Subgroups — #16792713

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	#13131643	C₂	1	1	yes	maximal, characteristic
2	#13131640	C₂	1	1	yes	maximal, characteristic
3	#13130058	C₂	1	2	yes	maximal, complemented
4	#13130054	C₂	1	2	yes	maximal, complemented
5	#13130052	C₂	1	2	yes	maximal, complemented
6	#13130043	C₂	1	1	yes	maximal, characteristic, complemented
7	#13130039	C₂	1	1	yes	maximal, characteristic, complemented
8	#13130038	C₂	1	2	yes	maximal, complemented
9	M₁₆×#1057	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1057⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123479	C₂	1	1	yes	maximal, characteristic
12	#11719227	C₂²	1	1	yes	characteristic
13	#11719119	C₂²	1	1	yes	characteristic
14	#11719119	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
17	#11719080	C₄	1	2	yes	—
18	#11719076	C₄	1	2	yes	—
19	#11719074	C₄	1	2	yes	—
20	#11719073	C₄	1	2	yes	—
21	#11719064	C₄	1	4	yes	—
22	#11719063	C₄	1	4	yes	—
23	#11719051	C₂²	1	2	yes	—
24	#11719049	C₂²	1	2	yes	—
25	C₄×#6118	C₂²	1	2	yes	—
26	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
27	C₁₃×#424	C₄	1	1	yes	characteristic
28	#11718790	C₄	1	1	yes	characteristic
29	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
30	#11718771	C₄	1	1	yes	characteristic
31	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
32	#11718660	C₄	1	1	yes	characteristic
33	#11718501	C₄	1	2	yes	—
34	#11718501	C₄	1	2	yes	—
35	#11718494	C₂²	1	2	yes	—
36	#11718494	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×#6127	C₂×C₄	1	1	yes	characteristic
48	C₂²×#1057	C₂×C₄	1	2	yes	—
49	C₂²×#1057	C₂×C₄	1	2	yes	—
50	C₂×#6118	C₂×C₄	1	4	yes	—
51	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
52	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂×#6099	C₂×C₄	1	1	yes	characteristic
57	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
58	#88147	C₂×C₄	1	1	yes	characteristic
59	#88147	C₂×C₄	1	1	yes	characteristic
60	#88141	C₂×C₄	1	1	yes	characteristic
61	#88141	C₂×C₄	1	1	yes	characteristic
62	#88135	C₂×C₄	1	2	yes	—
63	#88135	C₂×C₄	1	2	yes	—
64	#88126	C₂×C₄	1	2	yes	—
65	C₄²×D₁₃	C₂³	1	2	yes	—
66	C₂²⋊#1057	C₂×C₄	1	2	yes	—
67	C₂²⋊#1057	C₂×C₄	1	2	yes	—
68	#88112	C₂×C₄	1	4	yes	—
69	#88111	C₂×C₄	1	4	yes	—
70	#88111	C₂×C₄	1	4	yes	—
71	#88110	C₂×C₄	1	4	yes	—
72	C₄⋊#1057	C₂×C₄	1	4	yes	—
73	C₄⋊#1057	C₂×C₄	1	2	yes	—
74	C₄⋊#1057	C₂×C₄	1	2	yes	—
75	C₄×#1057	C₂×C₄	1	4	yes	—
76	C₄×#1057	C₂×C₄	1	2	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×C₄⋊C₈	C₂×C₄	1	2	yes	—
85	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
86	#88060	C₂×C₄	1	2	yes	—
87	#88060	C₂×C₄	1	2	yes	—
88	#88060	C₂×C₄	1	2	yes	—
89	#88060	C₂×C₄	1	2	yes	—
90	#88055	C₂×C₄	1	2	yes	—
91	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
92	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
93	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
94	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
95	C₈×Dic₁₃	C₂³	1	2	yes	—
96	C₄⋊#1055	C₂×C₄	1	2	yes	—
97	C₄⋊#1055	C₂×C₄	1	2	yes	—
98	#17331	—	13	1	no	maximal, Sylow
99	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊D₁₃	C₄²	1	2	yes	—
101	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
103	#6127	—	2	2	no	—
104	#6127	—	2	2	no	—
105	C₂×D₅₂	C₄²	1	1	yes	characteristic
106	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
