Subgroups — #16795061

Order 3328.

618 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131514	C₂	1	1	yes	maximal, characteristic
2	D₁₃⋊#432	C₂	1	1	yes	maximal, characteristic
3	#13130410	C₂	1	4	yes	maximal, complemented
4	#13130408	C₂	1	4	yes	maximal, complemented
5	#13130400	C₂	1	4	yes	maximal, complemented
6	#13123468	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
8	#11719130	C₂²	1	2	yes	—
9	#11719129	C₂²	1	2	yes	—
10	#11719128	—	2	1	no	—
11	#11719126	—	2	1	no	—
12	D₁₃⋊(C₂²⋊C₈)	—	2	1	no	—
13	C₄×#6122	—	2	1	no	—
14	#11719055	—	2	1	no	—
15	#11719055	—	2	1	no	—
16	#11719054	C₂²	1	2	yes	—
17	#11719054	C₂²	1	2	yes	—
18	#11719051	C₂²	1	2	yes	—
19	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
20	#11719049	—	2	1	no	—
21	#11719047	—	2	1	no	—
22	C₁₃×#423	C₄	1	1	yes	characteristic
23	#11718787	C₄	1	2	yes	—
24	M₁₆×Dic₁₃	C₂²	1	2	yes	—
25	#11718585	C₄	1	1	yes	characteristic
26	#11718496	C₄	1	2	yes	—
27	#11718496	C₄	1	2	yes	—
28	#11718494	C₂²	1	2	yes	—
29	#11718494	C₂²	1	2	yes	—
30	#11718256	C₂²	1	4	yes	complemented
31	#11718255	C₂²	1	4	yes	complemented
32	#11718254	C₂²	1	4	yes	complemented
33	#11718253	C₂²	1	4	yes	complemented
34	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
37	C₂×#6122	—	2	2	no	—
38	C₂×D₁₃⋊C₈	—	2	2	no	—
39	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
44	C₂×#6099	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₁₃	C₂×C₄	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	Q₈	1	1	yes	characteristic
51	C₄²×D₁₃	C₂³	1	1	yes	characteristic
52	C₄²×D₁₃	C₂³	1	1	yes	characteristic
53	C₂²⋊#1057	—	2	2	no	—
54	#88112	—	2	2	no	—
55	#88112	—	2	2	no	—
56	#88111	—	2	2	no	—
57	#88111	—	2	2	no	—
58	#88111	—	2	2	no	—
59	#88110	—	2	2	no	—
60	#88110	—	2	2	no	—
61	#88110	—	2	2	no	—
62	C₄⋊#1057	C₂³	1	4	yes	—
63	C₄⋊#1057	C₂³	1	4	yes	—
64	C₄×#1057	—	2	2	no	—
65	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	#88060	C₂×C₄	1	4	yes	—
68	#88060	C₂×C₄	1	4	yes	—
69	#88055	C₂×C₄	1	4	yes	—
70	C₈×Dic₁₃	C₂³	1	4	yes	—
71	C₄⋊#1055	C₂×C₄	1	2	yes	—
72	C₄⋊#1055	C₂×C₄	1	2	yes	—
73	#17821	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	4	2	no	—
82	C₂×C₄×D₁₃	—	4	1	no	—
83	C₂×C₄×D₁₃	—	4	1	no	—
84	C₂×C₄×D₁₃	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
97	#6122	—	4	4	no	—
98	C₂×#1057	—	2	4	no	—
99	C₂×#1057	—	2	4	no	—
100	D₁₃⋊C₈	—	4	4	no	—
101	C₁₃×M₁₆	—	4	2	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
117	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
118	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
119	#6099	—	4	2	no	—
120	C₂×#1055	C₂²×C₄	1	4	yes	—
121	#2920	—	13	1	no	—
122	#2920	—	13	1	no	—
123	C₂×#432	—	13	1	no	—
124	#1573	—	13	4	no	—
125	#1572	—	13	4	no	—
126	#1569	—	13	4	no	—
127	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₁₃	—	2	2	no	—
