Subgroups — #16016391

Order 3328.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718854	C₂	1	1	yes	maximal, characteristic
2	#13123753	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123687	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116731	C₂	1	8	yes	maximal, complemented
5	#13116608	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718108	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116440	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2145	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	#11719524	—	2	1	no	—
11	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
12	#11719226	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#427	C₂²	1	1	yes	characteristic
15	#11718897	—	2	1	no	—
16	#11718858	C₂²	1	4	yes	complemented
17	#11718857	—	2	2	no	—
18	#11718854	C₄	1	4	yes	complemented
19	#11718854	C₂²	1	4	yes	—
20	C₂×#88068	C₄	1	2	yes	complemented
21	C₂×#88066	C₄	1	2	yes	complemented
22	C₂×#88059	C₂²	1	1	yes	characteristic
23	#11718762	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	#11718662	—	2	1	no	—
26	#11718533	—	2	2	no	—
27	#11718530	C₂²	1	4	yes	complemented
28	C₁₃×#360	C₂²	1	4	yes	complemented
29	#11718108	C₂²	1	4	yes	complemented
30	#11718104	C₂²	1	4	yes	complemented
31	#11718079	C₂²	1	4	yes	complemented
32	C₂×C₁₃×#41	—	2	1	no	—
33	C₂²×D₅₂	—	2	1	no	—
34	C₁₃×#135	—	2	1	no	—
35	C₁₃×#135	—	2	1	no	—
36	C₁₃×#135	C₂×C₄	1	2	yes	complemented
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×C₁₃×SD₁₆	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	—	2	1	no	—
43	#88202	—	2	4	no	—
44	#88202	—	4	2	no	—
45	#88180	D₄	1	1	yes	characteristic
46	C₂×#6099	—	2	1	no	—
47	C₂×#6099	—	4	2	no	—
48	C₂×#6099	D₄	1	1	yes	characteristic
49	C₂×D₁₀₄	—	2	1	no	—
50	C₂×D₁₀₄	—	2	1	no	—
51	C₂×D₁₀₄	—	2	1	no	—
52	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
53	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
54	C₂×D₁₀₄	C₂³	1	2	yes	—
55	#88147	—	2	4	no	—
56	C₂²⋊C₄×D₁₃	—	4	2	no	—
57	C₄×D₅₂	—	4	2	no	—
58	C₁₃×M₃₂	—	2	1	no	—
59	C₁₃×M₃₂	C₂³	1	2	yes	—
60	C₂×C₁₃×C₁₆	—	2	1	no	—
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#106	C₂³	1	2	yes	—
63	#88077	—	2	4	no	—
64	#88068	C₂×C₄	1	8	yes	complemented
65	#88066	C₂×C₄	1	8	yes	complemented
66	#88059	—	2	1	no	—
67	#88059	C₂³	1	2	yes	—
68	#88055	C₂³	1	2	yes	—
69	C₈×Dic₁₃	D₄	1	2	yes	—
70	#88052	C₂×C₄	1	2	yes	complemented
71	#88052	C₂³	1	2	yes	—
72	C₂×#6090	C₂×C₄	1	2	yes	complemented
73	#88049	—	2	4	no	—
74	#33471	—	13	1	no	maximal, Sylow
75	C₂³×D₁₃	—	4	1	no	—
76	C₁₃×#41	—	2	2	no	—
77	C₁₃×#41	—	2	2	no	—
78	C₁₃×#41	—	4	1	no	—
79	C₁₃×#41	—	4	1	no	—
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×D₅₂	—	2	2	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×D₅₂	—	2	2	no	—
88	C₂×D₅₂	—	4	1	no	—
89	C₂×D₅₂	—	4	1	no	—
90	C₂×D₅₂	—	4	1	no	—
91	C₂×C₄×D₁₃	—	4	4	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
94	C₁₃×SD₁₆	—	2	2	no	—
95	C₁₃×SD₁₆	—	2	2	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₂²⋊Dic₁₃	—	2	1	no	—
108	#6103	—	4	4	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
111	#6099	—	2	1	no	—
112	#6099	—	2	1	no	—
113	#6099	—	2	2	no	—
114	#6099	—	4	4	no	—
115	#6099	—	4	1	no	—
116	C₂×#1055	—	4	4	no	—
117	C₂×#1055	—	4	1	no	—
118	C₂×#1055	C₂×D₄	1	2	yes	—
119	D₁₀₄	—	2	2	no	—
120	D₁₀₄	—	2	2	no	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	C₂²×C₄	1	4	yes	—
123	C₁₃×C₁₆	—	2	2	no	—
124	#6090	C₂²×C₄	1	4	yes	—
125	C₂×#509	—	13	1	no	—