109	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
110	C₂×#1057	—	2	4	no	—
111	C₂×#1057	—	4	2	no	—
112	C₂×#1057	C₂²×C₄	1	2	yes	—
113	C₂×#1057	C₂²×C₄	1	2	yes	—
114	C₂×#1057	C₂²×C₄	1	4	yes	—
115	C₂×#1057	C₂²×C₄	1	2	yes	—
116	C₂×#1057	C₂²×C₄	1	4	yes	—
117	C₂×#1057	C₂²×C₄	1	2	yes	—
118	#6118	—	2	8	no	—
119	D₁₃⋊C₈	—	4	4	no	—
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	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
126	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
127	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
128	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
129	#6103	C₄²	1	2	yes	—
130	#6103	C₄²	1	2	yes	—
131	#6101	C₄²	1	2	yes	—
132	#6101	C₄²	1	2	yes	—
133	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
134	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
136	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
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	#1452	—	13	2	no	—
150	#1451	—	13	1	no	—
151	#1450	—	13	1	no	—
152	C₈×M₁₆	—	13	1	no	—
153	C₈×M₁₆	—	13	2	no	—
154	C₂×#322	—	13	1	no	—
155	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
156	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
160	C₂×C₄×C₁₃	#131	1	2	yes	—
161	C₂×C₄×C₁₃	#131	1	2	yes	—
162	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	2	4	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	2	2	no	—
166	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
167	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
168	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
169	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
170	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
171	D₅₂	—	2	2	no	—
172	C₄×D₁₃	—	2	2	no	—
173	C₄×D₁₃	—	2	2	no	—
174	C₄×D₁₃	—	2	4	no	—
175	C₄×D₁₃	—	2	4	no	—
176	Dic₂₆	—	2	2	no	—
177	#1057	—	2	4	no	—
178	#1057	—	2	2	no	—
179	#1057	—	2	2	no	—
180	#1057	—	2	4	no	—
181	#1057	—	2	2	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	4	no	—
195	C₈×D₄	—	13	2	no	—
196	#433	—	13	2	no	—
197	#433	—	13	2	no	—
198	#431	—	13	2	no	—
199	#431	—	13	2	no	—
200	#431	—	13	1	no	—
201	#431	—	13	1	no	—
202	#431	—	13	2	no	—
203	#431	—	13	2	no	—
204	#424	—	13	1	no	—
205	#424	#261	1	1	yes	characteristic
206	#408	—	13	1	no	—
207	#408	—	13	1	no	—
208	C₄×M₁₆	—	13	2	no	—
209	C₄×M₁₆	—	13	1	no	—
210	C₂×C₄×C₈	—	13	2	no	—
211	C₂×C₄×C₈	—	13	1	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	#405	1	2	yes	—
227	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
228	C₂×D₁₃	—	2	2	no	—
229	C₂×D₁₃	—	2	2	no	—
230	C₂×D₁₃	—	2	2	no	—
231	C₄×C₁₃	—	2	2	no	—
232	C₄×C₁₃	—	2	2	no	—
233	C₄×C₁₃	#405	1	2	yes	—
234	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
235	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
236	Dic₁₃	—	2	2	no	—
237	Dic₁₃	—	2	1	no	—
238	Dic₁₃	—	2	1	no	—
239	Dic₁₃	C₂×#131	1	2	yes	—
240	Dic₁₃	#568	1	2	yes	—
241	C₂×#41	—	13	1	no	—
242	C₂×M₁₆	—	13	4	no	—
243	C₂×M₁₆	—	13	1	no	—
244	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
245	C₂²×C₈	—	13	2	no	—
246	C₂²×C₈	—	13	1	no	—
247	C₂²×C₈	—	13	4	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	1	no	—
253	C₄×D₄	—	13	1	no	—
254	C₄×D₄	—	13	2	no	—
255	C₄×D₄	—	13	2	no	—
256	#117	—	13	2	no	—
257	#117	C₂×#261	1	1	yes	characteristic
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	C₂×#261	1	2	yes	—
