Subgroups — #15834608

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123925	C₂	1	2	yes	maximal
2	#13113733	C₂	1	2	yes	maximal, complemented
3	#13113708	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	#13113599	C₂	1	1	yes	maximal, characteristic
6	#13113283	C₂	1	2	yes	maximal
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
12	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
14	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₃×#103	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718685	—	2	4	no	—
21	#11718685	C₂²	1	4	yes	—
22	Q₈×#1055	C₂²	1	4	yes	—
23	#11718587	—	2	2	no	—
24	#11718587	C₂²	1	2	yes	—
25	#11718586	—	2	2	no	—
26	#11718586	C₂²	1	1	yes	characteristic
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	Q₈⋊#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	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×Q₁₆	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×Q₁₆	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	—	2	2	no	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	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	C₁₃×#107	C₂×C₄	1	2	yes	—
53	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
54	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₃×#103	C₂×C₄	1	2	yes	—
56	C₁₃×#103	C₂×C₄	1	2	yes	—
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#103	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊#1055	—	2	2	no	—
64	C₂²⋊#1055	D₄	1	1	yes	characteristic
65	C₂²⋊#1055	D₄	1	1	yes	characteristic
66	C₄⋊#1055	—	2	4	no	—
67	C₄⋊#1055	—	2	4	no	—
68	C₄⋊#1055	D₄	1	2	yes	—
69	C₄⋊#1055	C₂³	1	4	yes	—
70	#88043	—	2	2	no	—
71	#88043	—	2	1	no	—
72	#88043	—	2	4	no	—
73	C₄×#1055	—	2	4	no	—
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	#18058	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	4	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
87	C₁₃×Q₁₆	—	2	4	no	—
88	C₁₃×Q₁₆	—	2	2	no	—
89	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
102	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₃	—	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	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	4	no	—
114	C₂×#1055	C₂×D₄	1	2	yes	—
115	C₂×#1055	C₂×D₄	1	2	yes	—
116	#2984	—	13	2	no	—
117	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
118	#1591	—	13	2	no	—
119	#1587	—	13	4	no	—
120	C₈×Q₁₆	—	13	2	no	—
121	#1568	—	13	1	no	—
122	#1491	—	13	2	no	—
123	C₈×M₁₆	—	13	1	no	—
124	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	—	2	2	no	—
127	Q₈×C₁₃	—	2	2	no	—
128	Q₈×C₁₃	—	4	4	no	—
129	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	2	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	4	no	—
139	C₂×C₄×C₁₃	—	4	4	no	—
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
150	#1055	—	2	4	no	—
151	#1055	—	2	2	no	—
152	#1055	—	2	2	no	—
153	#1055	—	4	4	no	—
154	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
156	#446	—	13	4	no	—
157	#446	—	26	4	no	—
158	C₈×Q₈	—	13	4	no	—
159	C₄×Q₁₆	Dic₁₃	1	2	yes	—
160	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
161	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
162	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
163	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
164	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
165	#432	—	13	2	no	—
166	#432	—	26	2	no	—
167	#431	—	13	1	no	—
168	#431	—	26	2	no	—
169	C₂×#107	Dic₁₃	1	1	yes	characteristic
170	C₂×#103	Dic₁₃	1	2	yes	—
171	C₄×M₁₆	—	13	1	no	—
172	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
173	#335	—	13	2	no	—
174	#334	—	13	2	no	—
175	Q₈⋊C₈	—	13	4	no	—
176	Q₈⋊C₈	—	13	4	no	—
177	#322	—	13	2	no	—
178	C₈²	—	13	2	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	#443	1	2	yes	—
182	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	4	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
192	C₄×C₁₃	#568	1	1	yes	characteristic
193	C₄×C₁₃	#578	1	2	yes	—
194	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
197	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
198	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
199	C₂×M₁₆	—	26	1	no	—
200	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
205	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
206	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
207	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	#107	C₂×Dic₁₃	1	2	yes	—
213	#107	C₂×Dic₁₃	1	1	yes	characteristic
214	#107	C₂×Dic₁₃	1	1	yes	characteristic
215	C₄⋊C₈	—	13	2	no	—
216	C₄⋊C₈	—	13	4	no	—
217	C₄⋊C₈	—	26	4	no	—
218	C₄⋊C₈	—	26	4	no	—
219	#103	C₂×Dic₁₃	1	2	yes	—
220	#103	C₂×Dic₁₃	1	2	yes	—
221	#103	C₂×Dic₁₃	1	2	yes	—
222	#103	C₂×Dic₁₃	1	2	yes	—
223	C₂²⋊C₈	—	13	1	no	—
224	C₂²⋊C₈	—	13	1	no	—
225	C₂²⋊C₈	—	26	2	no	—
226	#97	—	26	4	no	—
227	#97	—	26	2	no	—
228	#97	—	26	1	no	—
229	C₄×C₈	—	13	2	no	—
230	C₄×C₈	—	13	2	no	—
231	C₄×C₈	—	26	4	no	—
232	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
238	C₂×C₁₃	#2950	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
244	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	#6129	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	#6099	1	4	yes	complemented
253	M₁₆	—	26	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	—	26	4	no	—
260	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
270	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	D₄×D₁₃	1	1	yes	characteristic
276	C₄²	#6129	1	1	yes	characteristic
277	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
278	C₄²	C₂³×D₁₃	1	1	yes	characteristic
279	C₁₃	#18058	1	1	yes	characteristic, complemented, Sylow
280	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	2	2	no	—
284	Q₈	—	4	4	no	—
285	Q₈	C₂×#6099	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
297	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
298	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×#6129	1	1	yes	characteristic
301	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	26	4	no	—
306	C₈	—	26	2	no	—
307	C₈	—	26	2	no	—
308	C₈	—	52	4	no	—
309	C₈	C₂×#6099	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#11718857	1	2	yes	—
313	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	#11719471	1	2	yes	—
323	C₄	C₂²×#6099	1	1	yes	characteristic
324	C₄	#11719524	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	D₄×#6099	1	1	yes	characteristic
327	C₂	#13123704	1	1	yes	characteristic
328	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	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#18058
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13113733	C₂	semidirect
3. #13113708	C₂	semidirect
4. Q₁₆×#1055	C₂	semidirect
89. C₁₃×Q₁₆	M₁₆	semidirect
252. Q₁₆	#6099	semidirect
279. C₁₃	#18058	semidirect

Maximal subgroups

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

subgroup	index
#13123925	2
#13113733	2
#13113708	2
Q₁₆×#1055	2
#13113599	2
#13113283	2
C₈×#6099	2
C₂×C₄×C₁₃×Q₁₆	2
#18058	13