Subgroups — #15834388

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123856	C₂	1	2	yes	maximal, complemented
2	#13113721	C₂	1	2	yes	maximal, complemented
3	#13113680	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113270	C₂	1	2	yes	maximal, complemented
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
10	C₄×C₁₃×#41	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×SD₁₆	C₄	1	2	yes	—
14	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
16	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₄	1	1	yes	characteristic
18	C₁₃×#416	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	Q₈×#1055	C₂²	1	4	yes	—
21	#11718673	—	2	4	no	—
22	#11718673	C₂²	1	4	yes	—
23	#11718587	—	2	2	no	—
24	#11718587	C₂²	1	2	yes	—
25	#11718586	C₂²	1	1	yes	characteristic
26	C₄×#6099	—	2	2	no	—
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	D₄⋊#1055	C₂²	1	4	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	#11718046	C₂²	1	2	yes	—
33	C₈×#1055	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
35	C₁₃×#135	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
43	C₄×D₄×C₁₃	—	2	2	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₃	C₂³	1	2	yes	—
47	C₁₃×#117	—	2	2	no	—
48	C₁₃×#117	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×#6099	—	2	4	no	—
52	C₂×#6099	D₄	1	2	yes	—
53	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
54	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₃×#106	C₂×C₄	1	2	yes	—
56	C₁₃×#103	C₂×C₄	1	2	yes	—
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#102	C₂×C₄	1	2	yes	—
59	C₁₃×#102	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²⋊#1055	—	2	2	no	—
65	C₂²⋊#1055	—	2	1	no	—
66	C₂²⋊#1055	—	2	4	no	—
67	C₄⋊#1055	—	2	1	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	C₂³	1	4	yes	—
70	#88043	—	2	2	no	—
71	#88043	D₄	1	1	yes	characteristic
72	#88043	D₄	1	1	yes	characteristic
73	C₄×#1055	—	2	4	no	—
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	#18039	—	13	1	no	maximal, Sylow
77	C₁₃×#41	—	2	4	no	—
78	C₁₃×#41	—	4	2	no	—
79	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
89	C₁₃×SD₁₆	—	2	4	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	2	no	—
103	C₁₃×C₂²⋊C₄	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	#41	1	1	yes	characteristic
110	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
111	#6099	—	2	4	no	—
112	#6099	—	4	8	no	—
113	C₂×#1055	—	2	4	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	C₂×D₄	1	2	yes	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	#2971	—	13	2	no	—
118	C₄×#135	D₁₃	1	1	yes	characteristic
119	#1589	—	13	2	no	—
120	#1583	—	13	4	no	—
121	C₈×Q₁₆	—	13	2	no	—
122	#1568	—	13	1	no	—
123	#1488	—	13	2	no	—
124	C₈×M₁₆	—	13	1	no	—
125	C₂³×C₁₃	—	2	2	no	—
126	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	2	no	—
128	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
129	D₄×C₁₃	—	2	2	no	—
130	D₄×C₁₃	—	2	2	no	—
131	D₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	4	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	8	2	no	—
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
152	#1055	—	2	2	no	—
153	#1055	—	4	2	no	—
154	#1055	—	4	4	no	—
155	#1055	#135	1	4	yes	complemented
156	C₂×#135	Dic₁₃	1	1	yes	characteristic
157	C₄×#41	C₂×D₁₃	1	2	yes	—
158	C₈×Q₈	—	13	4	no	—
159	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
160	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
161	C₄×SD₁₆	Dic₁₃	1	2	yes	—
162	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
163	C₄×D₈	Dic₁₃	1	1	yes	characteristic
164	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
165	#436	—	13	4	no	—
166	#436	—	26	4	no	—
167	#432	—	13	2	no	—
168	#432	—	26	2	no	—
169	#431	—	13	1	no	—
170	#427	Dic₁₃	1	1	yes	characteristic
171	#416	Dic₁₃	1	2	yes	—
172	C₄×M₁₆	—	13	1	no	—
