Subgroups — #16796865

Order 3328.

908 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131399	C₂	1	2	yes	maximal
2	#13130558	C₂	1	4	yes	maximal, complemented
3	#13130557	C₂	1	4	yes	maximal, complemented
4	C₂×#11718269	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718267	C₂	1	1	yes	maximal, characteristic
6	#13130506	C₂	1	2	yes	maximal
7	#13123252	C₂	1	1	yes	maximal, characteristic
8	C₂×#88237	C₂²	1	2	yes	—
9	C₂×D₁₃⋊M₁₆	C₂²	1	2	yes	—
10	C₂×M₁₆×D₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6094	C₂²	1	1	yes	characteristic
12	D₁₃×#117	C₂²	1	1	yes	characteristic
13	#11719099	C₄	1	8	yes	complemented
14	M₁₆×#261	C₄	1	8	yes	complemented
15	#11719089	C₂²	1	8	yes	—
16	C₁₃×#405	C₄	1	1	yes	characteristic, complemented
17	#11718783	C₄	1	1	yes	characteristic, complemented
18	#11718762	C₄	1	1	yes	characteristic, complemented
19	#11718658	C₄	1	1	yes	characteristic, complemented
20	#11718369	C₂²	1	2	yes	—
21	#11718368	C₄	1	2	yes	complemented
22	#11718350	C₂²	1	2	yes	—
23	C₄×#6094	C₄	1	2	yes	complemented
24	#11718269	C₂²	1	2	yes	—
25	#11718269	C₂²	1	2	yes	—
26	#11718267	C₂²	1	2	yes	—
27	#11718267	C₂²	1	2	yes	—
28	#11718235	C₂²	1	8	yes	—
29	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊#261	—	2	2	no	—
31	C₂×#6122	—	2	2	no	—
32	#88237	—	2	2	no	—
33	#88237	—	2	2	no	—
34	#88237	C₂×C₄	1	4	yes	—
35	#88237	C₂³	1	4	yes	—
36	C₂×C₄⋊#261	—	2	2	no	—
37	C₂×C₄×#261	C₂×C₄	1	4	yes	—
38	D₁₃⋊M₁₆	—	2	2	no	—
39	D₁₃⋊M₁₆	—	2	2	no	—
40	D₁₃⋊M₁₆	C₂×C₄	1	4	yes	—
41	D₁₃⋊M₁₆	C₂³	1	4	yes	—
42	C₂×#6118	—	2	2	no	—
43	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
44	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
47	#88180	C₂×C₄	1	1	yes	characteristic
48	C₂×#6099	C₂×C₄	1	1	yes	characteristic
49	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
50	M₁₆×D₁₃	C₂×C₄	1	4	yes	—
51	M₁₆×D₁₃	C₂×C₄	1	4	yes	—
52	C₂×#6094	—	2	2	no	—
53	C₂×#6094	—	2	2	no	—
54	C₂×#6094	C₂×C₄	1	1	yes	characteristic
55	C₂×#6094	C₂×C₄	1	1	yes	characteristic
56	C₂×#6094	C₂×C₄	1	2	yes	—
57	C₂×#6094	C₂³	1	1	yes	characteristic
58	C₂×#6094	C₂³	1	1	yes	characteristic
59	C₂×C₈×D₁₃	C₂³	1	2	yes	—
60	#88146	—	2	1	no	—
61	C₄⋊C₄×D₁₃	—	2	1	no	—
62	C₂²⋊C₄×D₁₃	—	2	1	no	—
63	#88131	—	2	1	no	—
64	#88126	C₂³	1	2	yes	—
65	C₄²×D₁₃	C₂×C₄	1	2	yes	—
66	#88100	C₂×C₄	1	16	yes	complemented
67	C₈×#261	C₂×C₄	1	16	yes	complemented
68	C₁₃×#97	C₂×C₄	1	2	yes	complemented
69	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	complemented
70	#88055	C₂×C₄	1	2	yes	complemented
71	#88055	C₂×C₄	1	2	yes	complemented
72	C₈×Dic₁₃	C₂×C₄	1	2	yes	complemented
73	C₈×Dic₁₃	C₂×C₄	1	2	yes	complemented
74	#88043	C₂×C₄	1	2	yes	complemented
75	C₄×#1055	C₂×C₄	1	2	yes	complemented
76	#20370	—	13	1	no	maximal, Sylow
77	C₂³×D₁₃	#41	1	1	yes	characteristic
78	C₂²×#261	—	2	4	no	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	#41	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	2	no	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
