Subgroups — #16016164

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127938	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123737	C₂	1	1	yes	maximal, characteristic
3	#13116676	C₂	1	4	yes	maximal, complemented
4	#13116671	C₂	1	4	yes	maximal, complemented
5	#13116601	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718153	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116435	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	#11719527	C₄	1	1	yes	characteristic
12	C₂×#88155	C₂²	1	1	yes	characteristic
13	C₁₃×#504	—	2	2	no	—
14	C₁₃×#504	C₂²	1	4	yes	complemented
15	C₂×C₁₃×M₃₂	—	2	2	no	—
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
18	#11718897	C₄	1	1	yes	characteristic
19	#11718806	C₄	1	4	yes	—
20	#11718805	C₂²	1	4	yes	—
21	#11718800	C₄	1	1	yes	characteristic
22	#11718785	C₄	1	1	yes	characteristic
23	#11718762	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718568	—	2	2	no	—
27	D₁₃×M₃₂	—	2	2	no	—
28	#11718562	C₂²	1	4	yes	complemented
29	#11718153	C₂²	1	2	yes	complemented
30	#11718153	C₂²	1	2	yes	complemented
31	#11718107	C₂²	1	2	yes	complemented
32	#11718107	C₂²	1	2	yes	complemented
33	#11718103	C₂²	1	4	yes	complemented
34	#11718094	C₂²	1	2	yes	complemented
35	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
36	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
37	C₂×#6127	C₂×C₄	1	1	yes	characteristic
38	C₁₃×#131	—	2	1	no	—
39	C₁₃×#131	—	2	1	no	—
40	C₁₃×#131	C₂×C₄	1	2	yes	—
41	C₁₃×#131	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	—	2	1	no	—
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	#88205	C₈	1	2	yes	—
47	#88204	C₂×C₄	1	4	yes	—
48	#88203	C₈	1	2	yes	—
49	#88196	C₈	1	2	yes	—
50	#88186	C₈	1	2	yes	—
51	#88180	C₂×C₄	1	1	yes	characteristic
52	C₂×#6099	C₂×C₄	1	1	yes	characteristic
53	C₂×#6099	C₂×C₄	1	1	yes	characteristic
54	#88155	—	2	1	no	—
55	#88155	—	2	1	no	—
56	#88155	C₂×C₄	1	2	yes	—
57	#88155	C₂³	1	2	yes	—
58	C₂×#6094	—	2	1	no	—
59	C₂×C₈×D₁₃	—	2	1	no	—
60	C₁₃×M₃₂	—	2	1	no	—
61	C₁₃×M₃₂	—	2	4	no	—
62	C₁₃×M₃₂	D₄	1	2	yes	—
63	C₁₃×M₃₂	D₄	1	2	yes	—
64	C₁₃×M₃₂	C₂³	1	2	yes	—
65	C₂×C₁₃×C₁₆	—	2	1	no	—
66	C₂×C₁₃×C₁₆	—	2	4	no	—
67	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
68	#88077	C₂×C₄	1	4	yes	—
69	#88065	C₂×C₄	1	4	yes	—
70	#88062	C₂×C₄	1	4	yes	—
71	#88055	C₂×C₄	1	2	yes	—
72	C₈×Dic₁₃	C₂³	1	2	yes	—
73	#88052	—	2	1	no	—
74	#88052	C₂³	1	2	yes	—
75	C₂×#6090	—	2	1	no	—
76	C₂×#6090	—	2	1	no	—
77	C₂×#6090	—	2	1	no	—
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	C₂×#6090	C₂³	1	1	yes	characteristic
80	#88038	—	2	4	no	—
81	C₁₆×D₁₃	—	2	4	no	—
82	#33442	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	2	1	no	—
84	C₁₃×#41	—	2	1	no	—
85	C₁₃×#41	C₂×C₈	1	2	yes	—
86	C₁₃×#41	C₂²×C₄	1	2	yes	—
87	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
88	C₂×D₄×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	—	2	1	no	—
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	2	1	no	—
93	#6127	—	2	1	no	—
94	#6127	—	2	1	no	—
95	#6127	C₂×C₈	1	2	yes	—
96	#6127	C₂²×C₄	1	2	yes	—
97	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	—	2	1	no	—
99	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
100	C₁₃×M₁₆	—	2	1	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₃	—	2	1	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	—	2	1	no	—
114	#6107	C₂×C₈	1	4	yes	—
115	#6106	C₂×C₈	1	4	yes	—
116	#6105	C₂×C₈	1	4	yes	—
117	#6104	C₂×C₈	1	4	yes	—
118	C₄⋊Dic₁₃	—	2	1	no	—
119	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
120	#6099	—	2	1	no	—
121	#6099	C₂×C₈	1	2	yes	—
122	#6099	C₂×C₈	1	2	yes	—
123	#6099	C₂²×C₄	1	2	yes	—
124	C₂×#1055	—	2	1	no	—
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	#6094	—	2	2	no	—
127	#6094	—	2	2	no	—
128	C₈×D₁₃	—	2	2	no	—
129	C₈×D₁₃	—	2	2	no	—