173	C₄×M₁₆	—	26	2	no	—
174	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
175	#335	—	13	2	no	—
176	#334	—	13	2	no	—
177	Q₈⋊C₈	—	13	4	no	—
178	D₄⋊C₈	—	13	4	no	—
179	#322	—	13	2	no	—
180	C₈²	—	13	2	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	4	2	no	—
185	C₂²×C₁₃	—	4	2	no	—
186	C₂²×C₁₃	—	4	2	no	—
187	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	4	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	—	4	2	no	—
195	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
196	C₄×C₁₃	#568	1	1	yes	characteristic
197	C₄×C₁₃	C₂×#135	1	2	yes	—
198	C₂×#41	C₂×Dic₁₃	1	2	yes	—
199	#135	—	2	4	no	—
200	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
201	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
202	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
203	C₂×M₁₆	—	13	2	no	—
204	C₂×M₁₆	—	26	4	no	—
205	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
207	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
208	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
212	C₄×D₄	C₂²×D₁₃	1	2	yes	—
213	#117	—	2	2	no	—
214	#117	C₂×Dic₁₃	1	2	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	#107	C₂×Dic₁₃	1	1	yes	characteristic
218	#107	C₂×Dic₁₃	1	1	yes	characteristic
219	#106	C₂×Dic₁₃	1	2	yes	—
220	C₄⋊C₈	—	13	4	no	—
221	C₄⋊C₈	—	26	4	no	—
222	C₄⋊C₈	—	26	1	no	—
223	#103	C₂×Dic₁₃	1	2	yes	—
224	#103	C₂×Dic₁₃	1	2	yes	—
225	#102	C₂×Dic₁₃	1	2	yes	—
226	#102	C₂×Dic₁₃	1	2	yes	—
227	C₂²⋊C₈	—	26	4	no	—
228	C₂²⋊C₈	—	26	2	no	—
229	C₂²⋊C₈	—	26	1	no	—
230	#97	—	13	1	no	—
231	#97	—	13	1	no	—
232	#97	—	26	2	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	2	no	—
235	C₄×C₈	—	26	4	no	—
236	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	4	1	no	—
241	C₂×C₁₃	—	8	2	no	—
242	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
243	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
244	C₂×C₁₃	#2957	1	1	yes	characteristic
245	#41	—	2	4	no	—
246	#41	—	4	2	no	—
247	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	#6129	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
256	Q₁₆	#6099	1	4	yes	complemented
257	SD₁₆	—	2	4	no	—
258	D₈	—	2	2	no	—
259	M₁₆	—	26	4	no	—
260	M₁₆	—	52	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	26	4	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
274	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	D₄×D₁₃	1	1	yes	characteristic
282	C₄²	#6129	1	1	yes	characteristic
283	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
284	C₄²	C₂³×D₁₃	1	1	yes	characteristic
285	C₁₃	#18039	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	2	2	no	—
287	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	C₂×#6099	1	4	yes	—
290	D₄	—	2	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂×#6129	1	1	yes	characteristic
308	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	13	4	no	—
313	C₈	—	26	2	no	—
314	C₈	—	52	2	no	—
315	C₈	—	52	4	no	—
316	C₈	C₂×#6099	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	D₁₃×#135	1	2	yes	—
332	C₄	C₂²×#6099	1	1	yes	characteristic
333	C₄	#11719524	1	1	yes	characteristic
334	C₂	—	4	1	no	—
335	C₂	—	8	2	no	—
336	C₂	D₄×#6099	1	1	yes	characteristic
337	C₂	#135×Dic₁₃	1	1	yes	characteristic
338	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#18039
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13123856	C₂	semidirect
2. #13113721	C₂	semidirect
3. #13113680	C₂	semidirect
4. Q₁₆×#1055	C₂	semidirect
5. #13113599	C₂	semidirect
6. #13113270	C₂	semidirect
7. C₈×#6099	C₂	semidirect
88. C₁₃×Q₁₆	M₁₆	semidirect
155. #1055	#135	semidirect
256. Q₁₆	#6099	semidirect
285. C₁₃	#18039	semidirect

Maximal subgroups

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

subgroup	index
#13123856	2
#13113721	2
#13113680	2
Q₁₆×#1055	2
#13113599	2
#13113270	2
C₈×#6099	2
C₄×C₁₃×#135	2
#18039	13