129	C₂²×D₁₃	—	4	1	no	—
130	C₂²×D₁₃	—	4	1	no	—
131	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	4	1	no	—
145	C₂×Dic₁₃	—	4	1	no	—
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	—	2	1	no	—
155	C₄×D₁₃	—	2	1	no	—
156	C₄×D₁₃	—	2	1	no	—
157	C₄×D₁₃	—	2	1	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	4	1	no	—
161	C₄×D₁₃	—	4	1	no	—
162	C₄×D₁₃	—	4	1	no	—
163	C₄×D₁₃	—	4	1	no	—
164	C₄×D₁₃	—	4	1	no	—
165	C₄×D₁₃	—	4	1	no	—
166	#1057	—	4	4	no	—
167	#1057	—	4	4	no	—
168	C₈×C₁₃	—	2	4	no	—
169	#1055	—	2	4	no	—
170	C₂²×C₄²	—	13	1	no	—
171	#432	—	13	2	no	—
172	#432	—	13	2	no	—
173	#432	—	13	2	no	—
174	#432	—	13	2	no	—
175	#432	—	13	2	no	—
176	#432	—	26	1	no	—
177	#432	—	26	1	no	—
178	#432	—	26	1	no	—
179	#431	—	13	2	no	—
180	#431	—	13	2	no	—
181	#431	—	13	2	no	—
182	#431	—	26	1	no	—
183	#423	—	13	1	no	—
184	#423	—	13	2	no	—
185	#423	#261	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	13	2	no	—
187	#407	—	13	2	no	—
188	#407	—	26	1	no	—
189	C₂×C₂²⋊C₈	—	26	1	no	—
190	C₄×M₁₆	—	13	2	no	—
191	C₄×M₁₆	—	26	1	no	—
192	C₂×#97	—	26	1	no	—
193	#335	—	13	4	no	—
194	#335	—	13	4	no	—
195	#334	—	13	4	no	—
196	#334	—	13	4	no	—
197	C₂²×C₁₃	—	2	1	no	—
198	C₂²×C₁₃	—	2	1	no	—
199	C₂²×C₁₃	—	2	1	no	—
200	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₂×D₁₃	—	2	1	no	—
202	C₂×D₁₃	—	2	1	no	—
203	C₂×D₁₃	—	2	1	no	—
204	C₂×D₁₃	—	2	1	no	—
205	C₂×D₁₃	—	2	1	no	—
206	C₂×D₁₃	—	4	4	no	—
207	C₂×D₁₃	#430	1	2	yes	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	#568	1	1	yes	characteristic
214	C₄×C₁₃	#568	1	1	yes	characteristic
215	Dic₁₃	—	2	1	no	—
216	Dic₁₃	—	2	1	no	—
217	Dic₁₃	—	2	1	no	—
218	Dic₁₃	—	2	1	no	—
219	Dic₁₃	—	2	1	no	—
220	Dic₁₃	#575	1	2	yes	—
221	C₂³×C₄	—	13	1	no	—
222	C₂³×C₄	—	13	1	no	—
223	C₂³×C₄	—	13	1	no	—
224	C₂×M₁₆	—	13	2	no	—
225	C₂×M₁₆	—	26	2	no	—
226	C₂×M₁₆	C₂×#261	1	2	yes	—
227	C₂²×C₈	—	26	2	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	—	13	1	no	—
237	C₂×C₄²	—	13	1	no	—
238	C₂×C₄²	—	13	1	no	—
239	C₂×C₄²	C₂×#261	1	1	yes	characteristic
240	C₄⋊C₈	—	13	4	no	—
241	C₄⋊C₈	—	13	4	no	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	13	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	—	26	2	no	—
246	C₄⋊C₈	C₂×#261	1	2	yes	—
247	C₄⋊C₈	C₂×#261	1	2	yes	—
248	C₂²⋊C₈	—	13	4	no	—
249	C₂²⋊C₈	—	13	4	no	—
250	C₂²⋊C₈	—	26	2	no	—
251	C₂²⋊C₈	—	26	2	no	—
252	C₂²⋊C₈	—	26	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	#97	—	13	4	no	—
255	#97	—	26	2	no	—
256	#97	—	26	2	no	—
257	#97	—	26	2	no	—
258	C₄×C₈	—	13	4	no	—
259	C₄×C₈	—	26	2	no	—
260	C₂×C₁₃	—	4	1	no	—
261	C₂×C₁₃	#2905	1	1	yes	characteristic
262	C₂×C₁₃	#2911	1	1	yes	characteristic
263	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
264	D₁₃	—	2	2	no	—
265	D₁₃	—	4	1	no	—
