Subgroups — #16795000

Order 3328.

668 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131513	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719051	C₂	1	1	yes	maximal, characteristic
3	#13130414	C₂	1	4	yes	maximal, complemented
4	#13130411	C₂	1	4	yes	maximal, complemented
5	#13130386	C₂	1	4	yes	maximal, complemented
6	D₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
8	#11719130	C₂²	1	2	yes	—
9	#11719127	—	2	1	no	—
10	C₂×#88111	—	2	1	no	—
11	#11719119	—	2	1	no	—
12	C₂×C₄⋊#1057	C₂²	1	2	yes	—
13	C₂×C₄×#1057	—	2	1	no	—
14	#11719054	—	2	1	no	—
15	#11719052	C₂²	1	2	yes	—
16	#11719051	—	2	1	no	—
17	#11719051	—	2	1	no	—
18	#11719051	C₂²	1	2	yes	—
19	#11719051	C₂²	1	2	yes	—
20	#11719051	C₂²	1	2	yes	—
21	C₄×#6118	—	2	1	no	—
22	C₁₃×#431	C₄	1	1	yes	characteristic
23	#11718586	C₄	1	1	yes	characteristic
24	#11718494	C₂²	1	2	yes	—
25	D₁₃×C₄⋊C₈	C₄	1	2	yes	—
26	D₁₃×C₂²⋊C₈	C₄	1	2	yes	—
27	#11718379	C₂²	1	2	yes	—
28	#11718346	C₄	1	2	yes	—
29	C₄×C₈×D₁₃	C₂²	1	2	yes	—
30	#11718255	C₂²	1	8	yes	complemented
31	#11718254	C₂²	1	8	yes	complemented
32	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
35	C₂²×#1057	—	2	2	no	—
36	C₂×#6118	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
42	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₁₃	D₄	1	1	yes	characteristic
45	C₄²×D₁₃	D₄	1	1	yes	characteristic
46	C₄²×D₁₃	Q₈	1	1	yes	characteristic
47	C₄²×D₁₃	Q₈	1	1	yes	characteristic
48	C₄²×D₁₃	C₂³	1	1	yes	characteristic
49	C₄²×D₁₃	C₂³	1	1	yes	characteristic
50	C₂²⋊#1057	—	2	2	no	—
51	#88111	—	2	2	no	—
52	#88111	—	2	2	no	—
53	#88111	—	2	2	no	—
54	#88110	—	2	2	no	—
55	C₄⋊#1057	—	2	2	no	—
56	C₄⋊#1057	—	2	2	no	—
57	C₄⋊#1057	C₂³	1	4	yes	—
58	C₄⋊#1057	C₂³	1	4	yes	—
59	C₄×#1057	—	2	2	no	—
60	C₄×#1057	—	2	2	no	—
61	C₄×#1057	—	2	2	no	—
62	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
63	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
65	C₂²⋊#1055	C₂×C₄	1	2	yes	—
66	#88060	C₂×C₄	1	4	yes	—
67	#88054	C₂×C₄	1	4	yes	—
68	C₈×Dic₁₃	C₂³	1	4	yes	—
69	C₄⋊#1055	C₂×C₄	1	2	yes	—
70	C₄×#1055	C₂×C₄	1	2	yes	—
71	#17806	—	13	1	no	maximal, Sylow
72	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	—	2	4	no	—
80	C₂×C₄×D₁₃	—	2	4	no	—
81	C₂×C₄×D₁₃	—	4	2	no	—
82	C₂×C₄×D₁₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	C₂×D₄	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	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×#1057	—	2	4	no	—
95	C₂×#1057	—	2	4	no	—
96	C₂×#1057	—	4	4	no	—
97	#6118	—	4	4	no	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
99	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
115	C₂×#1055	C₂²×C₄	1	4	yes	—
116	C₈×D₁₃	—	2	8	no	—
117	#2920	—	13	1	no	—
118	C₂×#431	—	13	1	no	—
119	C₂×#431	—	13	1	no	—
120	#1574	—	13	4	no	—