130	C₁₃×C₁₆	—	2	2	no	—
131	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
132	#6090	—	2	2	no	—
133	#6090	—	2	1	no	—
134	#6090	—	2	1	no	—
135	#3407	—	13	1	no	—
136	#3407	D₁₃	1	1	yes	characteristic, complemented
137	#2955	—	13	1	no	—
138	#2171	—	13	4	no	—
139	#2170	—	13	4	no	—
140	#2153	—	13	1	no	—
141	C₂×#350	—	13	1	no	—
142	C₂×#350	—	13	1	no	—
143	#2108	—	13	1	no	—
144	C₂³×C₁₃	—	2	1	no	—
145	C₂³×C₁₃	#131	1	1	yes	characteristic
146	C₂²×D₁₃	—	2	1	no	—
147	Q₈×C₁₃	—	2	1	no	—
148	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
149	D₄×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	2	2	no	—
151	D₄×C₁₃	—	2	1	no	—
152	D₄×C₁₃	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	2	2	no	—
163	C₂²⋊D₁₃	—	2	2	no	—
164	C₂×Dic₁₃	—	2	1	no	—
165	C₂×Dic₁₃	—	2	2	no	—
166	D₅₂	—	2	1	no	—
167	D₅₂	C₂²×C₈	1	2	yes	—
168	C₄×D₁₃	—	2	2	no	—
169	C₄×D₁₃	—	2	2	no	—
170	Dic₂₆	—	2	1	no	—
171	Dic₂₆	C₂²×C₈	1	2	yes	—
172	C₈×C₁₃	—	2	2	no	—
173	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
174	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
175	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
176	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
177	#1055	—	2	2	no	—
178	#1055	C₂²×C₈	1	4	yes	—
179	#575	—	13	1	no	—
180	C₂×#131	—	13	1	no	—
181	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
182	#504	—	2	2	no	—
183	#504	—	13	4	no	—
184	#504	—	26	2	no	—
185	#504	C₂×D₁₃	1	4	yes	complemented
186	C₂×M₃₂	—	2	2	no	—
187	C₂×M₃₂	—	13	1	no	—
188	C₂×M₃₂	—	26	2	no	—
189	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
190	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
191	C₂²×C₁₆	—	13	1	no	—
192	#444	—	13	4	no	—
193	#443	—	13	4	no	—
194	#430	—	13	1	no	—
195	#421	—	13	1	no	—
196	#421	—	13	1	no	—
197	#405	—	13	1	no	—
198	#364	—	13	4	no	—
199	#350	—	13	2	no	—
200	#350	—	13	2	no	—
201	#350	—	13	2	no	—
202	#350	—	13	2	no	—
203	#346	—	13	2	no	—
204	C₄×C₁₆	—	13	2	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	—	4	1	no	—
207	C₂²×C₁₃	—	8	1	no	—
208	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
209	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
210	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
211	C₂×D₁₃	—	2	2	no	—
212	C₂×D₁₃	—	8	1	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
215	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
216	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
218	Dic₁₃	—	2	2	no	—
219	Dic₁₃	—	4	2	no	—
220	C₂×#41	—	13	1	no	—
221	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
222	#137	—	13	2	no	—
223	#137	—	13	2	no	—
224	AGL(1,ℤ/8)	—	13	2	no	—
225	AGL(1,ℤ/8)	—	13	2	no	—
226	#135	—	13	4	no	—
227	#131	—	2	1	no	—
228	#131	—	2	1	no	—
229	#131	—	13	2	no	—
230	#131	—	13	2	no	—
231	#131	—	26	1	no	—
232	#131	—	26	1	no	—
233	#131	C₄×D₁₃	1	2	yes	—
234	#131	C₂²×D₁₃	1	2	yes	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	13	1	no	—
238	C₂×M₁₆	—	26	1	no	—
239	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	26	1	no	—
242	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	#117	—	13	1	no	—
244	M₃₂	—	2	1	no	—
245	M₃₂	—	2	4	no	—
246	M₃₂	—	13	2	no	—
247	M₃₂	—	26	1	no	—
248	M₃₂	—	26	4	no	—
249	M₃₂	C₂²⋊D₁₃	1	2	yes	—
250	M₃₂	C₂²⋊D₁₃	1	2	yes	—
251	M₃₂	C₂²×D₁₃	1	2	yes	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	2	4	no	—
254	C₂×C₁₆	—	13	1	no	—
255	C₂×C₁₆	—	13	1	no	—
256	C₂×C₁₆	—	26	1	no	—
257	C₂×C₁₆	—	26	1	no	—
258	C₂×C₁₆	—	26	1	no	—
259	C₂×C₁₆	—	26	4	no	—
260	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
261	#108	—	13	4	no	—
262	#104	—	13	4	no	—
263	#104	—	13	4	no	—
264	#97	—	13	2	no	—
265	C₄×C₈	—	13	2	no	—
266	C₂×C₁₃	—	2	1	no	—
267	C₂×C₁₃	—	2	1	no	—
268	C₂×C₁₃	—	2	1	no	—
269	C₂×C₁₃	—	4	2	no	—
270	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