126	#2957	—	13	1	no	—
127	#2948	—	13	1	no	—
128	AGL(1,ℤ/16)	—	13	8	no	—
129	#2155	—	13	1	no	—
130	C₂×#361	—	13	1	no	—
131	#2145	D₁₃	1	1	yes	characteristic, complemented
132	#2110	—	13	1	no	—
133	C₂³×C₁₃	—	4	1	no	—
134	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂²×D₁₃	—	4	2	no	—
136	C₂²×D₁₃	—	4	2	no	—
137	C₂²×D₁₃	—	4	2	no	—
138	C₂²×D₁₃	—	16	2	no	—
139	Q₈×C₁₃	—	2	2	no	—
140	Q₈×C₁₃	—	4	1	no	—
141	D₄×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₁₃	—	2	2	no	—
156	D₅₂	—	2	4	no	—
157	D₅₂	—	4	2	no	—
158	D₅₂	—	4	2	no	—
159	D₅₂	—	4	2	no	—
160	C₄×D₁₃	—	16	4	no	—
161	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
162	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
163	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
164	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
165	#1055	—	2	2	no	—
166	#1055	—	4	2	no	—
167	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
168	C₂²×D₈	—	13	1	no	—
169	#568	—	26	1	no	—
170	#518	—	26	1	no	—
171	#509	—	13	4	no	—
172	#509	—	13	4	no	—
173	C₂×SD₃₂	—	13	2	no	—
174	C₂×D₁₆	—	13	2	no	—
175	C₂×M₃₂	—	13	1	no	—
176	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
177	#444	—	13	4	no	—
178	#443	—	26	2	no	—
179	#442	—	13	4	no	—
180	C₄×D₈	—	26	2	no	—
181	C₂×#108	—	13	1	no	—
182	#427	C₂×D₁₃	1	1	yes	characteristic
183	#421	—	26	1	no	—
184	#418	—	26	1	no	—
185	#405	—	13	1	no	—
186	#365	—	13	4	no	—
187	#361	—	13	4	no	—
188	#360	C₂×D₁₃	1	4	yes	complemented
189	#347	—	13	4	no	—
190	C₂²×C₁₃	—	4	1	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	16	1	no	—
193	C₂²×C₁₃	#442	1	1	yes	characteristic
194	C₂²×C₁₃	#442	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₂×D₁₃	—	4	4	no	—
197	C₂×D₁₃	—	8	4	no	—
198	C₂×D₁₃	—	16	2	no	—
199	C₂×D₁₃	—	16	2	no	—
200	C₄×C₁₃	—	4	2	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	#442	1	1	yes	characteristic
203	C₄×C₁₃	#442	1	1	yes	characteristic
204	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
205	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	Dic₁₃	—	4	2	no	—
207	C₂×#41	—	2	1	no	—
208	C₂²×D₄	—	26	1	no	—
209	#135	—	2	1	no	—
210	#135	—	2	1	no	—
211	#135	C₄×D₁₃	1	2	yes	complemented
212	#135	C₂²×D₁₃	1	2	yes	—
213	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic, complemented
214	C₂×SD₁₆	—	2	1	no	—
215	C₂×D₈	—	13	2	no	—
216	C₂×D₈	—	13	2	no	—
217	C₂×D₈	—	13	2	no	—
218	C₂×D₈	—	26	1	no	—
219	C₂×D₈	—	26	1	no	—
220	C₂×D₈	—	26	1	no	—
221	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic, complemented
222	#131	—	26	4	no	—
223	#131	—	52	2	no	—
224	C₂×M₁₆	—	13	1	no	—
225	C₂×M₁₆	—	26	1	no	—
226	C₂×M₁₆	—	52	2	no	—
227	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₄×D₄	—	26	4	no	—
229	C₄×D₄	—	52	2	no	—
230	#117	—	2	1	no	—
231	#117	—	13	1	no	—
232	C₂×C₂²⋊C₄	—	52	2	no	—
233	SD₃₂	—	13	8	no	—
234	D₁₆	—	13	8	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	13	2	no	—
237	M₃₂	—	13	2	no	—
238	M₃₂	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	13	2	no	—
241	#108	—	13	2	no	—
242	#108	—	26	1	no	—
243	#107	—	2	1	no	—
244	#106	C₂²×D₁₃	1	2	yes	—
245	#104	—	26	4	no	—
246	#102	—	26	4	no	—
247	#97	—	13	2	no	—
248	C₄×C₈	—	13	2	no	—
249	C₂×C₁₃	—	2	1	no	—
250	C₂×C₁₃	—	2	1	no	—
251	C₂×C₁₃	—	2	1	no	—
252	C₂×C₁₃	—	8	2	no	—
253	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
254	D₁₃	—	8	4	no	—
255	C₂⁴	—	52	1	no	—
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	26	2	no	—
264	C₂×D₄	—	26	2	no	—
265	C₂×D₄	—	26	2	no	—
266	C₂×D₄	—	52	1	no	—