121	#1573	—	13	4	no	—
122	#1564	—	13	4	no	—
123	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×D₁₃	—	2	2	no	—
125	C₂²×D₁₃	—	2	2	no	—
126	C₂²×D₁₃	—	4	1	no	—
127	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	#108	1	4	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
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	—	2	2	no	—
154	C₄×D₁₃	—	2	2	no	—
155	C₄×D₁₃	—	2	2	no	—
156	C₄×D₁₃	—	4	1	no	—
157	C₄×D₁₃	—	4	1	no	—
158	C₄×D₁₃	#131	1	4	yes	—
159	#1057	—	4	4	no	—
160	#1057	—	4	4	no	—
161	C₈×C₁₃	—	2	4	no	—
162	#1055	—	2	4	no	—
163	C₂²×C₄²	—	13	1	no	—
164	#432	—	13	2	no	—
165	#432	—	26	1	no	—
166	#431	—	13	1	no	—
167	#431	—	13	2	no	—
168	#431	—	13	2	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	#261	1	1	yes	characteristic
177	#423	—	13	2	no	—
178	C₂×C₄⋊C₈	—	13	2	no	—
179	C₂×C₄⋊C₈	—	13	2	no	—
180	#407	—	26	1	no	—
181	C₂×C₂²⋊C₈	—	13	2	no	—
182	C₂×C₂²⋊C₈	—	26	1	no	—
183	C₄×M₁₆	—	26	1	no	—
184	C₂×C₄×C₈	—	13	2	no	—
185	C₂×C₄×C₈	—	26	1	no	—
186	#335	—	13	8	no	—
187	#334	—	13	8	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	C₂×#108	1	2	yes	—
191	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₂×D₁₃	—	2	1	no	—
193	C₂×D₁₃	—	2	1	no	—
194	C₂×D₁₃	—	2	1	no	—
195	C₂×D₁₃	—	2	1	no	—
196	C₂×D₁₃	—	2	1	no	—
197	C₂×D₁₃	—	4	4	no	—
198	C₂×D₁₃	#424	1	2	yes	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	C₂×#108	1	2	yes	—
203	C₄×C₁₃	C₂×#108	1	2	yes	—
204	C₄×C₁₃	C₂×#131	1	2	yes	—
205	Dic₁₃	—	2	1	no	—
206	Dic₁₃	—	2	1	no	—
207	Dic₁₃	—	2	1	no	—
208	Dic₁₃	—	2	1	no	—
209	Dic₁₃	#424	1	2	yes	—
210	Dic₁₃	#575	1	2	yes	—
211	C₂³×C₄	—	13	1	no	—
212	C₂³×C₄	—	13	1	no	—
213	C₂³×C₄	—	13	1	no	—
214	C₂×M₁₆	—	26	2	no	—
215	C₂²×C₈	—	13	4	no	—
216	C₂²×C₈	—	26	2	no	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	1	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	C₂×#261	1	1	yes	characteristic
229	C₄⋊C₈	—	13	4	no	—
230	C₄⋊C₈	—	13	4	no	—
231	C₄⋊C₈	—	13	4	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₄⋊C₈	—	26	2	no	—
234	C₄⋊C₈	—	26	2	no	—
235	C₄⋊C₈	C₂×#261	1	2	yes	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	—	13	4	no	—
238	C₂²⋊C₈	—	26	2	no	—
239	C₂²⋊C₈	—	26	2	no	—
240	C₂²⋊C₈	—	26	2	no	—
241	C₂²⋊C₈	—	26	2	no	—
242	C₂²⋊C₈	C₂×#261	1	2	yes	—
243	#97	—	26	2	no	—
244	C₄×C₈	—	13	4	no	—
245	C₄×C₈	—	13	2	no	—
246	C₄×C₈	—	26	2	no	—
247	C₄×C₈	—	26	2	no	—
248	C₄×C₈	—	26	2	no	—
249	C₄×C₈	C₂×#261	1	2	yes	—
250	C₂×C₁₃	—	2	2	no	—
251	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
252	C₂×C₁₃	#2911	1	1	yes	characteristic
253	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
254	D₁₃	—	2	2	no	—
255	D₁₃	—	4	1	no	—
256	C₂⁴	—	13	1	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	13	1	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	13	1	no	—
