Subgroups — #16792702

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131645	C₂	1	1	yes	maximal, characteristic
2	#41×#1057	C₂	1	1	yes	maximal, characteristic
3	#13130057	C₂	1	2	yes	maximal, complemented
4	#13130053	C₂	1	2	yes	maximal, complemented
5	#13130047	C₂	1	2	yes	maximal, complemented
6	#13130043	C₂	1	1	yes	maximal, characteristic, complemented
7	#13130041	C₂	1	2	yes	maximal, complemented
8	#13130039	C₂	1	1	yes	maximal, characteristic, complemented
9	M₁₆×#1057	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1057⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123476	C₂	1	1	yes	maximal, characteristic
12	#41×Dic₁₃	C₂²	1	1	yes	characteristic
13	#11719158	C₄	1	2	yes	—
14	Q₈×#1057	C₄	1	2	yes	—
15	#11719145	C₄	1	2	yes	—
16	#11719144	C₄	1	4	yes	—
17	D₄×#1057	C₄	1	2	yes	—
18	D₄×#1057	C₄	1	4	yes	—
19	#11719125	C₂²	1	2	yes	—
20	C₄×#6122	C₂²	1	2	yes	—
21	#11719119	C₂²	1	1	yes	characteristic
22	#11719119	C₂²	1	1	yes	characteristic
23	#11719119	C₂²	1	2	yes	—
24	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
25	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
26	C₂×C₄×#1057	C₂²	1	2	yes	—
27	C₁₃×#408	C₄	1	1	yes	characteristic
28	#11718895	C₄	1	1	yes	characteristic
29	#11718780	C₄	1	1	yes	characteristic
30	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
31	#11718763	C₄	1	1	yes	characteristic
32	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
33	#11718381	C₄	1	2	yes	—
34	#11718381	C₄	1	2	yes	—
35	#11718379	C₂²	1	2	yes	—
36	#11718379	C₂²	1	2	yes	—
37	#1057⋊C₈	C₂²	1	1	yes	characteristic
38	#1057⋊C₈	C₂²	1	1	yes	characteristic
39	#1057⋊C₈	C₂²	1	1	yes	characteristic
40	#1057⋊C₈	C₂²	1	1	yes	characteristic
41	#1057⋊C₈	C₂²	1	2	yes	—
42	#1057⋊C₈	C₂²	1	2	yes	—
43	#11718251	C₂²	1	2	yes	—
44	#11718251	C₂²	1	2	yes	—
45	C₈×#1057	C₂²	1	2	yes	—
46	C₈×#1057	C₂²	1	2	yes	—
47	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
48	C₂×#6122	C₂×C₄	1	4	yes	—
49	C₂²×#1057	C₂×C₄	1	2	yes	—
50	C₂²×#1057	C₂×C₄	1	4	yes	—
51	C₂²×#1057	C₂×C₄	1	2	yes	—
52	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
55	Q₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
56	D₄×Dic₁₃	C₂×C₄	1	2	yes	—
57	D₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	D₄×Dic₁₃	C₂×C₄	1	2	yes	—
59	D₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	#88180	C₂×C₄	1	1	yes	characteristic
61	#88180	C₂×C₄	1	2	yes	—
62	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
64	C₂×#6099	C₂×C₄	1	1	yes	characteristic
65	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊#1057	C₂×C₄	1	2	yes	—
67	C₂²⋊#1057	C₂×C₄	1	4	yes	—
68	C₂²⋊#1057	C₂×C₄	1	2	yes	—
69	C₂²⋊#1057	C₂×C₄	1	4	yes	—
70	#88112	C₂×C₄	1	4	yes	—
71	#88110	C₂×C₄	1	4	yes	—
72	C₄⋊#1057	C₂×C₄	1	2	yes	—
73	C₄⋊#1057	C₂×C₄	1	4	yes	—
74	C₄⋊#1057	C₂×C₄	1	2	yes	—
75	C₄×#1057	C₂×C₄	1	2	yes	—
76	C₄×#1057	C₂×C₄	1	4	yes	—
77	C₄×#1057	C₂×C₄	1	2	yes	—
78	C₄×#1057	C₂³	1	2	yes	—
79	C₄×#1057	C₂³	1	1	yes	characteristic
80	C₄×#1057	C₂³	1	2	yes	—
81	C₄×#1057	C₂³	1	1	yes	characteristic
82	C₄×#1057	C₂³	1	1	yes	characteristic
83	C₄×#1057	C₂³	1	1	yes	characteristic
84	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