271	D₁₃	—	4	2	no	—
272	#41	—	2	1	no	—
273	#41	—	2	1	no	—
274	#41	—	13	2	no	—
275	#41	—	13	2	no	—
276	#41	—	26	1	no	—
277	#41	—	26	1	no	—
278	#41	C₈×D₁₃	1	2	yes	—
279	#41	C₂×C₄×D₁₃	1	2	yes	—
280	C₂×Q₈	—	13	1	no	—
281	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	13	1	no	—
284	C₂×D₄	—	26	1	no	—
285	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	26	1	no	—
288	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
289	Q₁₆	—	13	4	no	—
290	SD₁₆	—	13	4	no	—
291	SD₁₆	—	13	4	no	—
292	D₈	—	13	4	no	—
293	M₁₆	—	2	1	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	13	2	no	—
297	M₁₆	—	13	2	no	—
298	M₁₆	—	13	2	no	—
299	M₁₆	—	26	1	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	2	no	—
302	M₁₆	C₂×C₄×D₁₃	1	2	yes	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	13	2	no	—
307	C₂×C₈	—	26	1	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
311	C₂×C₈	#6127	1	1	yes	characteristic
312	C₂×C₈	#6127	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
316	C₄⋊C₄	—	26	1	no	—
317	C₂²⋊C₄	—	26	1	no	—
318	C₄²	—	13	2	no	—
319	C₁₆	—	2	2	no	—
320	C₁₆	—	26	1	no	—
321	C₁₆	—	26	1	no	—
322	C₁₆	—	26	2	no	—
323	C₁₆	C₂×C₂²⋊D₁₃	1	4	yes	—
324	C₁₃	#33442	1	1	yes	characteristic, complemented, Sylow
325	C₂³	—	2	1	no	—
326	C₂³	—	26	1	no	—
327	C₂³	#88155	1	1	yes	characteristic
328	Q₈	—	2	1	no	—
329	Q₈	—	13	2	no	—
330	Q₈	—	26	1	no	—
331	Q₈	C₂×C₈×D₁₃	1	2	yes	—
332	D₄	—	2	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	1	no	—
335	D₄	—	13	2	no	—
336	D₄	—	26	2	no	—
337	D₄	—	26	2	no	—
338	D₄	—	26	1	no	—
339	D₄	C₂×C₈×D₁₃	1	2	yes	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	26	1	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
349	C₂×C₄	#88155	1	1	yes	characteristic
350	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
351	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	13	4	no	—
355	C₈	—	26	2	no	—
356	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
357	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
358	C₈	C₂×#6127	1	1	yes	characteristic
359	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
360	C₂²	—	2	2	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	26	2	no	—
364	C₂²	—	104	1	no	—
365	C₂²	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
366	C₂²	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
367	C₂²	C₂×#88155	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	26	2	no	—
370	C₄	—	52	2	no	—
371	C₄	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
372	C₄	C₈×C₂²⋊D₁₃	1	1	yes	characteristic
373	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
374	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	52	2	no	—
380	C₂	C₂×C₈×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3407	C₂
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	#11719527	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#3407	D₁₃
hypocenter	C₁₃	#33442
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #3407

Complements

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

N	H ≅ G/N	type
1. #13127938	C₂	semidirect
3. #13116676	C₂	semidirect
4. #13116671	C₂	semidirect
5. #13116601	C₂	semidirect
6. C₂×#11718107	C₂	semidirect
7. C₂×#11718153	C₂	semidirect
8. #13116435	C₂	semidirect
9. C₁₃×#3407	C₂	semidirect
14. C₁₃×#504	C₂²	semidirect
28. #11718562	C₂²	semidirect
29. #11718153	C₂²	semidirect
30. #11718153	C₂²	semidirect
31. #11718107	C₂²	semidirect
32. #11718107	C₂²	semidirect
33. #11718103	C₂²	semidirect
34. #11718094	C₂²	semidirect
35. C₁₆×Dic₁₃	C₂²	semidirect
136. #3407	D₁₃	semidirect
185. #504	C₂×D₁₃	semidirect
324. C₁₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#13127938	2
#13123737	2
#13116676	2
#13116671	2
#13116601	2
C₂×#11718107	2
C₂×#11718153	2
#13116435	2
C₁₃×#3407	2
#33442	13