264	C₂²×C₄	—	13	1	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	—	26	4	no	—
273	C₂²×C₄	—	26	4	no	—
274	C₂²×C₄	—	52	2	no	—
275	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
276	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
277	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
278	M₁₆	—	52	4	no	—
279	C₂×C₈	—	13	4	no	—
280	C₂×C₈	—	26	8	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	26	4	no	—
283	C₂×C₈	—	52	4	no	—
284	C₂×C₈	C₂²×#261	1	4	yes	—
285	C₄²	—	13	1	no	—
286	C₄²	—	13	1	no	—
287	C₄²	—	13	1	no	—
288	C₄²	—	13	1	no	—
289	C₄²	—	13	1	no	—
290	C₄²	—	13	1	no	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	1	no	—
293	C₄²	—	13	1	no	—
294	C₄²	—	13	1	no	—
295	C₄²	—	13	1	no	—
296	C₄²	—	13	1	no	—
297	C₄²	C₄⋊#261	1	1	yes	characteristic
298	C₄²	C₄⋊#261	1	1	yes	characteristic
299	C₄²	C₂²×#261	1	1	yes	characteristic
300	C₄²	C₂²×#261	1	1	yes	characteristic
301	C₁₃	#17806	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	13	1	no	—
303	C₂³	—	13	1	no	—
304	C₂³	—	26	2	no	—
305	C₂³	—	26	2	no	—
306	C₂³	—	52	1	no	—
307	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	13	1	no	—
311	C₂×C₄	—	13	1	no	—
312	C₂×C₄	—	13	4	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	1	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	—	26	2	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	52	1	no	—
330	C₂×C₄	—	52	1	no	—
331	C₂×C₄	#88104	1	4	yes	—
332	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
337	C₂×C₄	C₂³×#261	1	1	yes	characteristic
338	C₈	—	2	4	no	—
339	C₈	—	26	4	no	—
340	C₈	—	52	4	no	—
341	C₈	—	52	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	13	2	no	—
345	C₂²	—	26	1	no	—
346	C₂²	—	26	1	no	—
347	C₂²	—	26	1	no	—
348	C₂²	—	26	1	no	—
349	C₂²	—	26	1	no	—
350	C₂²	—	52	4	no	—
351	C₂²	C₂×#88104	1	2	yes	—
352	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	13	2	no	—
357	C₄	—	13	2	no	—
358	C₄	—	26	1	no	—
359	C₄	—	26	1	no	—
360	C₄	—	26	1	no	—
361	C₄	—	26	1	no	—
362	C₄	C₂×#88104	1	2	yes	—
363	C₄	C₂×#88104	1	2	yes	—
364	C₄	D₁₃⋊#131	1	2	yes	—
365	C₂	—	2	2	no	—
366	C₂	—	26	2	no	—
367	C₂	—	52	1	no	—
368	C₂	D₁₃⋊#424	1	1	yes	characteristic
369	C₂	#13131489	1	1	yes	characteristic
370	C₂	C₂²×#88104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#431	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#431	#261
hypocenter	C₁₃	#17806
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #13130414	C₂	semidirect
4. #13130411	C₂	semidirect
5. #13130386	C₂	semidirect
30. #11718255	C₂²	semidirect
31. #11718254	C₂²	semidirect
301. C₁₃	#17806	semidirect

Maximal subgroups

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

subgroup	index
#13131513	2
C₂×#11719051	2
#13130414	2
#13130411	2
#13130386	2
D₁₃×#431	2
#17806	13