86	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
87	C₂×C₄×D₁₃	#41	1	2	yes	—
88	C₂×C₄×D₁₃	#41	1	2	yes	—
89	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊#261	—	2	4	no	—
91	C₂²⋊#261	—	2	4	no	—
92	#6122	—	2	4	no	—
93	#6122	—	2	4	no	—
94	C₂×#1057	—	2	4	no	—
95	C₄⋊#261	—	2	4	no	—
96	C₄⋊#261	—	2	4	no	—
97	C₄×#261	—	2	4	no	—
98	C₄×#261	C₂²×C₄	1	8	yes	—
99	#6118	—	2	4	no	—
100	#6118	—	2	4	no	—
101	D₁₃⋊C₈	—	2	4	no	—
102	D₁₃⋊C₈	C₂²×C₄	1	8	yes	—
103	C₁₃×M₁₆	C₄²	1	2	yes	—
104	C₁₃×M₁₆	C₄²	1	2	yes	—
105	C₂×C₈×C₁₃	—	2	1	no	—
106	C₂×C₈×C₁₃	—	2	1	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₂²⋊C₄	—	2	1	no	—
112	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
113	C₂²⋊Dic₁₃	—	2	1	no	—
114	#6103	—	2	2	no	—
115	C₄⋊Dic₁₃	—	2	1	no	—
116	#6101	—	2	2	no	—
117	C₄×Dic₁₃	—	2	2	no	—
118	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
119	#6099	C₄²	1	2	yes	—
120	#6099	C₄²	1	2	yes	—
121	C₂×#1055	—	2	1	no	—
122	C₂×#1055	—	2	1	no	—
123	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
124	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	#6094	—	2	2	no	—
127	#6094	—	2	2	no	—
128	#6094	—	2	2	no	—
129	#6094	—	2	2	no	—
130	#6094	C₄²	1	8	yes	—
131	C₈×D₁₃	C₂²×C₄	1	8	yes	—
132	#2873	—	13	2	no	—
133	#2873	—	13	1	no	—
134	#1747	—	13	4	no	—
135	#1747	—	13	4	no	—
136	C₂×#338	—	13	1	no	—
137	C₂×H(3,ℤ/4)	—	13	2	no	—
138	C₂×H(3,ℤ/4)	—	13	1	no	—
139	C₂³×C₁₃	#117	1	1	yes	characteristic
140	C₂²×D₁₃	—	8	1	no	—
141	C₂²×D₁₃	#117	1	2	yes	—
142	C₂²×D₁₃	#117	1	2	yes	—
143	C₂²×D₁₃	#117	1	1	yes	characteristic
144	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
145	C₂×#261	—	2	8	no	—
146	C₂×#261	—	8	4	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	#117	1	2	yes	—
156	C₂×Dic₁₃	#117	1	2	yes	—
157	C₂×Dic₁₃	#117	1	1	yes	characteristic
158	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
159	C₄×D₁₃	—	8	2	no	—
160	C₄×D₁₃	C₂×C₄²	1	2	yes	—
161	C₄×D₁₃	#117	1	2	yes	—
162	C₄×D₁₃	#117	1	2	yes	—
163	C₄×D₁₃	#117	1	2	yes	—
164	C₄×D₁₃	#117	1	2	yes	—
165	C₄×D₁₃	#117	1	2	yes	—
166	C₄×D₁₃	C₂³×C₄	1	2	yes	—
167	C₄×D₁₃	C₂×#41	1	2	yes	—
168	#1057	—	2	8	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	C₂×C₄²	1	4	yes	—
172	#1055	—	2	1	no	—
173	#1055	—	2	1	no	—
174	#1055	C₂×C₄²	1	4	yes	—
175	C₂²×M₁₆	—	13	1	no	—
176	C₂²×M₁₆	—	13	2	no	—
177	C₂²×M₁₆	—	13	1	no	—
178	C₂×#117	—	13	2	no	—
179	C₂×#117	—	13	1	no	—
180	#405	—	13	8	no	—
181	#405	—	13	8	no	—
182	#405	—	13	1	no	—
183	#405	—	13	1	no	—
184	#405	—	13	1	no	—
185	#405	—	13	2	no	—
186	#405	—	13	2	no	—
187	#405	#261	1	1	yes	characteristic, complemented
188	C₄×M₁₆	—	13	8	no	—
189	C₄×M₁₆	—	13	2	no	—
190	C₄×M₁₆	—	13	2	no	—
191	#338	—	13	2	no	—
192	#338	—	13	2	no	—
193	H(3,ℤ/4)	—	13	8	no	—
194	H(3,ℤ/4)	—	13	2	no	—
195	H(3,ℤ/4)	—	13	2	no	—
196	C₂²×C₁₃	—	4	1	no	—
197	C₂²×C₁₃	#376	1	1	yes	characteristic
198	C₂²×C₁₃	#376	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
