Subgroups — #16016264

Order 3328.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 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	#13123712	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123687	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116709	C₂	1	2	yes	maximal, complemented
5	#13116703	C₂	1	2	yes	maximal, complemented
6	#13116685	C₂	1	2	yes	maximal, complemented
7	#13116680	C₂	1	2	yes	maximal, complemented
8	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718101	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	#11719532	—	2	1	no	—
14	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
15	#11719226	—	2	1	no	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	#11718893	—	2	1	no	—
19	#11718854	C₄	1	8	yes	complemented
20	Q₁₆×Dic₁₃	C₂²	1	2	yes	complemented
21	C₂×#88068	C₂²	1	2	yes	—
22	#11718825	—	2	2	no	—
23	D₈×Dic₁₃	C₂²	1	2	yes	complemented
24	C₂×#88066	C₂²	1	2	yes	—
25	C₂×#88058	C₂²	1	1	yes	characteristic
26	#11718762	C₂²	1	1	yes	characteristic
27	C₂×#88052	C₂²	1	1	yes	characteristic
28	#11718662	—	2	1	no	—
29	#11718533	C₂²	1	2	yes	—
30	#11718533	C₂²	1	2	yes	—
31	#11718530	—	2	2	no	—
32	C₁₃×#358	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#358	C₂²	1	1	yes	characteristic, complemented
34	C₁₃×#357	C₂²	1	1	yes	characteristic, complemented
35	C₁₃×#357	C₂²	1	1	yes	characteristic, complemented
36	#11718101	C₂²	1	2	yes	—
37	#11718101	C₂²	1	2	yes	—
38	#11718094	C₂²	1	2	yes	complemented
39	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
40	#11718082	C₂²	1	2	yes	complemented
41	#11718081	C₂²	1	2	yes	complemented
42	C₂×C₁₃×#41	—	2	1	no	—
43	C₂²×D₅₂	—	2	1	no	—
44	C₁₃×#135	—	2	2	no	—
45	C₁₃×#135	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×SD₁₆	—	2	1	no	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×#117	—	2	1	no	—
51	Q₈×Dic₁₃	—	2	2	no	—
52	D₄×Dic₁₃	—	2	2	no	—
53	D₄×Dic₁₃	—	4	2	no	—
54	#88180	—	4	1	no	—
55	#88180	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₁₃	—	2	1	no	—
57	C₂×C₄⋊Dic₁₃	—	4	1	no	—
58	C₂×#6099	D₄	1	1	yes	characteristic
59	C₂×D₁₀₄	—	2	1	no	—
60	C₂×D₁₀₄	—	2	2	no	—
61	C₂×D₁₀₄	C₂×C₄	1	4	yes	complemented
62	C₂×D₁₀₄	C₂³	1	2	yes	—
63	#88147	—	2	2	no	—
64	#88147	—	2	2	no	—
65	C₂²⋊C₄×D₁₃	—	4	2	no	—
66	C₄×D₅₂	—	4	1	no	—
67	C₄×D₅₂	—	4	1	no	—
68	C₂×C₁₃×C₁₆	—	2	2	no	—
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₁₃×#107	C₂³	1	1	yes	characteristic
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#106	—	2	1	no	—
74	#88075	—	2	2	no	—
75	#88072	—	2	2	no	—
76	#88068	C₂×C₄	1	8	yes	complemented
77	#88066	C₂×C₄	1	8	yes	complemented
78	#88058	—	2	1	no	—
79	#88058	C₂³	1	2	yes	—
80	#88055	D₄	1	2	yes	—
81	C₈×Dic₁₃	C₂³	1	2	yes	—
82	#88052	C₂×C₄	1	4	yes	complemented
83	C₂×#6090	C₂³	1	2	yes	—
84	#88049	—	2	2	no	—
85	#88049	—	2	2	no	—
86	#33454	—	13	1	no	maximal, Sylow
87	C₂³×D₁₃	—	4	1	no	—
88	C₁₃×#41	—	2	4	no	—
89	C₁₃×#41	—	4	2	no	—
90	C₂×Q₈×C₁₃	—	2	1	no	—
91	C₂×D₄×C₁₃	—	2	1	no	—
92	C₂×D₄×C₁₃	—	4	1	no	—
93	C₂²×C₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×Dic₁₃	—	4	1	no	—
96	C₂²×Dic₁₃	—	4	2	no	—
97	C₂×D₅₂	—	2	4	no	—
98	C₂×D₅₂	—	2	2	no	—
99	C₂×D₅₂	—	4	1	no	—
100	C₂×D₅₂	—	4	2	no	—
101	C₂×C₄×D₁₃	—	4	2	no	—
102	C₂×C₄×D₁₃	—	4	2	no	—
103	C₁₃×Q₁₆	—	2	1	no	—
104	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
105	C₁₃×SD₁₆	—	2	4	no	—
106	C₁₃×D₈	—	2	1	no	—
107	C₁₃×D₈	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	2	yes	—
111	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₂²⋊C₄	—	4	1	no	—
115	C₄²×C₁₃	—	4	1	no	—
116	C₂²⋊Dic₁₃	—	4	2	no	—
117	C₂²⋊Dic₁₃	D₈	1	2	yes	—
118	#6103	—	4	2	no	—
119	#6103	—	4	2	no	—
120	C₄⋊Dic₁₃	—	2	2	no	—
121	C₄⋊Dic₁₃	—	4	1	no	—
122	C₄⋊Dic₁₃	—	4	2	no	—