85	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
86	C₂²⋊#1055	C₂×C₄	1	2	yes	—
87	C₂²⋊#1055	C₂×C₄	1	2	yes	—
88	#88055	C₂×C₄	1	2	yes	—
89	#88054	C₂×C₄	1	2	yes	—
90	#88054	C₂×C₄	1	2	yes	—
91	#88054	C₂×C₄	1	2	yes	—
92	#88054	C₂×C₄	1	2	yes	—
93	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
96	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
97	C₈×Dic₁₃	C₂³	1	2	yes	—
98	#17330	—	13	1	no	maximal, Sylow
99	C₁₃×#41	—	2	2	no	—
100	C₁₃×#41	—	2	2	no	—
101	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
102	C₂×D₄×C₁₃	C₄²	1	2	yes	—
103	C₂×D₄×C₁₃	C₄²	1	1	yes	characteristic
104	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
107	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
110	#6122	—	4	4	no	—
111	C₂×#1057	—	2	4	no	—
112	C₂×#1057	—	2	8	no	—
113	C₂×#1057	—	4	2	no	—
114	C₂×#1057	C₂²×C₄	1	4	yes	—
115	C₂×#1057	C₂²×C₄	1	2	yes	—
116	C₂×#1057	C₂²×C₄	1	2	yes	—
117	C₂×#1057	C₂²×C₄	1	2	yes	—
118	C₂×#1057	C₂²×C₄	1	4	yes	—
119	C₂×#1057	C₂²×C₄	1	2	yes	—
120	C₁₃×M₁₆	—	2	2	no	—
121	C₂×C₈×C₁₃	—	2	2	no	—
122	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
125	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
126	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
127	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
128	C₄⋊Dic₁₃	C₄²	1	2	yes	—
129	C₄⋊Dic₁₃	C₄²	1	2	yes	—
130	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
131	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
132	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
133	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
134	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
135	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
136	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
137	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
138	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
139	#6099	—	2	2	no	—
140	C₂×#1055	—	2	2	no	—
141	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
142	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
143	C₂×#1055	C₂²×C₄	1	2	yes	—
144	#2965	—	13	1	no	—
145	C₈×#41	—	13	1	no	—
146	#2922	—	13	1	no	—
147	#1454	—	13	2	no	—
148	#1453	—	13	2	no	—
149	#1451	—	13	1	no	—
150	#1451	—	13	2	no	—
151	#1450	—	13	1	no	—
152	#1450	—	13	2	no	—
153	C₈×M₁₆	—	13	1	no	—
154	C₂×#322	—	13	1	no	—
155	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
156	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
157	Q₈×C₁₃	—	2	2	no	—
158	D₄×C₁₃	—	2	4	no	—
159	D₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
163	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
164	C₂×C₄×C₁₃	#131	1	2	yes	—
165	C₂×C₄×C₁₃	#131	1	2	yes	—
166	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×Dic₁₃	—	2	2	no	—
168	C₂×Dic₁₃	—	2	4	no	—
169	C₂×Dic₁₃	—	2	2	no	—
170	C₂×Dic₁₃	—	2	4	no	—
171	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
172	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
173	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
174	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
175	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
