Subgroups — #15832205

Order 3328.

588 conjugacy classes of subgroups.

The subgroup lattice has 370 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	#13123503	C₂	1	2	yes	maximal
3	#13113626	C₂	1	2	yes	maximal, complemented
4	#13113620	C₂	1	2	yes	maximal, complemented
5	#13113617	C₂	1	2	yes	maximal, complemented
6	#13113600	C₂	1	2	yes	maximal, complemented
7	#13113585	C₂	1	2	yes	maximal, complemented
8	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#6127	C₂²	1	1	yes	characteristic
11	C₁₃×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718772	C₄	1	2	yes	—
14	#11718772	C₄	1	2	yes	—
15	#11718586	C₂²	1	2	yes	—
16	#11718586	C₂²	1	1	yes	characteristic
17	#11718498	C₂²	1	2	yes	—
18	#11718496	C₂²	1	2	yes	—
19	#11718494	C₂²	1	2	yes	—
20	D₁₃×C₄⋊C₈	C₂²	1	2	yes	—
21	#11718388	—	2	2	no	—
22	#11718381	—	2	2	no	—
23	#11718370	—	2	2	no	—
24	#11718369	—	2	2	no	—
25	#11718349	C₄	1	1	yes	characteristic
26	#11718349	C₄	1	1	yes	characteristic
27	C₄×#6094	C₂²	1	2	yes	—
28	#11718346	C₂²	1	2	yes	—
29	#11718340	C₄	1	1	yes	characteristic
30	#11718340	C₄	1	1	yes	characteristic
31	C₁₃×#335	C₂²	1	1	yes	characteristic
32	C₁₃×#335	C₂²	1	1	yes	characteristic
33	C₁₃×#334	C₂²	1	2	yes	—
34	#11718072	C₂²	1	2	yes	—
35	#11718072	C₂²	1	2	yes	—
36	#11718071	C₂²	1	2	yes	—
37	#11718071	C₂²	1	2	yes	—
38	#11718050	C₂²	1	1	yes	characteristic
39	#11718050	C₂²	1	1	yes	characteristic
40	#11718049	C₂²	1	2	yes	—
41	C₂×#6127	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₄×C₂²⋊D₁₃	Q₈	1	2	yes	—
47	#88180	D₄	1	1	yes	characteristic
48	C₂×#6094	—	2	2	no	—
49	C₂×#6094	C₂×C₄	1	2	yes	—
50	C₂×#6094	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₁₃	—	2	2	no	—
52	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
53	C₄×D₅₂	Q₈	1	1	yes	characteristic
54	#88126	D₄	1	2	yes	—
55	C₄²×D₁₃	C₂³	1	2	yes	—
56	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₂₆	Q₈	1	1	yes	characteristic
58	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
60	C₁₃×C₂²⋊C₈	—	2	2	no	—
61	C₁₃×#97	—	2	2	no	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₂²⋊#1055	—	2	2	no	—
67	C₂²⋊#1055	C₂×C₄	1	2	yes	—
68	#88060	—	2	2	no	—
69	#88060	—	2	2	no	—
70	#88060	C₂×C₄	1	2	yes	—
71	#88060	C₂×C₄	1	2	yes	—
72	#88055	—	2	2	no	—
73	#88055	C₂×C₄	1	2	yes	—
74	#88055	C₂×C₄	1	2	yes	—
75	#88054	—	2	2	no	—
76	#88054	—	2	2	no	—
77	#88054	C₂×C₄	1	2	yes	—
78	#88054	C₂×C₄	1	2	yes	—
79	C₈×Dic₁₃	—	2	2	no	—
80	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
81	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
82	C₄⋊#1055	C₂³	1	2	yes	—
83	C₄⋊#1055	C₂³	1	2	yes	—
84	C₄×#1055	—	2	2	no	—
85	C₄×#1055	C₂³	1	2	yes	—
86	#17833	—	13	1	no	maximal, Sylow
87	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₁₃	C₄⋊C₄	1	2	yes	—
91	#6127	—	4	1	no	—
92	#6127	—	4	1	no	—
93	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
97	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	4	1	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
111	#6103	C₄⋊C₄	1	2	yes	—
112	#6103	C₄⋊C₄	1	2	yes	—
113	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