267	C₂×D₄	—	52	1	no	—
268	C₂×D₄	—	52	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	52	4	no	—
271	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	—	13	4	no	—
278	D₈	—	26	2	no	—
279	D₈	—	26	2	no	—
280	D₈	—	26	2	no	—
281	D₈	C₂×C₄×D₁₃	1	2	yes	—
282	M₁₆	—	26	1	no	—
283	M₁₆	—	26	1	no	—
284	M₁₆	—	26	2	no	—
285	M₁₆	—	52	4	no	—
286	M₁₆	—	52	1	no	—
287	C₂×C₈	—	13	2	no	—
288	C₂×C₈	—	52	4	no	—
289	C₂×C₈	—	52	1	no	—
290	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
293	C₂×C₈	#6129	1	1	yes	characteristic
294	C₂×C₈	#6129	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	26	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	26	1	no	—
300	C₂²⋊C₄	—	52	4	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	13	2	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	13	4	no	—
305	C₁₃	#33471	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	4	1	no	—
307	C₂³	—	52	2	no	—
308	C₂³	—	52	2	no	—
309	C₂³	—	52	2	no	—
310	C₂³	—	208	2	no	—
311	C₂³	#88165	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	26	4	no	—
319	D₄	—	52	2	no	—
320	D₄	—	52	2	no	—
321	D₄	—	52	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	208	4	no	—
328	C₂×C₄	#88135	1	1	yes	characteristic
329	C₂×C₄	#88135	1	1	yes	characteristic
330	C₂×C₄	#88165	1	1	yes	characteristic
331	C₂×C₄	#88168	1	1	yes	characteristic
332	C₂×C₄	#88168	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
334	C₈	—	26	2	no	—
335	C₈	—	52	2	no	—
336	C₈	#88135	1	1	yes	characteristic
337	C₈	#88135	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
339	C₈	C₂×#6129	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	16	1	no	—
343	C₂²	—	52	4	no	—
344	C₂²	—	104	4	no	—
345	C₂²	—	208	2	no	—
346	C₂²	—	208	2	no	—
347	C₂²	#11718446	1	1	yes	characteristic
348	C₂²	#11718446	1	1	yes	characteristic
349	C₂²	C₂×#88165	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	52	2	no	—
353	C₄	#11718446	1	1	yes	characteristic
354	C₄	#11718446	1	1	yes	characteristic
355	C₄	C₂×#88135	1	1	yes	characteristic
356	C₄	C₂×#88168	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	2	no	—
361	C₂	—	104	4	no	—
362	C₂	C₂×#11718446	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718446
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2145	C₂
socle	C₂×C₁₃	C₂×#442
prime radical	C₂×#11718854	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2145	D₁₃
hypocenter	C₁₃	#33471
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
2. #13123753	C₂	semidirect
3. #13123687	C₂	semidirect
4. #13116731	C₂	semidirect
5. #13116608	C₂	semidirect
6. C₂×#11718108	C₂	semidirect
7. #13116440	C₂	semidirect
8. C₁₃×#2145	C₂	semidirect
16. #11718858	C₂²	semidirect
18. #11718854	C₄	semidirect
20. C₂×#88068	C₄	semidirect
21. C₂×#88066	C₄	semidirect
27. #11718530	C₂²	semidirect
28. C₁₃×#360	C₂²	semidirect
29. #11718108	C₂²	semidirect
30. #11718104	C₂²	semidirect
31. #11718079	C₂²	semidirect
36. C₁₃×#135	C₂×C₄	semidirect
38. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
40. C₂×C₁₃×D₈	C₂×C₄	semidirect
64. #88068	C₂×C₄	semidirect
65. #88066	C₂×C₄	semidirect
70. #88052	C₂×C₄	semidirect
72. C₂×#6090	C₂×C₄	semidirect
131. #2145	D₁₃	semidirect
188. #360	C₂×D₁₃	semidirect
211. #135	C₄×D₁₃	semidirect
213. C₂×Q₁₆	C₄×D₁₃	semidirect
221. C₂×D₈	C₄×D₁₃	semidirect
305. C₁₃	#33471	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718854	2
#13123753	2
#13123687	2
#13116731	2
#13116608	2
C₂×#11718108	2
#13116440	2
C₁₃×#2145	2
#33471	13