177	#1057	—	2	4	no	—
178	#1057	—	2	2	no	—
179	#1057	—	2	2	no	—
180	#1057	—	2	2	no	—
181	#1057	—	2	4	no	—
182	#1057	#131	1	4	yes	—
183	C₈×C₁₃	—	2	1	no	—
184	C₈×C₁₃	—	2	1	no	—
185	C₈×C₁₃	—	2	2	no	—
186	#1055	—	2	1	no	—
187	#1055	—	2	1	no	—
188	#1055	—	2	2	no	—
189	C₄×#41	—	13	1	no	—
190	#446	—	13	2	no	—
191	C₈×Q₈	—	13	2	no	—
192	#436	—	13	2	no	—
193	#435	—	13	4	no	—
194	C₈×D₄	—	13	2	no	—
195	C₈×D₄	—	13	4	no	—
196	#433	—	13	2	no	—
197	#433	—	13	2	no	—
198	#431	—	13	1	no	—
199	#431	—	13	2	no	—
200	#431	—	13	1	no	—
201	#431	—	13	2	no	—
202	#431	—	13	2	no	—
203	#431	—	13	2	no	—
204	#424	—	13	1	no	—
205	#424	—	13	1	no	—
206	#408	—	13	1	no	—
207	#408	#261	1	1	yes	characteristic
208	C₄×M₁₆	—	13	2	no	—
209	C₄×M₁₆	—	13	1	no	—
210	C₂×C₄×C₈	—	13	1	no	—
211	C₂×C₄×C₈	—	13	2	no	—
212	C₂×C₄×C₈	—	13	1	no	—
213	C₂×C₄×C₈	—	13	1	no	—
214	#322	—	13	2	no	—
215	#322	—	13	2	no	—
216	#322	—	13	1	no	—
217	#322	—	13	1	no	—
218	#322	—	13	1	no	—
219	#322	—	13	1	no	—
220	#322	—	13	2	no	—
221	#322	—	13	2	no	—
222	C₈²	—	13	2	no	—
223	C₈²	—	13	2	no	—
224	C₂²×C₁₃	—	2	1	no	—
225	C₂²×C₁₃	—	2	1	no	—
226	C₂²×C₁₃	—	2	2	no	—
227	C₂²×C₁₃	—	2	2	no	—
228	C₂²×C₁₃	—	2	2	no	—
229	C₂²×C₁₃	#405	1	2	yes	—
230	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
231	C₄×C₁₃	—	2	2	no	—
232	C₄×C₁₃	#405	1	2	yes	—
233	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
234	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
235	Dic₁₃	—	2	1	no	—
236	Dic₁₃	—	2	2	no	—
237	Dic₁₃	—	2	1	no	—
238	Dic₁₃	—	2	2	no	—
239	Dic₁₃	C₂×#131	1	2	yes	—
240	Dic₁₃	#568	1	2	yes	—
241	C₂×#41	C₂×#261	1	1	yes	characteristic
242	C₂×M₁₆	—	13	1	no	—
243	C₂×M₁₆	—	13	4	no	—
244	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
245	C₂²×C₈	—	13	2	no	—
246	C₂²×C₈	—	13	4	no	—
247	C₂²×C₈	—	13	1	no	—
248	C₂²×C₈	—	13	2	no	—
249	C₂²×C₈	C₂×#261	1	1	yes	characteristic
250	C₄×Q₈	—	13	1	no	—
251	C₄×Q₈	—	13	1	no	—
252	C₄×D₄	—	13	2	no	—
253	C₄×D₄	—	13	1	no	—
254	C₄×D₄	—	13	2	no	—
255	C₄×D₄	—	13	1	no	—
256	#117	—	13	1	no	—
257	#117	—	13	2	no	—
258	C₂×C₄²	—	13	1	no	—
259	C₂×C₄²	—	13	2	no	—
260	C₄⋊C₈	—	13	2	no	—
261	C₄⋊C₈	—	13	4	no	—
262	C₄⋊C₈	—	13	2	no	—
263	C₄⋊C₈	—	13	4	no	—
264	C₄⋊C₈	—	13	2	no	—
265	C₄⋊C₈	—	13	2	no	—
266	C₄⋊C₈	—	13	2	no	—
267	C₄⋊C₈	—	13	2	no	—
268	C₂²⋊C₈	—	13	2	no	—
269	C₂²⋊C₈	—	13	4	no	—
270	C₂²⋊C₈	—	13	2	no	—
271	C₂²⋊C₈	—	13	4	no	—
272	C₂²⋊C₈	—	13	2	no	—
273	C₂²⋊C₈	—	13	2	no	—
274	C₂²⋊C₈	C₂×#261	1	2	yes	—
275	C₂²⋊C₈	C₂×#261	1	2	yes	—
276	#97	—	13	4	no	—
277	#97	—	13	2	no	—
278	C₄×C₈	—	13	2	no	—
279	C₄×C₈	—	13	4	no	—
280	C₄×C₈	—	13	2	no	—
281	C₄×C₈	—	13	2	no	—
282	C₄×C₈	—	13	1	no	—
283	C₄×C₈	—	13	2	no	—
284	C₄×C₈	—	13	1	no	—
285	C₄×C₈	—	13	1	no	—
286	C₄×C₈	—	13	1	no	—
287	C₄×C₈	—	13	1	no	—
288	C₄×C₈	—	13	2	no	—
289	C₄×C₈	—	13	1	no	—
290	C₄×C₈	—	13	1	no	—
291	C₄×C₈	—	13	1	no	—
292	C₂×C₁₃	—	2	2	no	—
293	C₂×C₁₃	—	4	2	no	—
294	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
295	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
296	C₂×C₁₃	#2875	1	1	yes	characteristic
297	#41	—	2	2	no	—
298	#41	—	2	2	no	—
299	C₂×Q₈	C₄×#261	1	1	yes	characteristic