115	#6101	C₄⋊C₄	1	2	yes	—
116	#6101	C₄⋊C₄	1	2	yes	—
117	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
119	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
120	C₂×#1055	—	2	2	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	C₂²×C₄	1	2	yes	—
123	C₂×#1055	C₂²×C₄	1	2	yes	—
124	#6094	—	2	4	no	—
125	#6094	—	4	2	no	—
126	#6094	—	4	4	no	—
127	C₈×D₁₃	—	4	4	no	—
128	#2965	—	13	1	no	—
129	#2925	—	13	2	no	—
130	#1574	—	13	2	no	—
131	#1573	—	13	2	no	—
132	#1572	—	13	2	no	—
133	#1569	—	13	2	no	—
134	#1568	D₁₃	1	1	yes	characteristic, complemented
135	#1565	—	13	2	no	—
136	#1564	—	13	1	no	—
137	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
150	C₂²⋊D₁₃	—	4	2	no	—
151	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
154	D₅₂	—	4	1	no	—
155	C₄×D₁₃	—	2	2	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	4	2	no	—
160	Dic₂₆	—	4	1	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	4	2	no	—
165	C₈×C₁₃	—	4	2	no	—
166	C₈×C₁₃	#131	1	2	yes	—
167	#1055	—	2	2	no	—
168	#1055	—	2	2	no	—
169	#1055	—	4	2	no	—
170	#1055	—	4	2	no	—
171	C₄×#41	—	13	1	no	—
172	#446	—	13	1	no	—
173	#446	—	13	1	no	—
174	#436	—	13	1	no	—
175	#436	—	13	1	no	—
176	#435	—	13	2	no	—
177	#435	—	13	2	no	—
178	#433	—	26	2	no	—
179	#433	—	26	2	no	—
180	#432	—	13	2	no	—
181	#432	C₂×D₁₃	1	2	yes	—
182	#431	—	13	1	no	—
183	#431	—	13	2	no	—
184	#431	—	13	2	no	—
185	#431	—	13	2	no	—
186	#423	—	13	2	no	—
187	C₂×C₄⋊C₈	—	13	2	no	—
188	#408	—	26	2	no	—
189	#405	—	26	2	no	—
190	C₄×M₁₆	—	13	2	no	—
191	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
192	#335	—	13	2	no	—
193	#335	—	13	2	no	—
194	#335	—	13	1	no	—
195	#335	—	13	1	no	—
196	#335	C₂×D₁₃	1	1	yes	characteristic
197	#335	C₂×D₁₃	1	1	yes	characteristic
198	#334	—	13	2	no	—
199	#334	—	13	2	no	—
200	#334	—	13	2	no	—
201	#334	C₂×D₁₃	1	2	yes	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
206	C₂×D₁₃	—	2	2	no	—
207	C₂×D₁₃	—	2	2	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
214	C₄×C₁₃	#568	1	1	yes	characteristic
215	C₄×C₁₃	#575	1	1	yes	characteristic
216	C₄×C₁₃	#575	1	1	yes	characteristic
217	Dic₁₃	—	2	2	no	—
218	Dic₁₃	—	2	2	no	—
219	Dic₁₃	—	2	2	no	—
220	C₂×#41	—	13	1	no	—
221	C₂×M₁₆	—	13	2	no	—
222	C₂×M₁₆	—	13	2	no	—
223	C₂×M₁₆	—	26	2	no	—
224	C₂²×C₈	—	26	2	no	—
225	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
226	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
227	C₄×Q₈	—	13	1	no	—
228	C₄×Q₈	—	13	1	no	—
229	C₄×D₄	—	13	2	no	—
230	C₄×D₄	—	13	2	no	—
231	C₄×D₄	—	13	1	no	—
232	C₄×D₄	—	13	1	no	—
233	#117	—	13	1	no	—
234	#117	—	13	2	no	—
235	C₂×C₄²	—	13	2	no	—
236	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
237	C₄⋊C₈	—	13	2	no	—
238	C₄⋊C₈	—	13	2	no	—
239	C₄⋊C₈	—	13	1	no	—
240	C₄⋊C₈	—	13	1	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	26	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
246	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	13	2	no	—
249	C₂²⋊C₈	—	13	2	no	—
250	C₂²⋊C₈	—	13	2	no	—
251	C₂²⋊C₈	—	26	2	no	—
252	C₂²⋊C₈	—	26	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	#97	—	2	2	no	—
