Subgroups — #16016395

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	#13123753	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123688	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116735	C₂	1	8	yes	maximal, complemented
4	#13116609	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718155	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116572	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116440	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#509	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	#11719524	—	2	1	no	—
12	#11719503	—	2	1	no	—
13	C₁₃×#509	C₄	1	4	yes	complemented
14	C₁₃×#509	C₂²	1	4	yes	—
15	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₁₃×D₁₆	C₄	1	2	yes	complemented
17	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
18	#11718897	—	2	1	no	—
19	#11718876	—	2	1	no	—
20	#11718858	C₂²	1	4	yes	complemented
21	#11718857	—	2	2	no	—
22	#11718813	C₂²	1	4	yes	complemented
23	D₈×Dic₁₃	—	2	2	no	—
24	C₂×#88059	C₂²	1	1	yes	characteristic
25	#11718767	C₂²	1	1	yes	characteristic
26	#11718762	C₂²	1	1	yes	characteristic
27	C₂×#88052	C₂²	1	1	yes	characteristic
28	#11718158	C₂²	1	4	yes	complemented
29	#11718155	C₂²	1	4	yes	complemented
30	#11718105	C₂²	1	4	yes	complemented
31	#11718104	C₂²	1	4	yes	complemented
32	C₂×C₁₃×#41	—	2	1	no	—
33	C₂²×D₄×C₁₃	—	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	—	2	1	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
44	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
45	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×C₁₃×D₈	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊Dic₁₃	—	4	2	no	—
49	#88202	—	2	4	no	—
50	#88202	—	4	2	no	—
51	D₄×Dic₁₃	—	2	4	no	—
52	D₄×Dic₁₃	—	4	2	no	—
53	#88180	—	2	1	no	—
54	#88180	D₄	1	1	yes	characteristic
55	C₂×#6099	—	2	1	no	—
56	C₂×#6099	—	4	2	no	—
57	C₂×#6099	D₄	1	1	yes	characteristic
58	C₁₃×SD₃₂	C₂×C₄	1	8	yes	complemented
59	C₁₃×D₁₆	C₂×C₄	1	8	yes	complemented
60	C₁₃×M₃₂	C₂×C₄	1	2	yes	complemented
61	C₁₃×M₃₂	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
63	#88077	—	2	4	no	—
64	#88072	—	2	4	no	—
65	#88059	—	2	1	no	—
66	#88059	C₂³	1	2	yes	—
67	#88058	—	2	1	no	—
68	#88057	C₂³	1	2	yes	—
69	#88055	C₂³	1	2	yes	—
70	C₈×Dic₁₃	D₄	1	2	yes	—
71	#88052	—	2	1	no	—
72	#88052	C₂³	1	2	yes	—
73	C₂×#6090	—	2	1	no	—
74	#33471	—	13	1	no	maximal, Sylow
75	C₂⁴×C₁₃	—	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	2	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₁₃	—	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	2	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	—	2	1	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₁₃×D₈	C₂²×C₄	1	2	yes	—
101	C₁₃×D₈	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₁₃	—	2	1	no	—
109	C₂²⋊Dic₁₃	—	4	4	no	—
110	C₂²⋊Dic₁₃	—	4	1	no	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄×Dic₁₃	—	4	1	no	—
114	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
115	#6099	—	2	1	no	—
116	#6099	—	2	1	no	—
117	#6099	—	2	2	no	—
118	#6099	—	4	4	no	—
119	#6099	—	4	1	no	—
120	C₂×#1055	—	4	4	no	—
121	C₂×#1055	—	4	1	no	—
122	C₂×#1055	C₂×D₄	1	2	yes	—
123	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
124	#6090	—	2	2	no	—
125	C₂×#509	D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
132	#2110	—	13	1	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂³×C₁₃	—	4	1	no	—
135	C₂³×C₁₃	—	4	2	no	—
136	C₂³×C₁₃	—	4	2	no	—
137	C₂³×C₁₃	—	16	2	no	—
138	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	Q₈×C₁₃	—	2	2	no	—
140	Q₈×C₁₃	—	4	1	no	—
141	D₄×C₁₃	—	2	4	no	—
142	D₄×C₁₃	—	2	2	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	4	2	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	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	C₂×D₁₃	1	1	yes	characteristic
169	#568	—	26	1	no	—
170	#518	—	26	1	no	—
171	#509	Dic₁₃	1	4	yes	complemented
172	#509	C₂×D₁₃	1	4	yes	—
173	C₂×SD₃₂	Dic₁₃	1	2	yes	complemented
174	C₂×D₁₆	Dic₁₃	1	2	yes	complemented
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	—	13	1	no	—
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	—	13	4	no	—
189	#347	—	13	4	no	—
190	C₂²×C₁₃	—	4	1	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	4	4	no	—
193	C₂²×C₁₃	—	8	4	no	—
194	C₂²×C₁₃	—	16	1	no	—
195	C₂²×C₁₃	—	16	2	no	—
196	C₂²×C₁₃	—	16	2	no	—
197	C₂²×C₁₃	#442	1	1	yes	characteristic
198	C₂²×C₁₃	#442	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₁₃	—	4	2	no	—
201	C₄×C₁₃	#442	1	1	yes	characteristic
202	C₄×C₁₃	#442	1	1	yes	characteristic
203	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	Dic₁₃	—	4	2	no	—
206	Dic₁₃	—	8	2	no	—
207	C₂×#41	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	#135	—	2	1	no	—
210	#135	—	2	1	no	—
211	#135	C₂×Dic₁₃	1	2	yes	complemented
212	#135	C₂²×D₁₃	1	2	yes	—
213	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic, complemented
214	C₂×SD₁₆	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
219	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
220	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic, complemented
221	C₂×D₈	C₂²×D₁₃	1	2	yes	—
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	—	13	1	no	—
231	#117	—	26	1	no	—
232	C₂×C₂²⋊C₄	—	52	2	no	—
233	SD₃₂	C₂×Dic₁₃	1	8	yes	complemented
234	D₁₆	C₂×Dic₁₃	1	8	yes	complemented
235	M₃₂	—	13	2	no	—
236	M₃₂	—	26	1	no	—
237	M₃₂	C₂×Dic₁₃	1	2	yes	complemented
238	M₃₂	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₆	—	26	1	no	—
240	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	complemented
241	#108	—	13	2	no	—
242	#108	—	26	1	no	—
243	#107	—	26	1	no	—
244	#106	—	13	2	no	—
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	4	no	—
253	C₂×C₁₃	—	8	2	no	—
254	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
255	C₂⁴	—	4	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	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	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	C₂²×Dic₁₃	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	2	no	—
280	D₈	C₂²×Dic₁₃	1	2	yes	—
281	D₈	C₂²×Dic₁₃	1	4	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	D₄×D₁₃	1	1	yes	characteristic
291	C₂×C₈	#6129	1	1	yes	characteristic
292	C₂×C₈	#6129	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
296	C₄⋊C₄	—	26	1	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₂²⋊C₄	—	26	1	no	—
299	C₂²⋊C₄	—	52	4	no	—
300	C₂²⋊C₄	—	52	1	no	—
301	C₄²	—	13	2	no	—
302	C₄²	—	52	1	no	—
303	C₁₆	—	26	2	no	—
304	C₁₆	C₂²×Dic₁₃	1	4	yes	—
305	C₁₃	#33471	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	4	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	16	2	no	—
311	C₂³	#88165	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	208	4	no	—
328	C₂×C₄	#88165	1	1	yes	characteristic
329	C₂×C₄	#88165	1	1	yes	characteristic
330	C₂×C₄	#88165	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
332	C₂×C₄	D₄×Dic₁₃	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	D₄×Dic₁₃	1	1	yes	characteristic
337	C₈	D₄×Dic₁₃	1	1	yes	characteristic
338	C₈	C₂×#6129	1	1	yes	characteristic
339	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	16	1	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	16	2	no	—
347	C₂²	#11718813	1	1	yes	characteristic
348	C₂²	#11718813	1	1	yes	characteristic
349	C₂²	C₂×#88165	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	52	2	no	—
352	C₄	—	104	2	no	—
353	C₄	#11718813	1	1	yes	characteristic
354	C₄	#11718813	1	1	yes	characteristic
355	C₄	C₂×#88165	1	1	yes	characteristic
356	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#11718813	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718813
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#509	C₂
socle	C₂×C₁₃	C₂×#442
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#509	D₁₃
hypocenter	C₁₃	#33471
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#509

