Subgroups — #16016431

Order 3328.

530 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116743	C₂	1	2	yes	maximal, complemented
4	#13116742	C₂	1	2	yes	maximal, complemented
5	C₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116738	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116590	C₂	1	1	yes	maximal, characteristic
8	#13116590	C₂	1	2	yes	maximal
9	#13116429	C₂	1	2	yes	maximal
10	C₄×#88038	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×D₁₃	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	Q₈×#1055	C₄	1	1	yes	characteristic
15	C₂×#88038	C₂²	1	1	yes	characteristic
16	C₂×#88038	C₂²	1	2	yes	—
17	C₂×#88038	C₂²	1	1	yes	characteristic
18	D₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
19	D₁₃×C₄⋊C₈	C₄	1	2	yes	—
20	#11718491	C₄	1	2	yes	—
21	#11718491	C₄	1	1	yes	characteristic
22	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₁₃	C₂²	1	2	yes	—
24	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₆	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₁₃×#346	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	#11718094	C₂²	1	1	yes	characteristic
31	#11718094	C₂²	1	1	yes	characteristic
32	#11718093	C₂²	1	2	yes	—
33	#11718093	C₂²	1	2	yes	—
34	#11718093	C₂²	1	2	yes	—
35	C₁₆×Dic₁₃	C₂²	1	2	yes	—
36	C₁₆×Dic₁₃	C₂²	1	2	yes	—
37	C₄⋊#6090	C₂²	1	1	yes	characteristic
38	C₄⋊#6090	C₂²	1	2	yes	—
39	#11718053	C₂²	1	2	yes	—
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
48	C₄⋊C₄×D₁₃	C₈	1	2	yes	—
49	#88141	C₈	1	2	yes	—
50	#88141	C₈	1	1	yes	characteristic
51	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×D₁₃	C₂×C₄	1	2	yes	—
53	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	#88054	C₂×C₄	1	2	yes	—
63	#88054	C₂×C₄	1	2	yes	—
64	#88054	C₂×C₄	1	2	yes	—
65	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₃	C₂³	1	2	yes	—
68	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#6090	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	2	yes	—
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#1055	C₂×C₄	1	2	yes	—
74	C₄×#1055	C₂×C₄	1	1	yes	characteristic
75	C₄×#1055	C₂×C₄	1	2	yes	—
76	#88038	—	2	2	no	—
77	#88038	—	2	4	no	—
78	#88038	Q₈	1	4	yes	complemented
79	Q₈×M₃₂	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
81	Q₈×D₁₃	—	2	4	no	—
82	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
88	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
97	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
98	#6101	C₂×C₈	1	2	yes	—
99	#6101	C₂×C₈	1	2	yes	—
100	#6101	C₂×C₈	1	2	yes	—
101	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
103	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
107	C₂×#1055	C₂²×C₄	1	2	yes	—
108	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
109	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
110	C₈×D₁₃	—	2	2	no	—
111	C₈×D₁₃	—	2	4	no	—
112	C₈×D₁₃	C₂×Q₈	1	2	yes	—
113	C₈×D₁₃	#41	1	2	yes	—
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	—	2	2	no	—
116	C₁₃×C₁₆	C₂×Q₈	1	2	yes	complemented
117	#6090	—	2	1	no	—
118	#6090	—	2	2	no	—
119	#6090	C₂×Q₈	1	2	yes	complemented
120	C₂×C₈×Q₈	—	13	1	no	—
121	#2183	—	13	1	no	—
122	#2183	—	13	2	no	—
123	#2183	—	13	2	no	—
124	#2183	D₁₃	1	1	yes	characteristic, complemented
125	Q₈×C₁₆	—	13	1	no	—
126	Q₈×C₁₆	—	13	1	no	—
127	#2150	—	13	2	no	—
128	#2150	—	13	1	no	—
129	C₄×M₃₂	—	13	2	no	—
130	C₄×M₃₂	—	13	1	no	—
131	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
132	Q₈×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
139	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	—	2	4	no	—
145	C₄×D₁₃	—	4	1	no	—
146	C₄×D₁₃	—	4	2	no	—
147	C₄×D₁₃	C₄×Q₈	1	2	yes	—
148	C₄×D₁₃	#131	1	2	yes	—
149	Dic₂₆	—	2	4	no	—
150	Dic₂₆	M₃₂	1	4	yes	complemented
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	#1055	—	2	2	no	—
156	#1055	—	2	1	no	—
157	#1055	C₄×Q₈	1	2	yes	—
158	C₂×C₄×Q₈	—	13	1	no	—
159	C₂×M₃₂	—	13	1	no	—
160	C₂×M₃₂	—	13	2	no	—
161	C₂×M₃₂	—	13	1	no	—
162	C₈×Q₈	—	13	1	no	—
163	C₈×Q₈	—	13	2	no	—
164	C₈×Q₈	—	13	1	no	—
165	C₈×Q₈	—	13	1	no	—
166	C₈×Q₈	—	13	2	no	—
167	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	13	2	no	—
169	C₂×C₄⋊C₈	—	13	1	no	—
170	C₂×C₄×C₈	—	13	2	no	—
171	C₂×C₄×C₈	—	13	1	no	—
172	C₄⋊C₁₆	—	13	2	no	—
173	C₄⋊C₁₆	—	13	2	no	—
174	C₄⋊C₁₆	—	13	2	no	—
175	C₄⋊C₁₆	—	13	1	no	—
176	C₄⋊C₁₆	—	13	2	no	—
177	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
178	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