255	#97	—	13	2	no	—
256	#97	—	13	2	no	—
257	#97	—	26	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	26	2	no	—
260	C₄×C₈	—	26	2	no	—
261	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
262	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
265	C₂×C₁₃	—	4	1	no	—
266	C₂×C₁₃	#2906	1	1	yes	characteristic
267	C₂×C₁₃	#2911	1	1	yes	characteristic
268	C₂×C₁₃	#2916	1	1	yes	characteristic
269	D₁₃	—	4	2	no	—
270	#41	—	52	1	no	—
271	#41	—	52	1	no	—
272	C₂×Q₈	—	13	1	no	—
273	C₂×D₄	—	13	2	no	—
274	C₂×D₄	—	13	1	no	—
275	C₂²×C₄	—	13	2	no	—
276	C₂²×C₄	—	13	2	no	—
277	C₂²×C₄	—	13	2	no	—
278	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
280	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
281	M₁₆	—	26	4	no	—
282	M₁₆	—	52	2	no	—
283	M₁₆	—	52	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	13	2	no	—
289	C₂×C₈	—	13	2	no	—
290	C₂×C₈	—	26	2	no	—
291	C₂×C₈	—	26	2	no	—
292	C₂×C₈	—	52	4	no	—
293	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
297	C₄⋊C₄	—	13	2	no	—
298	C₄⋊C₄	—	13	2	no	—
299	C₄⋊C₄	—	13	1	no	—
300	C₄⋊C₄	—	13	1	no	—
301	C₂²⋊C₄	—	13	2	no	—
302	C₂²⋊C₄	—	13	2	no	—
303	C₂²⋊C₄	—	13	2	no	—
304	C₄²	—	13	2	no	—
305	C₄²	—	13	2	no	—
306	C₄²	—	13	2	no	—
307	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
308	C₄²	D₄×D₁₃	1	1	yes	characteristic
309	C₄²	Q₈×D₁₃	1	1	yes	characteristic
310	C₄²	C₂³×D₁₃	1	1	yes	characteristic
311	C₁₃	#17833	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	13	2	no	—
313	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
314	Q₈	—	52	1	no	—
315	D₄	—	52	1	no	—
316	D₄	—	52	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	13	2	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	13	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	26	2	no	—
342	C₈	—	26	2	no	—
343	C₈	—	52	2	no	—
344	C₈	—	52	2	no	—
345	C₈	#88161	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	26	2	no	—
350	C₂²	—	26	2	no	—
351	C₂²	—	26	2	no	—
352	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	26	2	no	—
358	C₄	—	26	2	no	—
359	C₄	—	26	2	no	—
360	C₄	C₂×#88161	1	1	yes	characteristic
361	C₄	#11719450	1	1	yes	characteristic
362	C₄	#11719473	1	1	yes	characteristic
363	C₄	#11719473	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	52	2	no	—
366	C₂	#13123405	1	1	yes	characteristic
367	C₂	#13123424	1	1	yes	characteristic
368	C₂	#13123444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1568	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6127	D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1568	D₁₃
hypocenter	C₁₃	#17833
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #13113626	C₂	semidirect
4. #13113620	C₂	semidirect
5. #13113617	C₂	semidirect
6. #13113600	C₂	semidirect
7. #13113585	C₂	semidirect
8. #13113584	C₂	semidirect
9. C₁₃×#1568	C₂	semidirect
134. #1568	D₁₃	semidirect
311. C₁₃	#17833	semidirect

Maximal subgroups

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

subgroup	index
#13123802	2
#13123503	2
#13113626	2
#13113620	2
#13113617	2
#13113600	2
#13113585	2
#13113584	2
C₁₃×#1568	2
#17833	13