267	C₄⋊C₈	C₂×#261	1	2	yes	—
268	C₂²⋊C₈	—	13	4	no	—
269	C₂²⋊C₈	—	13	4	no	—
270	C₂²⋊C₈	—	13	2	no	—
271	C₂²⋊C₈	—	13	2	no	—
272	C₂²⋊C₈	—	13	2	no	—
273	C₂²⋊C₈	—	13	2	no	—
274	C₂²⋊C₈	—	13	2	no	—
275	C₂²⋊C₈	—	13	2	no	—
276	#97	—	13	4	no	—
277	#97	—	13	2	no	—
278	C₄×C₈	—	13	4	no	—
279	C₄×C₈	—	13	2	no	—
280	C₄×C₈	—	13	2	no	—
281	C₄×C₈	—	13	1	no	—
282	C₄×C₈	—	13	2	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	C₂×#405	1	1	yes	characteristic
294	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
295	C₂×C₁₃	#2875	1	1	yes	characteristic
296	D₁₃	—	4	2	no	—
297	#41	—	26	2	no	—
298	#41	—	26	2	no	—
299	C₂×Q₈	—	13	1	no	—
300	C₂×D₄	—	13	2	no	—
301	C₂×D₄	—	13	1	no	—
302	C₂²×C₄	—	13	1	no	—
303	C₂²×C₄	—	13	1	no	—
304	C₂²×C₄	—	13	2	no	—
305	C₂²×C₄	—	13	2	no	—
306	C₂²×C₄	—	13	2	no	—
307	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
308	M₁₆	—	2	2	no	—
309	M₁₆	—	26	8	no	—
310	M₁₆	—	26	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	13	2	no	—
313	C₂×C₈	—	13	2	no	—
314	C₂×C₈	—	13	2	no	—
315	C₂×C₈	—	13	4	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	2	no	—
323	C₂×C₈	—	52	4	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	C₄×#261	1	1	yes	characteristic
331	C₄⋊C₄	C₄×#261	1	1	yes	characteristic
332	C₂²⋊C₄	—	13	2	no	—
333	C₂²⋊C₄	—	13	2	no	—
334	C₂²⋊C₄	C₄×#261	1	2	yes	—
335	C₄²	—	13	1	no	—
336	C₄²	—	13	1	no	—
337	C₄²	—	13	2	no	—
338	C₄²	—	13	2	no	—
339	C₄²	—	13	1	no	—
340	C₄²	—	13	1	no	—
341	C₄²	C₂²×#261	1	2	yes	—
342	C₁₃	#17331	1	1	yes	characteristic, complemented, Sylow
343	C₂³	—	13	2	no	—
344	C₂³	C₂×C₄×#261	1	1	yes	characteristic
345	Q₈	—	26	2	no	—
346	D₄	—	26	4	no	—
347	D₄	—	26	2	no	—
348	C₂×C₄	—	13	1	no	—
349	C₂×C₄	—	13	1	no	—
350	C₂×C₄	—	13	1	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	13	2	no	—
353	C₂×C₄	—	26	4	no	—
354	C₂×C₄	—	26	4	no	—
355	C₂×C₄	—	26	2	no	—
356	C₂×C₄	—	26	2	no	—
357	C₂×C₄	—	26	2	no	—
358	C₂×C₄	—	26	2	no	—
359	C₂×C₄	C₂×C₄×#261	1	2	yes	—
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	#88240	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	2	no	—
370	C₈	—	26	2	no	—
371	C₈	—	26	4	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	—	26	2	no	—
380	C₂²	—	26	2	no	—
381	C₂²	—	26	2	no	—
382	C₂²	#11719072	1	2	yes	—
383	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
384	C₄	—	2	2	no	—
385	C₄	—	2	2	no	—
386	C₄	—	13	2	no	—
387	C₄	—	13	2	no	—
388	C₄	—	26	2	no	—
389	C₄	—	26	1	no	—
390	C₄	—	26	1	no	—
391	C₄	#11719072	1	2	yes	—
392	C₄	C₂×#88238	1	1	yes	characteristic
393	C₄	C₂×#88240	1	1	yes	characteristic
394	C₂	—	2	2	no	—
395	C₂	—	52	2	no	—
396	C₂	C₂×#11719072	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×#424	C₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#424	#261
hypocenter	C₁₃	#17331
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
3. #13130058	C₂	semidirect
4. #13130054	C₂	semidirect
5. #13130052	C₂	semidirect
6. #13130043	C₂	semidirect
7. #13130039	C₂	semidirect
8. #13130038	C₂	semidirect
9. M₁₆×#1057	C₂	semidirect
10. C₂×#1057⋊C₈	C₂	semidirect
342. C₁₃	#17331	semidirect

Maximal subgroups

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

subgroup	index
#13131643	2
#13131640	2
#13130058	2
#13130054	2
#13130052	2
#13130043	2
#13130039	2
#13130038	2
M₁₆×#1057	2
C₂×#1057⋊C₈	2
#13123479	2
#17331	13