266	C₂⁴	—	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	—	13	1	no	—
280	C₂²×C₄	—	13	1	no	—
281	C₂²×C₄	—	13	1	no	—
282	C₂²×C₄	—	52	1	no	—
283	C₂²×C₄	—	52	1	no	—
284	C₂²×C₄	—	52	2	no	—
285	C₂²×C₄	—	52	2	no	—
286	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
287	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
288	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
289	M₁₆	—	4	2	no	—
290	M₁₆	—	52	2	no	—
291	M₁₆	—	52	4	no	—
292	C₂×C₈	—	13	4	no	—
293	C₂×C₈	—	26	4	no	—
294	C₂×C₈	—	26	4	no	—
295	C₂×C₈	—	52	4	no	—
296	C₂×C₈	C₂²×#261	1	4	yes	—
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	—	13	1	no	—
308	C₄²	—	13	1	no	—
309	C₄²	C₄⋊#261	1	1	yes	characteristic
310	C₄²	C₄⋊#261	1	1	yes	characteristic
311	C₄²	C₂²×#261	1	1	yes	characteristic
312	C₄²	C₂²×#261	1	1	yes	characteristic
313	C₁₃	#17821	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	13	1	no	—
315	C₂³	—	13	1	no	—
316	C₂³	—	26	2	no	—
317	C₂³	—	52	1	no	—
318	C₂³	—	52	1	no	—
319	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	13	1	no	—
328	C₂×C₄	—	13	1	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	1	no	—
336	C₂×C₄	—	26	1	no	—
337	C₂×C₄	—	26	1	no	—
338	C₂×C₄	—	26	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	—	52	1	no	—
344	C₂×C₄	—	52	1	no	—
345	C₂×C₄	—	52	1	no	—
346	C₂×C₄	—	52	1	no	—
347	C₂×C₄	C₂×C₄⋊#261	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×C₄⋊#261	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
352	C₂×C₄	C₂³×#261	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	26	4	no	—
355	C₈	—	52	4	no	—
356	C₈	—	52	4	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	13	2	no	—
361	C₂²	—	26	1	no	—
362	C₂²	—	26	1	no	—
363	C₂²	—	26	1	no	—
364	C₂²	—	26	1	no	—
365	C₂²	—	26	1	no	—
366	C₂²	—	52	4	no	—
367	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	13	2	no	—
374	C₄	—	26	1	no	—
375	C₄	—	26	1	no	—
376	C₄	—	26	1	no	—
377	C₄	—	26	1	no	—
378	C₄	—	26	1	no	—
379	C₄	#11719621	1	1	yes	characteristic
380	C₄	#11719625	1	1	yes	characteristic
381	C₂	—	4	1	no	—
382	C₂	—	26	2	no	—
383	C₂	—	52	1	no	—
384	C₂	#13131480	1	1	yes	characteristic
385	C₂	#13131489	1	1	yes	characteristic
386	C₂	D₁₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#423	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#423	#261
hypocenter	C₁₃	#17821
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
3. #13130410	C₂	semidirect
4. #13130408	C₂	semidirect
5. #13130400	C₂	semidirect
30. #11718256	C₂²	semidirect
31. #11718255	C₂²	semidirect
32. #11718254	C₂²	semidirect
33. #11718253	C₂²	semidirect
313. C₁₃	#17821	semidirect

Maximal subgroups

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

subgroup	index
#13131514	2
D₁₃⋊#432	2
#13130410	2
#13130408	2
#13130400	2
#13123468	2
#17821	13