Subgroups — #16016342

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718855	C₂	1	1	yes	maximal, characteristic
2	#13123753	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123752	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116719	C₂	1	4	yes	maximal, complemented
5	#13116714	C₂	1	4	yes	maximal, complemented
6	C₂×#11718085	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116565	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116435	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2143	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
11	#11719524	—	2	1	no	—
12	#11719450	—	2	1	no	—
13	C₂×#88158	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	#11718897	—	2	1	no	—
17	#11718858	—	2	2	no	—
18	#11718857	C₂²	1	4	yes	complemented
19	#11718855	C₄	1	2	yes	complemented
20	#11718855	C₄	1	2	yes	complemented
21	#11718855	C₂²	1	2	yes	—
22	#11718855	C₂²	1	2	yes	—
23	C₂×#88069	C₄	1	1	yes	characteristic, complemented
24	C₂×#88068	C₄	1	1	yes	characteristic, complemented
25	C₂×#88067	C₄	1	1	yes	characteristic, complemented
26	C₂×#88066	C₄	1	1	yes	characteristic, complemented
27	#11718800	—	2	1	no	—
28	C₂×#88059	C₂²	1	1	yes	characteristic
29	#11718762	C₂²	1	1	yes	characteristic
30	C₂²×#6090	C₂²	1	1	yes	characteristic
31	#11718556	C₂²	1	4	yes	complemented
32	#11718555	—	2	2	no	—
33	C₁₃×#359	C₂²	1	4	yes	complemented
34	#11718102	C₂²	1	4	yes	complemented
35	#11718094	C₂²	1	2	yes	complemented
36	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
37	#11718085	C₂²	1	2	yes	complemented
38	#11718085	C₂²	1	2	yes	complemented
39	C₂×C₁₃×#41	—	2	1	no	—
40	C₂×#6127	—	2	1	no	—
41	C₁₃×#135	—	2	1	no	—
42	C₁₃×#135	—	2	1	no	—
43	C₁₃×#135	C₂×C₄	1	2	yes	—
44	C₁₃×#135	C₂³	1	2	yes	—
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×SD₁₆	—	2	1	no	—
47	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₃×M₁₆	—	2	1	no	—
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	#88202	—	2	4	no	—
51	#88202	—	4	2	no	—
52	#88180	D₄	1	1	yes	characteristic
53	C₂×#6099	—	2	1	no	—
54	C₂×#6099	—	4	2	no	—
55	C₂×#6099	D₄	1	1	yes	characteristic
56	#88161	—	2	4	no	—
57	M₁₆×D₁₃	—	4	2	no	—
58	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
59	#88158	—	2	1	no	—
60	#88158	—	2	1	no	—
61	#88158	C₂×C₄	1	2	yes	—
62	#88158	C₂³	1	2	yes	—
63	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
64	C₂×#6095	—	2	1	no	—
65	#88155	—	4	2	no	—
66	C₁₃×M₃₂	—	2	1	no	—
67	C₁₃×M₃₂	—	2	1	no	—
68	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
69	C₁₃×#108	—	2	1	no	—
70	C₁₃×#108	C₂³	1	2	yes	—
71	#88077	—	2	4	no	—
72	#88069	C₂×C₄	1	2	yes	—
73	#88069	C₂×C₄	1	2	yes	—
74	#88068	C₂×C₄	1	2	yes	—
75	#88068	C₂×C₄	1	2	yes	—
76	#88067	C₂×C₄	1	2	yes	—
77	#88067	C₂×C₄	1	2	yes	—
78	#88066	C₂×C₄	1	2	yes	—
79	#88066	C₂×C₄	1	2	yes	—
80	#88065	—	2	4	no	—
81	#88059	—	2	1	no	—
82	#88059	C₂³	1	2	yes	—
83	#88055	D₄	1	2	yes	—
84	C₈×Dic₁₃	C₂³	1	2	yes	—
85	C₂×#6090	C₂×C₄	1	1	yes	characteristic
86	C₂×#6090	C₂×C₄	1	1	yes	characteristic
87	C₂×#6090	C₂×C₄	1	1	yes	characteristic
88	C₂×#6090	C₂×C₄	1	1	yes	characteristic
89	C₂×#6090	C₂³	1	1	yes	characteristic
90	C₂×#6090	C₂³	1	1	yes	characteristic
91	#33465	—	13	1	no	maximal, Sylow
92	C₁₃×#41	—	2	2	no	—
93	C₁₃×#41	—	2	2	no	—