200	C₂×D₁₃	—	4	1	no	—
201	C₂×D₁₃	—	4	2	no	—
202	C₂×D₁₃	—	4	2	no	—
203	C₂×D₁₃	—	4	1	no	—
204	C₂×D₁₃	#376	1	2	yes	—
205	C₂×D₁₃	C₂×#117	1	2	yes	—
206	#261	—	4	8	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	#376	1	1	yes	characteristic
209	C₄×C₁₃	#376	1	1	yes	characteristic
210	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
211	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
212	Dic₁₃	—	4	2	no	—
213	Dic₁₃	#376	1	2	yes	—
214	Dic₁₃	C₂×#117	1	2	yes	—
215	C₂³×C₄	—	13	1	no	—
216	C₂×M₁₆	—	13	1	no	—
217	C₂×M₁₆	—	13	4	no	—
218	C₂×M₁₆	—	13	4	no	—
219	C₂×M₁₆	—	13	4	no	—
220	C₂×M₁₆	—	13	4	no	—
221	C₂×M₁₆	—	13	2	no	—
222	C₂×M₁₆	—	13	1	no	—
223	C₂×M₁₆	—	13	1	no	—
224	C₂×M₁₆	—	13	1	no	—
225	C₂×M₁₆	—	13	1	no	—
226	C₂×M₁₆	—	26	2	no	—
227	C₂×M₁₆	—	26	2	no	—
228	C₂×M₁₆	—	26	2	no	—
229	C₂×M₁₆	—	26	2	no	—
230	C₂×M₁₆	—	26	2	no	—
231	C₂×M₁₆	—	26	2	no	—
232	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
233	C₂²×C₈	—	13	2	no	—
234	C₂²×C₈	—	13	4	no	—
235	C₂²×C₈	—	13	1	no	—
236	C₂²×C₈	C₂×#261	1	1	yes	characteristic
237	#117	—	13	4	no	—
238	#117	—	13	4	no	—
239	#117	—	13	1	no	—
240	#117	—	13	2	no	—
241	#117	—	26	1	no	—
242	#117	—	26	1	no	—
243	#117	—	26	2	no	—
244	#117	—	26	2	no	—
245	#117	C₂×#261	1	1	yes	characteristic
246	C₂×C₄⋊C₄	—	26	1	no	—
247	C₂×C₄⋊C₄	—	26	2	no	—
248	C₂×C₂²⋊C₄	—	26	1	no	—
249	C₂×C₂²⋊C₄	—	26	2	no	—
250	C₂×C₄²	—	13	4	no	—
251	C₂×C₄²	—	13	2	no	—
252	#97	—	13	16	no	—
253	#97	—	13	2	no	—
254	#97	—	13	2	no	—
255	#97	—	13	2	no	—
256	#97	C₂×#261	1	2	yes	complemented
257	C₄×C₈	—	13	16	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	13	2	no	—
260	C₄×C₈	—	13	2	no	—
261	C₄×C₈	C₂×#261	1	2	yes	complemented
262	C₂×C₁₃	—	2	1	no	—
263	C₂×C₁₃	—	2	1	no	—
264	C₂×C₁₃	—	2	1	no	—
265	C₂×C₁₃	C₂×#376	1	1	yes	characteristic
266	D₁₃	—	2	2	no	—
267	D₁₃	—	2	2	no	—
268	C₂⁴	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	2	no	—
271	C₂²×C₄	—	13	2	no	—
272	C₂²×C₄	—	13	2	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	13	2	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	26	2	no	—
279	C₂²×C₄	—	26	4	no	—
280	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
281	M₁₆	—	13	2	no	—
282	M₁₆	—	13	2	no	—
283	M₁₆	—	13	8	no	—
284	M₁₆	—	26	2	no	—
285	M₁₆	—	26	2	no	—
286	M₁₆	—	26	2	no	—
287	M₁₆	—	26	2	no	—
288	M₁₆	—	26	4	no	—
289	M₁₆	—	26	4	no	—
290	M₁₆	—	26	4	no	—
291	M₁₆	—	26	4	no	—
292	M₁₆	C₄×#261	1	2	yes	—
293	M₁₆	C₄×#261	1	2	yes	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	13	2	no	—
297	C₂×C₈	—	13	8	no	—
298	C₂×C₈	—	13	8	no	—
299	C₂×C₈	—	13	1	no	—
300	C₂×C₈	—	13	1	no	—
301	C₂×C₈	—	26	4	no	—
302	C₂×C₈	—	26	4	no	—
303	C₂×C₈	—	26	1	no	—
304	C₂×C₈	—	26	1	no	—
305	C₂×C₈	C₂²×#261	1	1	yes	characteristic
306	C₂×C₈	C₂²×#261	1	1	yes	characteristic
307	C₂×C₈	C₂²×#261	1	2	yes	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	4	no	—
311	C₄⋊C₄	—	26	4	no	—
312	C₄⋊C₄	—	26	1	no	—