179	#346	—	13	1	no	—
180	#346	—	13	1	no	—
181	#346	—	13	2	no	—
182	#346	C₂×D₁₃	1	2	yes	—
183	C₄×C₁₆	—	13	2	no	—
184	C₄×C₁₆	—	13	2	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
187	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
188	C₂×D₁₃	—	2	1	no	—
189	C₂×D₁₃	—	2	1	no	—
190	C₂×D₁₃	C₈×Q₈	1	2	yes	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
195	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
197	Dic₁₃	—	2	1	no	—
198	Dic₁₃	—	2	2	no	—
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	C₈×Q₈	1	2	yes	—
201	Dic₁₃	C₂×M₃₂	1	4	yes	—
202	C₂²×Q₈	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₄×Q₈	—	13	1	no	—
207	C₄×Q₈	—	13	1	no	—
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	1	no	—
210	C₄×Q₈	—	13	2	no	—
211	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	13	2	no	—
213	C₂×C₄⋊C₄	—	13	1	no	—
214	C₂×C₄²	—	13	2	no	—
215	C₂×C₄²	—	13	1	no	—
216	M₃₂	—	13	4	no	—
217	M₃₂	—	26	4	no	—
218	M₃₂	—	26	2	no	—
219	C₂×C₁₆	—	13	1	no	—
220	C₂×C₁₆	—	13	2	no	—
221	C₂×C₁₆	—	13	1	no	—
222	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
223	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
225	C₄⋊C₈	—	13	2	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	—	13	2	no	—
228	C₄⋊C₈	—	13	1	no	—
229	C₄⋊C₈	—	13	2	no	—
230	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
231	C₄⋊C₈	C₄×D₁₃	1	1	yes	characteristic
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	—	13	1	no	—
234	C₄×C₈	—	13	2	no	—
235	C₄×C₈	—	13	1	no	—
236	C₄×C₈	—	13	2	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	2	yes	—
240	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
241	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
242	C₂×C₁₃	#3407	1	1	yes	characteristic
243	D₁₃	—	2	2	no	—
244	C₂×Q₈	—	13	1	no	—
245	C₂×Q₈	—	13	2	no	—
246	C₂×Q₈	—	26	4	no	—
247	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	2	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	2	no	—
252	C₂²×C₄	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	—	26	4	no	—
260	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	13	2	no	—
264	C₄⋊C₄	—	13	2	no	—
265	C₄⋊C₄	—	13	2	no	—
266	C₄⋊C₄	—	13	2	no	—
267	C₄⋊C₄	—	13	1	no	—
268	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
269	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
270	C₄²	—	13	2	no	—
271	C₄²	—	13	2	no	—
272	C₄²	—	13	1	no	—
273	C₄²	—	13	1	no	—
274	C₄²	—	13	2	no	—
275	C₄²	—	13	1	no	—
276	C₄²	C₂×C₄×D₁₃	1	2	yes	—
277	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	—	13	2	no	—
281	C₁₆	—	26	2	no	—
282	C₁₆	—	26	1	no	—
283	C₁₆	Q₈×D₁₃	1	2	yes	complemented
284	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	13	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	13	4	no	—
288	Q₈	—	26	4	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	26	4	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	—	52	1	no	—
300	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
301	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
303	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	13	2	no	—
308	C₈	—	26	2	no	—
309	C₈	—	26	1	no	—
310	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
311	C₈	#41×D₁₃	1	1	yes	characteristic
312	C₂²	—	13	2	no	—
313	C₂²	—	26	1	no	—
314	C₂²	—	26	1	no	—
315	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	13	2	no	—
320	C₄	—	13	4	no	—
321	C₄	—	26	2	no	—
322	C₄	—	26	1	no	—
323	C₄	—	26	1	no	—
324	C₄	D₁₃×M₃₂	1	2	yes	—
325	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
326	C₄	D₁₃×#131	1	1	yes	characteristic
327	C₂	—	26	2	no	—
328	C₂	C₈×Q₈×D₁₃	1	1	yes	characteristic
329	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic
330	C₂	#13127937	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2183	C₂
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2183	D₁₃
hypocenter	C₁₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #13116745	C₂	semidirect
3. #13116743	C₂	semidirect
4. #13116742	C₂	semidirect
5. C₁₆×Dic₂₆	C₂	semidirect
6. #13116738	C₂	semidirect
11. C₁₃×#2183	C₂	semidirect
78. #88038	Q₈	semidirect
116. C₁₃×C₁₆	C₂×Q₈	semidirect
119. #6090	C₂×Q₈	semidirect
124. #2183	D₁₃	semidirect
150. Dic₂₆	M₃₂	semidirect
283. C₁₆	Q₈×D₁₃	semidirect
284. C₁₃	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₁₃	2
#13116745	2
#13116743	2
#13116742	2
C₁₆×Dic₂₆	2
#13116738	2
#13116590	2
#13116429	2
C₄×#88038	2
C₁₃×#2183	2
Q₈×M₃₂	13