94	C₁₃×#41	—	4	1	no	—
95	C₁₃×#41	—	4	1	no	—
96	C₂×Q₈×C₁₃	—	2	1	no	—
97	C₂×D₄×C₁₃	—	2	1	no	—
98	C₂×D₄×C₁₃	—	4	1	no	—
99	C₂²×C₄×C₁₃	—	4	1	no	—
100	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₂²⋊D₁₃	—	4	1	no	—
102	#6127	—	2	2	no	—
103	#6127	—	2	2	no	—
104	#6127	—	4	1	no	—
105	#6127	—	4	1	no	—
106	C₂×D₅₂	—	2	1	no	—
107	C₂×C₄×D₁₃	—	4	1	no	—
108	C₂×Dic₂₆	—	2	1	no	—
109	C₁₃×Q₁₆	—	2	1	no	—
110	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
111	C₁₃×SD₁₆	—	2	2	no	—
112	C₁₃×SD₁₆	—	2	2	no	—
113	C₁₃×D₈	—	2	1	no	—
114	C₁₃×D₈	C₂²×C₄	1	2	yes	—
115	C₁₃×M₁₆	—	2	2	no	—
116	C₁₃×M₁₆	—	4	1	no	—
117	C₂×C₈×C₁₃	—	4	1	no	—
118	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
122	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
123	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₁₃	—	2	1	no	—
125	C₄⋊Dic₁₃	—	2	1	no	—
126	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
127	#6099	—	2	2	no	—
128	#6099	—	4	1	no	—
129	#6099	—	4	4	no	—
130	#6099	D₈	1	2	yes	—
131	#6099	D₈	1	2	yes	—
132	C₂×#1055	—	4	1	no	—
133	C₂×#1055	—	4	4	no	—
134	C₂×#1055	C₂×D₄	1	2	yes	—
135	Dic₅₂	—	2	1	no	—
136	Dic₅₂	C₂²×C₄	1	2	yes	—
137	D₁₀₄	—	2	1	no	—
138	D₁₀₄	C₂²×C₄	1	2	yes	—
139	#6095	—	2	2	no	—
140	#6095	—	2	2	no	—
141	#6094	—	4	4	no	—
142	C₈×D₁₃	—	4	4	no	—
143	C₁₃×C₁₆	—	2	2	no	—
144	#6090	C₂²×C₄	1	1	yes	characteristic
145	#6090	C₂²×C₄	1	1	yes	characteristic
146	#6090	C₂²×C₄	1	1	yes	characteristic
147	#6090	C₂²×C₄	1	1	yes	characteristic
148	C₂×#508	—	13	1	no	—
149	#2957	—	13	1	no	—
150	#2957	—	13	1	no	—
151	#2179	—	13	4	no	—
152	#2178	—	13	4	no	—
153	C₂×#368	—	13	1	no	—
154	#2143	—	13	1	no	—
155	#2143	D₁₃	1	1	yes	characteristic, complemented
156	#2108	—	13	1	no	—
157	C₂³×C₁₃	—	4	1	no	—
158	C₂³×C₁₃	#135	1	1	yes	characteristic
159	C₂²×D₁₃	—	4	1	no	—
160	Q₈×C₁₃	—	2	2	no	—
161	Q₈×C₁₃	—	4	1	no	—
162	D₄×C₁₃	—	2	2	no	—
163	D₄×C₁₃	—	4	2	no	—
164	D₄×C₁₃	—	4	2	no	—
165	D₄×C₁₃	—	4	1	no	—
166	C₂×C₄×C₁₃	—	4	1	no	—
167	C₂×C₄×C₁₃	—	4	2	no	—
168	C₂×C₄×C₁₃	—	4	2	no	—
169	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
174	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₁₃	—	4	2	no	—
176	C₂²⋊D₁₃	—	4	2	no	—
177	C₂×Dic₁₃	—	2	2	no	—
178	C₂×Dic₁₃	—	4	1	no	—
179	D₅₂	—	2	2	no	—
180	D₅₂	—	4	1	no	—
181	C₄×D₁₃	—	4	2	no	—
182	C₄×D₁₃	—	4	2	no	—
183	Dic₂₆	—	2	2	no	—
184	Dic₂₆	—	4	1	no	—
185	C₈×C₁₃	—	4	2	no	—
186	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
187	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
188	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
189	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
190	#1055	—	4	2	no	—
191	#1055	C₂×D₈	1	4	yes	—
192	C₂×#135	—	13	1	no	—
193	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
194	#568	—	26	1	no	—
195	#568	—	26	1	no	—
196	#508	—	13	2	no	—
197	#508	—	13	2	no	—
198	#508	—	13	2	no	—
199	#508	—	13	2	no	—
200	C₂×Q₃₂	—	13	1	no	—
201	C₂×SD₃₂	—	13	1	no	—
202	C₂×SD₃₂	—	13	1	no	—
203	C₂×D₁₆	—	13	1	no	—
204	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
205	C₂²×C₁₆	—	13	1	no	—
206	#444	—	26	2	no	—
207	#444	—	26	2	no	—
208	#443	—	13	4	no	—
209	#443	—	13	4	no	—
210	C₂×#108	—	13	1	no	—
211	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
212	#421	—	26	1	no	—
213	#421	—	26	1	no	—
214	#405	—	13	1	no	—
