Subgroups — #16778940

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#431	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#13129661	C₂	1	4	yes	maximal, complemented
4	C₂×#11718346	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₃⋊M₁₆	C₄	1	2	yes	—
6	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
7	#11719125	—	2	1	no	—
8	D₁₃⋊(C₂²⋊C₈)	—	2	1	no	—
9	C₄×#6122	—	2	1	no	—
10	C₄×#6122	—	2	1	no	—
11	#11719119	C₄	1	2	yes	—
12	#11719051	—	2	1	no	—
13	D₁₃⋊(C₄⋊C₈)	—	2	1	no	—
14	#11719049	C₂²	1	2	yes	—
15	C₄×#6118	—	2	1	no	—
16	C₄×#6118	—	2	1	no	—
17	#11719047	C₂²	1	2	yes	—
18	C₄×D₁₃⋊C₈	C₄	1	2	yes	—
19	C₄×D₁₃⋊C₈	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
21	C₂×#88060	—	2	1	no	—
22	C₂×#88054	—	2	1	no	—
23	C₂×C₄×#1055	C₄	1	1	yes	characteristic
24	#11718346	—	2	1	no	—
25	#11718346	—	2	1	no	—
26	#11718346	C₄	1	2	yes	—
27	#11718346	C₂²	1	2	yes	—
28	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
31	C₂×#6122	—	2	2	no	—
32	D₁₃⋊M₁₆	—	2	4	no	—
33	D₁₃⋊M₁₆	C₈	1	8	yes	complemented
34	C₂×#6118	—	2	2	no	—
35	C₂×D₁₃⋊C₈	—	2	2	no	—
36	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
42	C₂²×#1055	—	2	1	no	—
43	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₁₃	D₄	1	1	yes	characteristic
47	C₄²×D₁₃	D₄	1	1	yes	characteristic
48	C₄²×D₁₃	D₄	1	1	yes	characteristic
49	C₄²×D₁₃	Q₈	1	1	yes	characteristic
50	C₄²×D₁₃	C₂³	1	1	yes	characteristic
51	C₂²⋊#1057	—	2	2	no	—
52	#88112	—	2	2	no	—
53	#88111	—	2	2	no	—
54	#88110	—	2	2	no	—
55	#88110	—	2	2	no	—
56	C₄⋊#1057	—	2	2	no	—
57	C₄×#1057	—	2	2	no	—
58	C₄×#1057	—	2	2	no	—
59	C₄×#1057	—	2	2	no	—
60	C₄×#1057	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₁₃	—	2	1	no	—
62	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
63	#88060	—	2	2	no	—
64	#88060	—	2	2	no	—
65	#88054	—	2	2	no	—
66	#88054	—	2	2	no	—
67	C₄×#1055	—	2	1	no	—
68	C₄×#1055	C₂×C₄	1	2	yes	—
69	#7860	—	13	1	no	maximal, Sylow
70	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₁₃	—	4	2	no	—
78	C₂×C₄×D₁₃	—	4	2	no	—
79	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
89	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
90	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
93	#6122	—	2	8	no	—
94	C₂×#1057	—	2	4	no	—
95	C₂×#1057	—	2	4	no	—
96	#6118	—	2	8	no	—
97	D₁₃⋊C₈	—	2	4	no	—
98	D₁₃⋊C₈	—	4	4	no	—
99	D₁₃⋊C₈	C₂×C₈	1	8	yes	complemented
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	C₄²	1	1	yes	characteristic
103	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	4	2	no	—
120	C₂×#431	—	13	1	no	—
121	C₂×#431	—	13	1	no	—
122	C₂×C₄×M₁₆	—	13	1	no	—
123	#1275	—	13	4	no	—
124	C₂³×C₁₃	#95	1	1	yes	characteristic
125	C₂²×D₁₃	—	4	1	no	—
126	C₂²×D₁₃	—	4	1	no	—
127	C₂²×D₁₃	#95	1	1	yes	characteristic
128	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
129	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
136	C₂×C₄×C₁₃	#97	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	#95	1	1	yes	characteristic
143	C₂×Dic₁₃	#95	1	1	yes	characteristic
