Subgroups — #15828600

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123802	C₂	1	1	yes	maximal, characteristic
2	#13123792	C₂	1	1	yes	maximal, characteristic
3	#13123481	C₂	1	1	yes	maximal, characteristic
4	#13113137	C₂	1	2	yes	maximal, complemented
5	#13113132	C₂	1	2	yes	maximal, complemented
6	#13113129	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113127	C₂	1	2	yes	maximal, complemented
8	#13113122	C₂	1	2	yes	maximal, complemented
9	M₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#6127	C₂²	1	1	yes	characteristic
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718790	C₄	1	2	yes	—
17	#11718780	C₄	1	2	yes	—
18	#11718772	C₄	1	4	yes	—
19	#11718586	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
22	#11718495	C₄	1	2	yes	—
23	#11718491	C₄	1	2	yes	—
24	#11718386	C₄	1	4	yes	—
25	#11718370	C₄	1	4	yes	—
26	#11718349	C₄	1	2	yes	—
27	C₄×#6094	C₂²	1	2	yes	—
28	#11718346	C₂²	1	2	yes	—
29	#11718346	C₂²	1	2	yes	—
30	#11718346	C₂²	1	2	yes	—
31	#11718346	C₂²	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	2	yes	—
33	#11718340	C₄	1	2	yes	—
34	C₁₃×#322	C₂²	1	2	yes	—
35	C₁₃×#322	C₂²	1	2	yes	—
36	#11718047	C₂²	1	2	yes	—
37	#11718047	C₂²	1	2	yes	—
38	#11718046	C₂²	1	1	yes	characteristic
39	#11718046	C₂²	1	1	yes	characteristic
40	#11718046	C₂²	1	1	yes	characteristic
41	#11718046	C₂²	1	1	yes	characteristic
42	#11718046	C₂²	1	2	yes	—
43	C₈×#1055	C₂²	1	2	yes	—
44	C₂×#6127	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
50	#88180	C₂×C₄	1	1	yes	characteristic
51	C₂²×#1055	C₂×C₄	1	2	yes	—
52	C₂²×#1055	C₂×C₄	1	2	yes	—
53	C₂×#6094	C₂×C₄	1	4	yes	—
54	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
55	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
56	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
57	#88126	C₂×C₄	1	2	yes	—
58	C₄²×D₁₃	C₂³	1	2	yes	—
59	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₁₃×#97	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	C₂²⋊#1055	C₂×C₄	1	2	yes	—
70	#88060	C₂×C₄	1	4	yes	—
71	#88060	C₂×C₄	1	4	yes	—
72	#88060	C₂×C₄	1	4	yes	—
73	#88060	C₂×C₄	1	4	yes	—
74	#88055	C₂×C₄	1	4	yes	—
75	#88054	C₂×C₄	1	4	yes	—
76	#88054	C₂×C₄	1	4	yes	—
77	#88054	C₂×C₄	1	4	yes	—
78	#88054	C₂×C₄	1	4	yes	—
79	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
80	C₄⋊#1055	C₂×C₄	1	2	yes	—
81	C₄×#1055	C₂×C₄	1	2	yes	—
82	C₄×#1055	C₂×C₄	1	2	yes	—
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	2	yes	—
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	C₄×#1055	C₂³	1	1	yes	characteristic
87	C₄×#1055	C₂³	1	1	yes	characteristic
88	#17329	—	13	1	no	maximal, Sylow
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	C₄²	1	2	yes	—
93	#6127	—	2	4	no	—
94	C₂×D₅₂	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
99	C₁₃×M₁₆	—	2	4	no	—
100	C₂×C₈×C₁₃	—	4	2	no	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
105	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
110	#6103	C₄²	1	2	yes	—
111	#6103	C₄²	1	2	yes	—
112	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
113	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
114	#6101	C₄²	1	2	yes	—
115	#6101	C₄²	1	2	yes	—
116	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
119	C₂×#1055	—	2	4	no	—
120	C₂×#1055	—	4	2	no	—
121	C₂×#1055	C₂²×C₄	1	4	yes	—