313	C₂²⋊C₄	—	2	1	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₂²⋊C₄	—	26	4	no	—
316	C₂²⋊C₄	—	26	4	no	—
317	C₂²⋊C₄	—	26	1	no	—
318	C₄²	—	13	8	no	—
319	C₄²	—	13	2	no	—
320	C₄²	—	26	2	no	—
321	C₄²	—	26	4	no	—
322	C₄²	C₂²×#261	1	2	yes	—
323	C₁₃	#20370	1	1	yes	characteristic, complemented, Sylow
324	C₂³	—	13	2	no	—
325	C₂³	—	13	2	no	—
326	C₂³	—	13	1	no	—
327	C₂³	—	13	1	no	—
328	C₂³	—	104	1	no	—
329	C₂³	#88237	1	1	yes	characteristic
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	13	2	no	—
332	C₂×C₄	—	13	2	no	—
333	C₂×C₄	—	13	1	no	—
334	C₂×C₄	—	13	1	no	—
335	C₂×C₄	—	13	2	no	—
336	C₂×C₄	—	13	2	no	—
337	C₂×C₄	—	13	2	no	—
338	C₂×C₄	—	13	2	no	—
339	C₂×C₄	—	13	2	no	—
340	C₂×C₄	—	13	2	no	—
341	C₂×C₄	—	13	2	no	—
342	C₂×C₄	—	13	2	no	—
343	C₂×C₄	—	26	8	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	104	2	no	—
346	C₂×C₄	—	104	4	no	—
347	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
349	C₂×C₄	#88237	1	1	yes	characteristic
350	C₂×C₄	#88237	1	1	yes	characteristic
351	C₂×C₄	#88237	1	1	yes	characteristic
352	C₂×C₄	C₂³×#261	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	13	4	no	—
356	C₈	—	26	8	no	—
357	C₈	—	26	1	no	—
358	C₈	—	26	1	no	—
359	C₈	C₂×C₄×#261	1	4	yes	—
360	C₂²	—	4	1	no	—
361	C₂²	—	13	2	no	—
362	C₂²	—	13	2	no	—
363	C₂²	—	52	1	no	—
364	C₂²	—	52	2	no	—
365	C₂²	—	52	1	no	—
366	C₂²	—	52	2	no	—
367	C₂²	#11719057	1	1	yes	characteristic
368	C₂²	#11719057	1	1	yes	characteristic
369	C₂²	C₂×#88237	1	1	yes	characteristic
370	C₄	—	4	2	no	—
371	C₄	—	13	2	no	—
372	C₄	—	13	2	no	—
373	C₄	—	52	2	no	—
374	C₄	—	52	8	no	—
375	C₄	#11719057	1	1	yes	characteristic
376	C₄	#11719057	1	1	yes	characteristic
377	C₄	C₂²×C₄×#261	1	1	yes	characteristic
378	C₄	C₂×#88237	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	26	2	no	—
383	C₂	—	26	2	no	—
384	C₂	C₂×#11719057	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#405	C₄
socle	C₂×C₁₃	C₂×#376
prime radical	C₂³×D₁₃	#41
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#20370
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13130558	C₂	semidirect
3. #13130557	C₂	semidirect
13. #11719099	C₄	semidirect
14. M₁₆×#261	C₄	semidirect
16. C₁₃×#405	C₄	semidirect
17. #11718783	C₄	semidirect
18. #11718762	C₄	semidirect
19. #11718658	C₄	semidirect
21. #11718368	C₄	semidirect
23. C₄×#6094	C₄	semidirect
66. #88100	C₂×C₄	semidirect
67. C₈×#261	C₂×C₄	semidirect
68. C₁₃×#97	C₂×C₄	semidirect
69. C₄×C₈×C₁₃	C₂×C₄	semidirect
70. #88055	C₂×C₄	semidirect
71. #88055	C₂×C₄	semidirect
72. C₈×Dic₁₃	C₂×C₄	semidirect
73. C₈×Dic₁₃	C₂×C₄	semidirect
74. #88043	C₂×C₄	semidirect
75. C₄×#1055	C₂×C₄	semidirect
187. #405	#261	semidirect
256. #97	C₂×#261	semidirect
261. C₄×C₈	C₂×#261	semidirect
323. C₁₃	#20370	semidirect

Maximal subgroups

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

subgroup	index
#13131399	2
#13130558	2
#13130557	2
C₂×#11718269	2
C₂×#11718267	2
#13130506	2
#13123252	2
#20370	13