215	#368	—	13	2	no	—
216	#368	—	13	2	no	—
217	#359	—	13	4	no	—
218	#359	C₂×D₁₃	1	4	yes	complemented
219	#346	—	13	2	no	—
220	C₄×C₁₆	—	13	2	no	—
221	C₂²×C₁₃	—	4	1	no	—
222	C₂²×C₁₃	—	4	2	no	—
223	C₂²×C₁₃	—	16	1	no	—
224	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
225	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
226	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
227	C₂×D₁₃	—	4	2	no	—
228	C₂×D₁₃	—	16	1	no	—
229	C₄×C₁₃	—	4	2	no	—
230	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
231	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
232	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
233	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
234	Dic₁₃	—	4	2	no	—
235	Dic₁₃	—	4	2	no	—
236	C₂×#41	—	2	1	no	—
237	C₂×#41	—	26	1	no	—
238	#135	—	2	1	no	—
239	#135	—	2	1	no	—
240	#135	—	13	2	no	—
241	#135	—	13	2	no	—
242	#135	—	26	1	no	—
243	#135	—	26	1	no	—
244	#135	C₄×D₁₃	1	2	yes	—
245	#135	C₂²×D₁₃	1	2	yes	—
246	C₂×Q₁₆	—	13	1	no	—
247	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
248	C₂×SD₁₆	—	2	1	no	—
249	C₂×SD₁₆	—	26	1	no	—
250	C₂×D₈	—	13	1	no	—
251	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
252	#131	—	26	4	no	—
253	#131	—	26	4	no	—
254	#131	—	52	2	no	—
255	#131	—	52	2	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂×M₁₆	—	13	1	no	—
258	C₂×M₁₆	—	26	1	no	—
259	C₂×M₁₆	—	52	2	no	—
260	C₂×M₁₆	—	52	2	no	—
261	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
262	#117	—	13	1	no	—
263	Q₃₂	—	13	2	no	—
264	Q₃₂	—	13	2	no	—
265	SD₃₂	—	13	2	no	—
266	SD₃₂	—	13	2	no	—
267	SD₃₂	—	13	2	no	—
268	SD₃₂	—	13	2	no	—
269	D₁₆	—	13	2	no	—
270	D₁₆	—	13	2	no	—
271	M₃₂	—	2	1	no	—
272	M₃₂	—	2	1	no	—
273	C₂×C₁₆	—	13	1	no	—
274	C₂×C₁₆	—	13	1	no	—
275	C₂×C₁₆	—	13	1	no	—
276	C₂×C₁₆	—	13	1	no	—
277	C₂×C₁₆	—	13	1	no	—
278	C₂×C₁₆	—	13	1	no	—
279	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
280	#108	—	2	1	no	—
281	#108	—	13	2	no	—
282	#108	—	26	1	no	—
283	#108	C₂²×D₁₃	1	2	yes	—
284	#104	—	26	4	no	—
285	#104	—	26	4	no	—
286	#97	—	13	2	no	—
287	C₄×C₈	—	13	2	no	—
288	C₂×C₁₃	—	2	1	no	—
289	C₂×C₁₃	—	2	1	no	—
290	C₂×C₁₃	—	2	1	no	—
291	C₂×C₁₃	—	8	2	no	—
292	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
293	D₁₃	—	8	2	no	—
294	#41	—	2	2	no	—
295	#41	—	2	2	no	—
296	#41	—	4	1	no	—
297	#41	—	4	1	no	—
298	#41	—	26	2	no	—
299	#41	—	26	2	no	—
300	#41	—	52	1	no	—
301	#41	—	52	1	no	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	26	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	4	1	no	—
306	C₂×D₄	—	26	1	no	—
307	C₂×D₄	—	52	1	no	—
308	C₂²×C₄	—	4	1	no	—
309	C₂²×C₄	—	52	1	no	—
310	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
311	Q₁₆	—	2	1	no	—
312	Q₁₆	—	13	2	no	—
313	Q₁₆	—	26	1	no	—
314	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
315	SD₁₆	—	2	2	no	—
316	SD₁₆	—	2	2	no	—
317	SD₁₆	—	26	2	no	—
318	SD₁₆	—	26	2	no	—
319	D₈	—	2	1	no	—
320	D₈	—	13	2	no	—
321	D₈	—	26	1	no	—
322	D₈	C₂×C₄×D₁₃	1	2	yes	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	4	1	no	—
325	M₁₆	—	13	2	no	—
326	M₁₆	—	13	2	no	—
327	M₁₆	—	26	2	no	—
328	M₁₆	—	52	4	no	—
329	M₁₆	—	52	4	no	—
330	M₁₆	—	52	1	no	—
331	C₂×C₈	—	4	1	no	—
332	C₂×C₈	—	13	2	no	—
333	C₂×C₈	—	52	4	no	—
334	C₂×C₈	—	52	4	no	—
335	C₂×C₈	—	52	1	no	—