300	C₂×D₄	C₄×#261	1	2	yes	—
301	C₂×D₄	C₄×#261	1	1	yes	characteristic
302	C₂²×C₄	—	13	2	no	—
303	C₂²×C₄	—	13	1	no	—
304	C₂²×C₄	—	13	1	no	—
305	C₂²×C₄	—	13	2	no	—
306	C₂²×C₄	C₂²×#261	1	2	yes	—
307	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
308	M₁₆	—	2	2	no	—
309	M₁₆	—	26	2	no	—
310	M₁₆	—	52	4	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	13	2	no	—
313	C₂×C₈	—	13	2	no	—
314	C₂×C₈	—	13	4	no	—
315	C₂×C₈	—	13	2	no	—
316	C₂×C₈	—	13	2	no	—
317	C₂×C₈	—	13	4	no	—
318	C₂×C₈	—	13	1	no	—
319	C₂×C₈	—	13	1	no	—
320	C₂×C₈	—	13	2	no	—
321	C₂×C₈	—	26	4	no	—
322	C₂×C₈	—	26	8	no	—
323	C₂×C₈	—	26	2	no	—
324	C₂×C₈	—	52	2	no	—
325	C₂×C₈	C₂²×#261	1	1	yes	characteristic
326	C₂×C₈	C₂²×#261	1	1	yes	characteristic
327	C₂×C₈	C₂²×#261	1	2	yes	—
328	C₄⋊C₄	—	13	2	no	—
329	C₄⋊C₄	—	13	2	no	—
330	C₄⋊C₄	—	13	1	no	—
331	C₄⋊C₄	—	13	1	no	—
332	C₂²⋊C₄	—	13	2	no	—
333	C₂²⋊C₄	—	13	2	no	—
334	C₂²⋊C₄	—	13	2	no	—
335	C₄²	—	13	2	no	—
336	C₄²	—	13	1	no	—
337	C₄²	—	13	1	no	—
338	C₄²	—	13	2	no	—
339	C₄²	—	13	1	no	—
340	C₄²	—	13	1	no	—
341	C₄²	—	13	2	no	—
342	C₁₃	#17330	1	1	yes	characteristic, complemented, Sylow
343	C₂³	C₂×C₄×#261	1	2	yes	—
344	C₂³	C₂×C₄×#261	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	D₄	—	2	4	no	—
347	D₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	13	1	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	13	1	no	—
353	C₂×C₄	—	13	2	no	—
354	C₂×C₄	—	13	1	no	—
355	C₂×C₄	—	13	2	no	—
356	C₂×C₄	—	26	2	no	—
357	C₂×C₄	—	26	4	no	—
358	C₂×C₄	—	26	2	no	—
359	C₂×C₄	—	26	4	no	—
360	C₂×C₄	C₂×C₄×#261	1	2	yes	—
361	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
362	C₂×C₄	#88238	1	2	yes	—
363	C₂×C₄	#88238	1	2	yes	—
364	C₂×C₄	C₂³×#261	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	2	2	no	—
368	C₈	—	13	4	no	—
369	C₈	—	26	4	no	—
370	C₈	—	26	2	no	—
371	C₈	—	26	2	no	—
372	C₈	—	26	2	no	—
373	C₈	—	26	4	no	—
374	C₈	—	26	1	no	—
375	C₈	—	26	1	no	—
376	C₈	—	26	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	#11719062	1	2	yes	—
383	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
384	C₄	—	2	2	no	—
385	C₄	—	13	2	no	—
386	C₄	—	13	2	no	—
387	C₄	—	26	1	no	—
388	C₄	—	26	2	no	—
389	C₄	—	26	1	no	—
390	C₄	—	26	2	no	—
391	C₄	#11719062	1	2	yes	—
392	C₄	C₂×#88238	1	1	yes	characteristic
393	C₄	C₂×#88238	1	1	yes	characteristic
394	C₂	—	2	2	no	—
395	C₂	—	4	2	no	—
396	C₂	C₂×#11719062	1	1	yes	characteristic
397	C₂	#131×#261	1	1	yes	characteristic
398	C₂	#13131406	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#408	C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#408	#261
hypocenter	C₁₃	#17330
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
3. #13130057	C₂	semidirect
4. #13130053	C₂	semidirect
5. #13130047	C₂	semidirect
6. #13130043	C₂	semidirect
7. #13130041	C₂	semidirect
8. #13130039	C₂	semidirect
9. M₁₆×#1057	C₂	semidirect
10. C₂×#1057⋊C₈	C₂	semidirect
342. C₁₃	#17330	semidirect

Maximal subgroups

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

subgroup	index
#13131645	2
#41×#1057	2
#13130057	2
#13130053	2
#13130047	2
#13130043	2
#13130041	2
#13130039	2
M₁₆×#1057	2
C₂×#1057⋊C₈	2
#13123476	2
#17330	13