Subgroups — #15832163

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123502	C₂	1	2	yes	maximal
2	#13113616	C₂	1	4	yes	maximal, complemented
3	#13113607	C₂	1	2	yes	maximal, complemented
4	#13113599	C₂	1	2	yes	maximal, complemented
5	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113584	C₂	1	2	yes	maximal, complemented
7	C₂×#11718050	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
12	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
13	C₈×D₄×C₁₃	C₄	1	2	yes	—
14	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
15	C₈×D₄×C₁₃	C₄	1	2	yes	—
16	C₁₃×#431	C₂²	1	1	yes	characteristic
17	C₁₃×#431	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
20	#11718895	—	2	2	no	—
21	#11718661	—	2	4	no	—
22	#11718658	—	2	2	no	—
23	#11718587	C₂²	1	4	yes	—
24	#11718586	C₂²	1	4	yes	—
25	#11718585	C₂²	1	2	yes	—
26	C₂×C₄⋊#1055	C₂²	1	2	yes	—
27	#11718050	C₂²	1	2	yes	—
28	#11718050	C₂²	1	2	yes	—
29	#11718050	C₂²	1	1	yes	characteristic
30	#11718050	C₂²	1	1	yes	characteristic
31	#11718050	C₂²	1	2	yes	—
32	#11718049	C₂²	1	4	yes	—
33	#11718049	C₂²	1	4	yes	—
34	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
41	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₁₃	Q₈	1	2	yes	—
44	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
45	C₁₃×#117	D₄	1	1	yes	characteristic
46	C₁₃×#117	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×#6099	—	2	2	no	—
50	C₂²×#1055	—	2	2	no	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
56	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	2	yes	—
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²⋊#1055	—	2	4	no	—
66	C₂²⋊#1055	—	2	4	no	—
67	C₄⋊#1055	—	2	2	no	—
68	C₄⋊#1055	—	2	2	no	—
69	C₄⋊#1055	C₂³	1	2	yes	—
70	C₄⋊#1055	C₂³	1	2	yes	—
71	C₄⋊#1055	C₂³	1	4	yes	—
72	#88043	—	2	4	no	—
73	C₄×#1055	—	2	4	no	—
74	#17829	—	13	1	no	maximal, Sylow
75	C₁₃×#41	—	2	2	no	—
76	C₁₃×#41	—	2	2	no	—
77	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	—	4	2	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
97	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
99	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
101	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
108	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6099	—	4	4	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	4	no	—
114	C₂×#1055	—	4	4	no	—
115	C₈×#41	D₁₃	1	1	yes	characteristic
116	#2925	—	13	2	no	—
117	#1572	—	13	4	no	—
118	#1570	—	13	2	no	—
119	#1568	—	13	2	no	—
120	#1567	—	13	1	no	—
121	#1564	—	13	2	no	—
122	C₂×#335	—	13	1	no	—
123	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	2	no	—
126	D₄×C₁₃	—	2	4	no	—
127	D₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#135	1	2	yes	—
143	C₂×C₄×C₁₃	#135	1	2	yes	—
144	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	#131	1	2	yes	—
153	#1055	—	4	2	no	—
154	#1055	—	4	2	no	—
155	#1055	—	4	4	no	—
156	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
157	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
158	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
159	C₈×D₄	Dic₁₃	1	1	yes	characteristic
160	C₈×D₄	Dic₁₃	1	2	yes	—
161	C₈×D₄	Dic₁₃	1	1	yes	characteristic
162	C₈×D₄	Dic₁₃	1	2	yes	—
163	#433	—	26	4	no	—
164	#432	—	13	4	no	—
165	#431	—	13	4	no	—
166	#431	C₂×D₁₃	1	1	yes	characteristic
167	#431	C₂×D₁₃	1	2	yes	—
168	#423	—	13	2	no	—
169	C₂×C₄⋊C₈	—	13	2	no	—
170	#408	—	26	2	no	—
171	#405	—	26	2	no	—
172	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
174	#335	—	13	1	no	—
175	#335	—	13	1	no	—
176	#335	—	13	2	no	—
177	#335	—	13	2	no	—
178	#335	—	13	2	no	—
179	#334	—	13	4	no	—
180	#334	—	13	4	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	2	no	—
184	C₂²×C₁₃	—	2	2	no	—
185	C₂²×C₁₃	—	2	2	no	—
186	C₂²×C₁₃	#427	1	2	yes	—
187	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	#427	1	2	yes	—
195	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
196	C₄×C₁₃	#568	1	1	yes	characteristic
197	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
198	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
199	C₂×#41	D₅₂	1	1	yes	characteristic
200	C₂×M₁₆	—	26	2	no	—
201	C₂²×C₈	—	26	2	no	—
202	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
204	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
