Subgroups — #15834616

Order 3328.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123968	C₂	1	2	yes	maximal, complemented
2	#13113721	C₂	1	2	yes	maximal, complemented
3	#13113715	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113275	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	#11718586	—	2	2	no	—
24	#11718586	C₂²	1	1	yes	characteristic
25	#11718585	—	2	2	no	—
26	#11718585	C₂²	1	2	yes	—
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	#11718050	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	1	no	—
52	C₂×#6099	—	2	2	no	—
53	C₂×#6099	—	2	4	no	—
54	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₃×#106	C₂×C₄	1	2	yes	—
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#103	C₂×C₄	1	2	yes	—
59	C₁₃×#102	C₂×C₄	1	2	yes	—
60	C₁₃×#102	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²⋊#1055	—	2	4	no	—
66	C₂²⋊#1055	D₄	1	2	yes	—
67	C₄⋊#1055	—	2	2	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	D₄	1	1	yes	characteristic
70	C₄⋊#1055	D₄	1	1	yes	characteristic
71	C₄⋊#1055	C₂³	1	4	yes	—
72	#88043	—	2	1	no	—
73	C₄×#1055	—	2	4	no	—
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	#18059	—	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	—	8	4	no	—
113	#6099	—	8	2	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	2	4	no	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	C₂×#1055	C₂×D₄	1	2	yes	—
118	#2991	—	13	2	no	—
119	C₄×#135	D₁₃	1	1	yes	characteristic
120	#1589	—	13	2	no	—
121	#1588	—	13	4	no	—
122	C₈×Q₁₆	—	13	2	no	—
123	#1567	—	13	1	no	—
124	#1489	—	13	2	no	—
125	C₈×M₁₆	—	13	1	no	—
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
130	D₄×C₁₃	—	2	2	no	—
131	D₄×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	4	4	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	4	no	—
145	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
155	#1055	—	2	4	no	—
156	#1055	—	2	2	no	—
157	#1055	—	2	2	no	—
158	#1055	—	4	4	no	—
159	C₂×#135	Dic₁₃	1	1	yes	characteristic
160	C₄×#41	C₂×D₁₃	1	2	yes	—
161	C₈×Q₈	—	13	4	no	—
162	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
163	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
164	C₄×SD₁₆	Dic₁₃	1	2	yes	—
165	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
166	C₄×D₈	Dic₁₃	1	1	yes	characteristic
167	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
168	#436	—	13	4	no	—
169	#436	—	26	4	no	—
170	#431	—	13	1	no	—
171	#431	—	26	2	no	—
172	#427	Dic₁₃	1	1	yes	characteristic
173	#423	—	13	2	no	—
174	#423	—	26	2	no	—
175	#416	Dic₁₃	1	2	yes	—
176	C₄×M₁₆	—	13	1	no	—
177	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
178	#335	—	13	2	no	—
179	#335	—	13	2	no	—
180	Q₈⋊C₈	—	13	4	no	—
181	D₄⋊C₈	—	13	4	no	—
182	#322	—	13	2	no	—
183	C₈²	—	13	2	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	4	2	no	—
188	C₂²×C₁₃	—	4	2	no	—
189	C₂²×C₁₃	—	4	2	no	—
190	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	4	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	4	2	no	—
196	C₄×C₁₃	—	4	2	no	—
197	C₄×C₁₃	#443	1	2	yes	—
198	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
199	C₄×C₁₃	#568	1	1	yes	characteristic
200	C₄×C₁₃	#576	1	1	yes	characteristic
201	C₄×C₁₃	#578	1	1	yes	characteristic
202	C₂×#41	C₂×Dic₁₃	1	2	yes	—
203	#135	—	2	4	no	—
204	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
206	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
207	C₂×M₁₆	—	26	4	no	—
208	C₂×M₁₆	—	26	1	no	—
209	C₂×M₁₆	—	26	2	no	—
210	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
212	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
213	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
217	C₄×D₄	C₂²×D₁₃	1	2	yes	—
218	#117	—	2	2	no	—
219	#117	C₂×Dic₁₃	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	#107	C₂×Dic₁₃	1	1	yes	characteristic
223	#107	C₂×Dic₁₃	1	1	yes	characteristic
224	#106	C₂×Dic₁₃	1	2	yes	—
225	C₄⋊C₈	—	13	1	no	—
226	C₄⋊C₈	—	13	1	no	—
227	C₄⋊C₈	—	13	4	no	—
228	C₄⋊C₈	—	26	4	no	—
229	C₄⋊C₈	—	26	2	no	—
230	#103	C₂×Dic₁₃	1	2	yes	—
231	#103	C₂×Dic₁₃	1	2	yes	—
232	#102	C₂×Dic₁₃	1	2	yes	—
233	#102	C₂×Dic₁₃	1	2	yes	—
234	C₂²⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	—	26	4	no	—
236	#97	—	26	1	no	—
237	C₄×C₈	—	13	2	no	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	—	26	4	no	—
240	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×C₁₃	—	4	1	no	—
245	C₂×C₁₃	—	8	2	no	—
246	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
247	C₂×C₁₃	#2951	1	1	yes	characteristic
248	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
249	#41	—	2	4	no	—
250	#41	—	4	2	no	—
251	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	#6129	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
260	Q₁₆	#6099	1	4	yes	complemented
261	SD₁₆	—	2	4	no	—
262	D₈	—	2	2	no	—
263	M₁₆	—	26	4	no	—
264	M₁₆	—	104	4	no	—
265	M₁₆	—	104	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	26	4	no	—
272	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
279	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	D₄×D₁₃	1	1	yes	characteristic
287	C₄²	#6129	1	1	yes	characteristic
288	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
289	C₄²	C₂³×D₁₃	1	1	yes	characteristic
290	C₁₃	#18059	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	2	2	no	—
292	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	C₂×#6099	1	4	yes	—
295	D₄	—	2	2	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
311	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
312	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂×#6129	1	1	yes	characteristic
315	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	26	4	no	—
320	C₈	—	26	2	no	—
321	C₈	—	26	2	no	—
322	C₈	—	52	4	no	—
323	C₈	C₂×#6099	1	2	yes	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	4	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	#11718857	1	2	yes	—
338	C₄	#11719474	1	1	yes	characteristic
339	C₄	#11719476	1	1	yes	characteristic
340	C₄	C₂²×#6099	1	1	yes	characteristic
341	C₄	#11719524	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	8	2	no	—
344	C₂	D₄×#6099	1	1	yes	characteristic
345	C₂	#13123712	1	1	yes	characteristic
346	C₂	C₂×#11718857	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	#18059
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. #13123968	C₂	semidirect
2. #13113721	C₂	semidirect
3. #13113715	C₂	semidirect
4. Q₁₆×#1055	C₂	semidirect
5. #13113595	C₂	semidirect
6. #13113275	C₂	semidirect
7. C₈×#6099	C₂	semidirect
88. C₁₃×Q₁₆	M₁₆	semidirect
260. Q₁₆	#6099	semidirect
290. C₁₃	#18059	semidirect

Maximal subgroups

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

subgroup	index
#13123968	2
#13113721	2
#13113715	2
Q₁₆×#1055	2
#13113595	2
#13113275	2
C₈×#6099	2
C₄×C₁₃×#135	2
#18059	13