144	C₂×Dic₁₃	#95	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂²⋊C₈	1	4	yes	—
146	C₂×Dic₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	2	2	no	—
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	4	1	no	—
154	C₄×D₁₃	—	4	1	no	—
155	C₄×D₁₃	—	4	1	no	—
156	C₄×D₁₃	—	4	1	no	—
157	C₄×D₁₃	C₄×C₈	1	2	yes	—
158	C₄×D₁₃	#97	1	2	yes	—
159	C₄×D₁₃	C₂²⋊C₈	1	4	yes	—
160	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
161	C₄×D₁₃	C₂²×C₈	1	2	yes	—
162	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
163	#1057	—	2	8	no	—
164	#1057	—	4	4	no	—
165	C₈×C₁₃	—	4	2	no	—
166	#1055	—	4	2	no	—
167	C₂²×M₁₆	—	13	2	no	—
168	C₂²×C₄²	—	13	1	no	—
169	#431	—	13	2	no	—
170	#431	—	13	2	no	—
171	#431	—	13	2	no	—
172	#431	—	13	2	no	—
173	#431	—	26	1	no	—
174	#431	—	26	1	no	—
175	#431	—	26	1	no	—
176	#431	—	26	1	no	—
177	C₂×C₄⋊C₈	—	26	1	no	—
178	C₂×C₄⋊C₈	—	26	1	no	—
179	C₂×C₂²⋊C₈	—	26	1	no	—
180	C₂×C₂²⋊C₈	—	26	1	no	—
181	C₄×M₁₆	—	26	1	no	—
182	C₄×M₁₆	—	26	1	no	—
183	C₄×M₁₆	—	26	1	no	—
184	C₄×M₁₆	—	26	1	no	—
185	C₂×#97	—	13	2	no	—
186	C₂×C₄×C₈	—	13	2	no	—
187	C₂×C₄×C₈	—	13	1	no	—
188	C₂×C₄×C₈	—	13	2	no	—
189	C₂×C₄×C₈	#261	1	1	yes	characteristic
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	#336	1	2	yes	—
193	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
194	C₂×D₁₃	—	2	1	no	—
195	C₂×D₁₃	—	2	1	no	—
196	C₂×D₁₃	—	2	1	no	—
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	—	4	4	no	—
199	C₂×D₁₃	#336	1	2	yes	—
200	C₂×D₁₃	C₂×C₄⋊C₈	1	2	yes	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	#336	1	2	yes	—
204	C₄×C₁₃	H(3,ℤ/4)	1	2	yes	—
205	C₄×C₁₃	#338	1	2	yes	—
206	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
207	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
208	Dic₁₃	—	2	1	no	—
209	Dic₁₃	—	2	1	no	—
210	Dic₁₃	—	2	1	no	—
211	Dic₁₃	—	2	1	no	—
212	Dic₁₃	#336	1	2	yes	—
213	Dic₁₃	C₂×C₂²⋊C₈	1	2	yes	—
214	C₂³×C₄	—	13	1	no	—
215	C₂³×C₄	—	13	1	no	—
216	C₂³×C₄	—	13	1	no	—
217	C₂×M₁₆	—	13	8	no	—
218	C₂×M₁₆	—	26	2	no	—
219	C₂×M₁₆	—	26	2	no	—
220	C₂×M₁₆	—	26	4	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	13	4	no	—
223	C₂²×C₈	—	26	2	no	—
224	C₂²×C₈	—	26	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	13	1	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	1	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	C₂×#261	1	1	yes	characteristic
237	C₄⋊C₈	—	26	2	no	—
238	C₄⋊C₈	—	26	2	no	—
239	C₄⋊C₈	—	26	2	no	—
240	C₄⋊C₈	—	26	2	no	—
241	C₂²⋊C₈	—	26	2	no	—
242	C₂²⋊C₈	—	26	2	no	—
243	C₂²⋊C₈	—	26	2	no	—
244	C₂²⋊C₈	—	26	2	no	—
245	#97	—	26	2	no	—
246	#97	—	26	2	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	13	4	no	—
249	C₄×C₈	—	13	2	no	—
250	C₄×C₈	—	26	2	no	—
251	C₄×C₈	—	26	2	no	—
252	C₄×C₈	—	26	2	no	—
253	C₄×C₈	—	26	1	no	—
254	C₄×C₈	C₂×#261	1	2	yes	—
255	C₂×C₁₃	—	2	2	no	—
256	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
257	C₂×C₁₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
258	C₂×C₁₃	C₂×#338	1	1	yes	characteristic
259	D₁₃	—	2	2	no	—
260	D₁₃	—	2	2	no	—
261	C₂⁴	—	13	1	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	13	4	no	—