122	C₂×#1055	C₂²×C₄	1	2	yes	—
123	C₂×#1055	C₂²×C₄	1	2	yes	—
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	#6094	—	4	4	no	—
127	C₈×D₁₃	—	4	4	no	—
128	#2965	—	13	1	no	—
129	C₈×#41	—	13	1	no	—
130	#2922	—	13	1	no	—
131	#1454	—	13	2	no	—
132	#1453	—	13	2	no	—
133	#1452	—	13	1	no	—
134	#1452	—	13	2	no	—
135	#1451	—	13	2	no	—
136	#1451	D₁₃	1	1	yes	characteristic, complemented
137	C₈×M₁₆	—	13	1	no	—
138	C₂×#322	—	13	1	no	—
139	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
140	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#131	1	4	yes	—
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂²⋊D₁₃	—	2	4	no	—
151	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
152	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
153	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
154	D₅₂	—	2	2	no	—
155	C₄×D₁₃	—	2	4	no	—
156	C₄×D₁₃	—	2	4	no	—
157	C₄×D₁₃	—	2	4	no	—
158	Dic₂₆	—	2	2	no	—
159	C₈×C₁₃	—	2	4	no	—
160	C₈×C₁₃	—	2	4	no	—
161	C₈×C₁₃	—	2	2	no	—
162	C₈×C₁₃	—	2	2	no	—
163	#1055	—	2	2	no	—
164	#1055	—	2	2	no	—
165	#1055	—	2	2	no	—
166	#1055	—	2	4	no	—
167	#1055	—	2	2	no	—
168	C₄×#41	—	13	1	no	—
169	#446	—	13	2	no	—
170	C₈×Q₈	—	13	2	no	—
171	#436	—	13	2	no	—
172	#435	—	13	4	no	—
173	C₈×D₄	—	13	2	no	—
174	C₈×D₄	—	13	4	no	—
175	#433	—	13	4	no	—
176	#431	—	13	1	no	—
177	#431	—	13	2	no	—
178	#431	—	13	2	no	—
179	#431	—	13	2	no	—
180	#431	—	13	2	no	—
181	#431	C₂×D₁₃	1	1	yes	characteristic
182	#424	—	13	2	no	—
183	#408	—	13	2	no	—
184	C₄×M₁₆	—	13	2	no	—
185	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
186	C₂×C₄×C₈	—	13	1	no	—
187	C₂×C₄×C₈	—	13	1	no	—
188	C₂×C₄×C₈	—	13	2	no	—
189	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
190	#322	—	13	1	no	—
191	#322	—	13	1	no	—
192	#322	—	13	1	no	—
193	#322	—	13	1	no	—
194	#322	—	13	2	no	—
195	#322	—	13	2	no	—
196	#322	—	13	2	no	—
197	#322	C₂×D₁₃	1	2	yes	—
198	#322	C₂×D₁₃	1	2	yes	—
199	C₈²	—	13	2	no	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	#405	1	2	yes	—
203	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
204	C₂×D₁₃	—	2	2	no	—
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	—	2	2	no	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	#405	1	2	yes	—
210	C₄×C₁₃	C₂×#131	1	2	yes	—
211	C₄×C₁₃	#568	1	2	yes	—
212	C₄×C₁₃	#568	1	2	yes	—
213	Dic₁₃	—	2	2	no	—
214	Dic₁₃	—	2	2	no	—
215	Dic₁₃	—	2	2	no	—
216	C₂×#41	—	13	1	no	—
217	C₂×M₁₆	—	13	4	no	—
218	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
219	C₂²×C₈	—	13	2	no	—
220	C₂²×C₈	—	13	2	no	—
221	C₂²×C₈	—	13	4	no	—
222	C₂²×C₈	C₄×D₁₃	1	2	yes	—
223	C₄×Q₈	—	13	1	no	—
224	C₄×Q₈	—	13	1	no	—
225	C₄×D₄	—	13	2	no	—
226	C₄×D₄	—	13	2	no	—
227	C₄×D₄	—	13	1	no	—
228	C₄×D₄	—	13	1	no	—
229	#117	—	13	1	no	—
230	#117	—	13	2	no	—
231	C₂×C₄²	—	13	2	no	—
232	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
233	C₄⋊C₈	—	13	4	no	—
234	C₄⋊C₈	—	13	4	no	—
235	C₄⋊C₈	—	13	4	no	—
236	C₄⋊C₈	—	13	4	no	—
237	C₄⋊C₈	—	13	2	no	—
238	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
239	C₂²⋊C₈	—	13	2	no	—
240	C₂²⋊C₈	—	13	4	no	—
241	C₂²⋊C₈	—	13	4	no	—
242	C₂²⋊C₈	—	13	4	no	—
243	C₂²⋊C₈	—	13	4	no	—
244	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
245	#97	—	13	4	no	—
246	#97	C₄×D₁₃	1	2	yes	—
247	C₄×C₈	—	13	4	no	—