Complements

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

N	H ≅ G/N	type
1. #13123753	C₂	semidirect
2. #13123688	C₂	semidirect
3. #13116735	C₂	semidirect
4. #13116609	C₂	semidirect
5. C₂×#11718155	C₂	semidirect
6. #13116572	C₂	semidirect
7. #13116440	C₂	semidirect
13. C₁₃×#509	C₄	semidirect
15. C₂×C₁₃×SD₃₂	C₄	semidirect
16. C₂×C₁₃×D₁₆	C₄	semidirect
20. #11718858	C₂²	semidirect
22. #11718813	C₂²	semidirect
28. #11718158	C₂²	semidirect
29. #11718155	C₂²	semidirect
30. #11718105	C₂²	semidirect
31. #11718104	C₂²	semidirect
36. C₁₃×#135	C₂×C₄	semidirect
38. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
45. C₂×C₁₃×D₈	C₂×C₄	semidirect
58. C₁₃×SD₃₂	C₂×C₄	semidirect
59. C₁₃×D₁₆	C₂×C₄	semidirect
60. C₁₃×M₃₂	C₂×C₄	semidirect
62. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
171. #509	Dic₁₃	semidirect
173. C₂×SD₃₂	Dic₁₃	semidirect
174. C₂×D₁₆	Dic₁₃	semidirect
211. #135	C₂×Dic₁₃	semidirect
213. C₂×Q₁₆	C₂×Dic₁₃	semidirect
220. C₂×D₈	C₂×Dic₁₃	semidirect
233. SD₃₂	C₂×Dic₁₃	semidirect
234. D₁₆	C₂×Dic₁₃	semidirect
237. M₃₂	C₂×Dic₁₃	semidirect
240. C₂×C₁₆	C₂×Dic₁₃	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
#13123753	2
#13123688	2
#13116735	2
#13116609	2
C₂×#11718155	2
#13116572	2
#13116440	2
C₂×C₁₃×#509	2
#33471	13