13	2	no	—
263	C₄²	—	13	2	no	—
264	C₄²	—	13	2	no	—
265	C₄²	—	13	1	no	—
266	C₄²	—	13	1	no	—
267	C₄²	—	52	8	no	—
268	C₄²	C₄⋊#261	1	2	yes	—
269	C₄²	C₂²×#261	1	1	yes	characteristic
270	C₄²	C₂²×#261	1	1	yes	characteristic
271	C₁₃	#20797	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	13	1	no	—
273	C₂³	—	13	2	no	—
274	C₂³	—	13	2	no	—
275	C₂³	—	13	1	no	—
276	C₂³	—	26	4	no	—
277	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	13	4	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	1	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	4	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	8	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	4	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	26	4	no	—
292	C₂×C₄	—	26	4	no	—
293	C₂×C₄	—	26	4	no	—
294	C₂×C₄	—	26	4	no	—
295	C₂×C₄	—	52	8	no	—
296	C₂×C₄	—	52	16	no	—
297	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
299	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
300	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
301	C₂×C₄	C₂³×#261	1	1	yes	characteristic
302	C₈	—	13	4	no	—
303	C₈	—	13	4	no	—
304	C₈	C₂×C₄⋊#261	1	4	yes	—
305	C₈	C₂×C₄⋊#261	1	4	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	13	1	no	—
308	C₂²	—	13	1	no	—
309	C₂²	—	13	4	no	—
310	C₂²	—	13	2	no	—
311	C₂²	—	26	4	no	—
312	C₂²	—	26	4	no	—
313	C₂²	—	26	2	no	—
314	C₂²	#11719059	1	2	yes	—
315	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	13	2	no	—
318	C₄	—	13	2	no	—
319	C₄	—	13	4	no	—
320	C₄	—	26	2	no	—
321	C₄	—	104	8	no	—
322	C₄	#11719059	1	2	yes	—
323	C₄	#11719092	1	4	yes	complemented
324	C₄	C₂²×C₄⋊#261	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	—	13	2	no	—
327	C₂	—	13	2	no	—
328	C₂	—	26	2	no	—
329	C₂	C₂×#11719059	1	1	yes	characteristic
330	C₂	C₂×#11719092	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#20797
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13130773	C₂	semidirect
4. #13130772	C₂	semidirect
5. #13130771	C₂	semidirect
17. C₂×C₄×C₈×C₁₃	C₄	semidirect
18. C₂×C₈×Dic₁₃	C₄	semidirect
19. C₂×C₄×#1055	C₄	semidirect
20. C₄×C₈×D₁₃	C₄	semidirect
21. C₄×C₈×D₁₃	C₄	semidirect
22. C₄×C₈×D₁₃	C₄	semidirect
85. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
98. C₂×#1055	C₄⋊C₄	semidirect
101. C₈×D₁₃	C₄⋊C₄	semidirect
102. C₈×D₁₃	C₄⋊C₄	semidirect
161. C₂×C₄×C₈	#261	semidirect
179. C₄×C₁₃	#427	semidirect
183. Dic₁₃	#427	semidirect
254. C₂×C₈	C₄⋊#261	semidirect
271. C₁₃	#20797	semidirect
323. C₄	#11719092	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719092	2
#13131285	2
#13130773	2
#13130772	2
#13130771	2
C₂×C₄×C₈×D₁₃	2
#20797	13