336	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
337	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
338	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
339	C₂×C₈	#6129	1	1	yes	characteristic
340	C₂×C₈	#6129	1	1	yes	characteristic
341	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
342	C₄⋊C₄	—	26	1	no	—
343	C₂²⋊C₄	—	26	1	no	—
344	C₄²	—	13	2	no	—
345	C₁₆	—	2	2	no	—
346	C₁₆	—	13	1	no	—
347	C₁₆	—	13	1	no	—
348	C₁₆	—	13	1	no	—
349	C₁₆	—	13	1	no	—
350	C₁₃	#33465	1	1	yes	characteristic, complemented, Sylow
351	C₂³	—	4	1	no	—
352	C₂³	—	52	1	no	—
353	C₂³	#88170	1	1	yes	characteristic
354	Q₈	—	2	2	no	—
355	Q₈	—	4	1	no	—
356	Q₈	—	26	2	no	—
357	Q₈	—	52	1	no	—
358	D₄	—	2	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	1	no	—
362	D₄	—	26	2	no	—
363	D₄	—	52	2	no	—
364	D₄	—	52	2	no	—
365	D₄	—	52	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	26	2	no	—
370	C₂×C₄	—	52	1	no	—
371	C₂×C₄	—	52	2	no	—
372	C₂×C₄	—	52	2	no	—
373	C₂×C₄	#88135	1	1	yes	characteristic
374	C₂×C₄	#88135	1	1	yes	characteristic
375	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
376	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
377	C₂×C₄	#88170	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
379	C₈	—	4	2	no	—
380	C₈	—	13	4	no	—
381	C₈	—	52	2	no	—
382	C₈	#88135	1	1	yes	characteristic
383	C₈	#88135	1	1	yes	characteristic
384	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
385	C₈	C₂×#6129	1	1	yes	characteristic
386	C₂²	—	4	1	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	16	1	no	—
389	C₂²	—	52	2	no	—
390	C₂²	—	208	1	no	—
391	C₂²	#11718417	1	1	yes	characteristic
392	C₂²	#11718417	1	1	yes	characteristic
393	C₂²	C₂×#88170	1	1	yes	characteristic
394	C₄	—	4	2	no	—
395	C₄	—	52	2	no	—
396	C₄	—	52	2	no	—
397	C₄	#11718417	1	1	yes	characteristic
398	C₄	#11718417	1	1	yes	characteristic
399	C₄	C₂×#88135	1	1	yes	characteristic
400	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	—	8	2	no	—
405	C₂	—	104	2	no	—
406	C₂	C₂×#11718417	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2143	C₂
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₂×#11718855	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2143	D₁₃
hypocenter	C₁₃	#33465
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2143

Complements

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

N	H ≅ G/N	type
2. #13123753	C₂	semidirect
3. #13123752	C₂	semidirect
4. #13116719	C₂	semidirect
5. #13116714	C₂	semidirect
6. C₂×#11718085	C₂	semidirect
7. #13116565	C₂	semidirect
8. #13116435	C₂	semidirect
9. C₁₃×#2143	C₂	semidirect
18. #11718857	C₂²	semidirect
19. #11718855	C₄	semidirect
20. #11718855	C₄	semidirect
23. C₂×#88069	C₄	semidirect
24. C₂×#88068	C₄	semidirect
25. C₂×#88067	C₄	semidirect
26. C₂×#88066	C₄	semidirect
31. #11718556	C₂²	semidirect
33. C₁₃×#359	C₂²	semidirect
34. #11718102	C₂²	semidirect
35. #11718094	C₂²	semidirect
36. C₁₆×Dic₁₃	C₂²	semidirect
37. #11718085	C₂²	semidirect
38. #11718085	C₂²	semidirect
155. #2143	D₁₃	semidirect
218. #359	C₂×D₁₃	semidirect
350. C₁₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718855	2
#13123753	2
#13123752	2
#13116719	2
#13116714	2
C₂×#11718085	2
#13116565	2
#13116435	2
C₁₃×#2143	2
#33465	13