264	C₂²×C₄	—	13	4	no	—
265	C₂²×C₄	—	13	1	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	1	no	—
271	C₂²×C₄	—	13	1	no	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	13	1	no	—
279	C₂²×C₄	—	52	2	no	—
280	C₂²×C₄	—	52	2	no	—
281	C₂²×C₄	C₄×#261	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
283	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
284	M₁₆	—	26	8	no	—
285	M₁₆	—	26	8	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	—	13	8	no	—
289	C₂×C₈	—	26	4	no	—
290	C₂×C₈	—	26	4	no	—
291	C₂×C₈	—	26	4	no	—
292	C₂×C₈	—	26	2	no	—
293	C₂×C₈	—	52	2	no	—
294	C₂×C₈	—	52	4	no	—
295	C₄²	—	13	1	no	—
296	C₄²	—	13	1	no	—
297	C₄²	—	13	1	no	—
298	C₄²	—	13	1	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	1	no	—
301	C₄²	—	13	1	no	—
302	C₄²	—	13	1	no	—
303	C₄²	—	13	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	13	1	no	—
307	C₄²	C₄×#261	1	1	yes	characteristic
308	C₄²	C₄⋊#261	1	1	yes	characteristic
309	C₄²	C₂²⋊#261	1	1	yes	characteristic
310	C₄²	C₂²×#261	1	1	yes	characteristic
311	C₁₃	#7860	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	13	1	no	—
313	C₂³	—	13	1	no	—
314	C₂³	—	13	4	no	—
315	C₂³	—	52	1	no	—
316	C₂³	—	52	1	no	—
317	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	13	4	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	13	4	no	—
324	C₂×C₄	—	13	4	no	—
325	C₂×C₄	—	13	2	no	—
326	C₂×C₄	—	13	2	no	—
327	C₂×C₄	—	13	2	no	—
328	C₂×C₄	—	13	2	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	—	52	1	no	—
341	C₂×C₄	—	52	1	no	—
342	C₂×C₄	—	52	1	no	—
343	C₂×C₄	C₈×#261	1	2	yes	—
344	C₂×C₄	#88100	1	2	yes	—
345	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
346	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
347	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
350	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	26	8	no	—
353	C₈	—	52	2	no	—
354	C₈	—	52	4	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	13	2	no	—
358	C₂²	—	13	2	no	—
359	C₂²	—	26	1	no	—
360	C₂²	—	26	1	no	—
361	C₂²	—	26	1	no	—
362	C₂²	—	26	1	no	—
363	C₂²	—	52	4	no	—
364	C₂²	(C₂×#261)⋊C₈	1	2	yes	—
365	C₂²	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	13	2	no	—
369	C₄	—	13	2	no	—
370	C₄	—	26	1	no	—
371	C₄	—	26	1	no	—
372	C₄	—	26	1	no	—
373	C₄	—	26	1	no	—
374	C₄	#11718235	1	2	yes	—
375	C₄	(C₂×#261)⋊C₈	1	2	yes	—
376	C₄	#11718269	1	2	yes	—
377	C₄	C₂×C₈×#261	1	1	yes	characteristic
378	C₄	C₂×#88100	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	26	2	no	—
381	C₂	—	26	2	no	—
382	C₂	C₂×(C₂×#261)⋊C₈	1	1	yes	characteristic
383	C₂	C₂×#11718235	1	1	yes	characteristic
384	C₂	C₂×#11718269	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄²×D₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#7860
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13129661	C₂	semidirect
33. D₁₃⋊M₁₆	C₈	semidirect
99. D₁₃⋊C₈	C₂×C₈	semidirect
311. C₁₃	#7860	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#431	2
C₄×D₁₃⋊M₁₆	2
#13129661	2
C₂×#11718346	2
#7860	13