205	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
206	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
207	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
208	C₄×D₄	Dic₂₆	1	2	yes	—
209	C₄×D₄	Dic₂₆	1	1	yes	characteristic
210	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
211	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
212	#117	D₅₂	1	1	yes	characteristic
213	#117	D₅₂	1	2	yes	—
214	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
216	C₄⋊C₈	—	13	2	no	—
217	C₄⋊C₈	—	13	2	no	—
218	C₄⋊C₈	—	13	4	no	—
219	C₄⋊C₈	—	26	2	no	—
220	C₄⋊C₈	—	26	2	no	—
221	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
222	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
223	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
224	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
225	C₂²⋊C₈	—	26	4	no	—
226	C₂²⋊C₈	—	26	4	no	—
227	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
228	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
229	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
230	#97	—	26	4	no	—
231	C₄×C₈	—	26	4	no	—
232	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
234	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
236	C₄×C₈	C₂²×D₁₃	1	2	yes	—
237	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₁₃	—	2	2	no	—
240	C₂×C₁₃	—	4	2	no	—
241	C₂×C₁₃	#2906	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
243	C₂×C₁₃	#2916	1	1	yes	characteristic
244	#41	—	2	2	no	—
245	#41	—	2	2	no	—
246	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
247	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
248	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
250	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₅₂	1	2	yes	—
252	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
255	M₁₆	—	52	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	4	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	52	4	no	—
264	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
266	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
268	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
271	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
272	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
273	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
274	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
275	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
276	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
277	C₄²	C₂×Dic₂₆	1	2	yes	—
278	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
279	C₄²	C₂×D₅₂	1	2	yes	—
280	C₄²	C₂×D₅₂	1	1	yes	characteristic
281	C₄²	C₂²×Dic₁₃	1	2	yes	—
282	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
283	C₄²	C₂³×D₁₃	1	1	yes	characteristic
284	C₁₃	#17829	1	1	yes	characteristic, complemented, Sylow
285	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
286	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	#88158	1	2	yes	—
299	C₂×C₄	#88158	1	2	yes	—
300	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
304	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
308	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	52	2	no	—
315	C₈	—	52	2	no	—
316	C₈	—	52	4	no	—
317	C₈	#88202	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	#11718767	1	2	yes	—
324	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	#11718767	1	2	yes	—
332	C₄	C₂×#88158	1	1	yes	characteristic
333	C₄	C₂×#88158	1	1	yes	characteristic
334	C₄	C₂×#88202	1	1	yes	characteristic
335	C₄	#11719524	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#13123404	1	1	yes	characteristic
339	C₂	C₂×#11718767	1	1	yes	characteristic
340	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×#41	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	#17829
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #13113616	C₂	semidirect
3. #13113607	C₂	semidirect
4. #13113599	C₂	semidirect
5. #13113595	C₂	semidirect
6. #13113584	C₂	semidirect
7. C₂×#11718050	C₂	semidirect
284. C₁₃	#17829	semidirect

Maximal subgroups

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

subgroup	index
#13123502	2
#13113616	2
#13113607	2
#13113599	2
#13113595	2
#13113584	2
C₂×#11718050	2
C₈×C₁₃×#41	2
#17829	13