248	C₄×C₈	—	13	2	no	—
249	C₄×C₈	—	13	2	no	—
250	C₄×C₈	—	13	1	no	—
251	C₄×C₈	—	13	2	no	—
252	C₄×C₈	—	13	1	no	—
253	C₄×C₈	—	13	1	no	—
254	C₄×C₈	—	13	1	no	—
255	C₄×C₈	C₄×D₁₃	1	2	yes	—
256	C₄×C₈	C₂²×D₁₃	1	2	yes	—
257	C₄×C₈	C₂²×D₁₃	1	2	yes	—
258	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
260	C₂×C₁₃	—	2	2	no	—
261	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
262	C₂×C₁₃	#2875	1	1	yes	characteristic
263	C₂×C₁₃	#2875	1	1	yes	characteristic
264	D₁₃	—	4	2	no	—
265	#41	—	26	4	no	—
266	C₂×Q₈	—	13	1	no	—
267	C₂×D₄	—	13	2	no	—
268	C₂×D₄	—	13	1	no	—
269	C₂²×C₄	—	13	2	no	—
270	C₂²×C₄	—	13	2	no	—
271	C₂²×C₄	—	13	2	no	—
272	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
275	M₁₆	—	2	4	no	—
276	M₁₆	—	52	4	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	13	2	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	13	4	no	—
283	C₂×C₈	—	26	4	no	—
284	C₂×C₈	—	52	4	no	—
285	C₂×C₈	—	52	2	no	—
286	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
287	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
290	C₄⋊C₄	—	13	2	no	—
291	C₄⋊C₄	—	13	2	no	—
292	C₄⋊C₄	—	13	1	no	—
293	C₄⋊C₄	—	13	1	no	—
294	C₂²⋊C₄	—	13	2	no	—
295	C₂²⋊C₄	—	13	2	no	—
296	C₂²⋊C₄	—	13	2	no	—
297	C₄²	—	13	2	no	—
298	C₄²	—	13	2	no	—
299	C₄²	—	13	2	no	—
300	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₄²	C₂³×D₁₃	1	1	yes	characteristic
304	C₁₃	#17329	1	1	yes	characteristic, complemented, Sylow
305	C₂³	—	13	2	no	—
306	C₂³	C₄²×D₁₃	1	1	yes	characteristic
307	Q₈	—	26	2	no	—
308	D₄	—	26	4	no	—
309	D₄	—	26	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	13	2	no	—
314	C₂×C₄	—	13	2	no	—
315	C₂×C₄	—	26	4	no	—
316	C₂×C₄	—	26	4	no	—
317	C₂×C₄	—	26	4	no	—
318	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
319	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
320	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
321	C₂×C₄	#88161	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	2	4	no	—
329	C₈	—	26	2	no	—
330	C₈	—	26	2	no	—
331	C₈	—	26	4	no	—
332	C₈	—	26	2	no	—
333	C₈	—	26	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	26	2	no	—
337	C₂²	—	26	2	no	—
338	C₂²	—	26	2	no	—
339	C₂²	#11718369	1	2	yes	—
340	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	26	2	no	—
344	C₄	—	26	2	no	—
345	C₄	—	26	2	no	—
346	C₄	#11718369	1	2	yes	—
347	C₄	C₂×#88161	1	2	yes	—
348	C₄	#11719450	1	2	yes	—
349	C₄	#11719455	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	52	2	no	—
352	C₂	C₂×#11718369	1	1	yes	characteristic
353	C₂	#13123263	1	1	yes	characteristic
354	C₂	#13123265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#1451	C₂
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1451	D₁₃
hypocenter	C₁₃	#17329
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #13113137	C₂	semidirect
5. #13113132	C₂	semidirect
6. #13113129	C₂	semidirect
7. #13113127	C₂	semidirect
8. #13113122	C₂	semidirect
9. M₁₆×#1055	C₂	semidirect
10. C₂×#11718046	C₂	semidirect
11. C₁₃×#1451	C₂	semidirect
136. #1451	D₁₃	semidirect
304. C₁₃	#17329	semidirect

Maximal subgroups

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

subgroup	index
#13123802	2
#13123792	2
#13123481	2
#13113137	2
#13113132	2
#13113129	2
#13113127	2
#13113122	2
M₁₆×#1055	2
C₂×#11718046	2
C₁₃×#1451	2
#17329	13