123	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
124	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
125	C₄×Dic₁₃	—	4	2	no	—
126	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
127	#6099	—	2	2	no	—
128	C₂×#1055	C₂×D₄	1	2	yes	—
129	D₁₀₄	—	2	4	no	—
130	D₁₀₄	—	2	2	no	—
131	D₁₀₄	C₂²×C₄	1	4	yes	—
132	C₁₃×C₁₆	—	2	1	no	—
133	C₁₃×C₁₆	—	2	1	no	—
134	#6090	C₂²×C₄	1	4	yes	—
135	C₂×#509	—	13	1	no	—
136	#2951	—	13	1	no	—
137	#2948	—	13	1	no	—
138	#2177	—	13	2	no	—
139	#2175	—	13	2	no	—
140	C₄×SD₃₂	—	13	2	no	—
141	C₄×D₁₆	—	13	2	no	—
142	#2159	—	13	1	no	—
143	#2138	D₁₃	1	1	yes	characteristic, complemented
144	C₂×#357	—	13	1	no	—
145	#2108	—	13	1	no	—
146	C₂³×C₁₃	—	4	1	no	—
147	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂²×D₁₃	—	4	2	no	—
149	C₂²×D₁₃	—	4	4	no	—
150	C₂²×D₁₃	—	16	1	no	—
151	C₂²×D₁₃	—	16	1	no	—
152	Q₈×C₁₃	—	2	2	no	—
153	Q₈×C₁₃	—	4	1	no	—
154	D₄×C₁₃	—	2	2	no	—
155	D₄×C₁₃	—	4	4	no	—
156	D₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	—	4	4	no	—
159	C₂×C₄×C₁₃	—	4	1	no	—
160	C₂×C₄×C₁₃	—	4	1	no	—
161	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#135	1	2	yes	—
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×Dic₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	16	2	no	—
167	C₂×Dic₁₃	—	16	1	no	—
168	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
169	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
170	D₅₂	—	2	4	no	—
171	D₅₂	—	4	2	no	—
172	D₅₂	—	4	4	no	—
173	C₄×D₁₃	—	16	2	no	—
174	C₄×D₁₃	—	16	2	no	—
175	C₈×C₁₃	C₄×D₄	1	2	yes	—
176	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
177	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
178	#1055	—	2	2	no	—
179	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
180	C₂²×D₈	—	13	1	no	—
181	#520	—	26	1	no	—
182	#518	—	26	1	no	—
183	#509	—	13	8	no	—
184	C₂×SD₃₂	—	13	2	no	—
185	C₂×D₁₆	—	13	2	no	—
186	C₂×M₃₂	—	13	1	no	—
187	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
188	#442	—	26	2	no	—
189	#440	—	26	2	no	—
190	C₄×Q₁₆	—	13	2	no	—
191	C₄×D₈	—	13	2	no	—
192	C₄×D₈	—	13	2	no	—
193	C₄×D₈	—	13	2	no	—
194	#427	C₂×D₁₃	1	1	yes	characteristic
195	C₂×#107	—	13	1	no	—
196	#418	—	26	1	no	—
197	#417	—	26	1	no	—
198	#405	—	13	1	no	—
199	#367	—	13	2	no	—
200	#366	—	13	2	no	—
201	#358	C₂×D₁₃	1	1	yes	characteristic, complemented
202	#358	C₂×D₁₃	1	1	yes	characteristic, complemented
203	#357	—	13	2	no	—
204	#357	—	13	2	no	—
205	#357	C₂×D₁₃	1	1	yes	characteristic, complemented
206	#357	C₂×D₁₃	1	1	yes	characteristic, complemented
207	#346	—	13	2	no	—
208	C₄×C₁₆	—	13	2	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	—	2	1	no	—
211	C₂²×C₁₃	—	4	2	no	—
212	C₂²×C₁₃	—	8	1	no	—
213	C₂²×C₁₃	—	8	1	no	—
214	C₂²×C₁₃	C₄×D₈	1	2	yes	—
215	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
216	C₂×D₁₃	—	4	4	no	—
217	C₂×D₁₃	—	8	2	no	—
218	C₂×D₁₃	—	8	2	no	—
219	C₂×D₁₃	—	16	4	no	—
220	C₄×C₁₃	—	4	2	no	—
221	C₄×C₁₃	—	8	1	no	—
222	C₄×C₁₃	—	8	1	no	—
223	C₄×C₁₃	C₄×D₈	1	2	yes	—
224	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
226	Dic₁₃	—	2	2	no	—
227	Dic₁₃	—	2	2	no	—
228	Dic₁₃	—	8	2	no	—
229	C₂×#41	—	2	1	no	—
230	C₂²×D₄	—	26	1	no	—
231	#135	—	2	2	no	—
232	#135	C₄×D₁₃	1	4	yes	complemented
233	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×SD₁₆	—	2	1	no	—
235	C₂×D₈	—	13	4	no	—
236	C₂×D₈	—	13	2	no	—
237	C₂×D₈	—	26	1	no	—
238	C₂×D₈	—	26	2	no	—
239	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×M₁₆	—	13	1	no	—
241	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄×Q₈	—	26	2	no	—
243	C₄×D₄	—	26	2	no	—
244	C₄×D₄	—	26	2	no	—
245	C₄×D₄	—	26	2	no	—
246	C₄×D₄	—	52	1	no	—
247	C₄×D₄	—	52	1	no	—
248	C₄×D₄	—	52	2	no	—
249	#117	—	2	1	no	—
250	#117	—	13	1	no	—
251	#117	—	52	1	no	—
252	C₂×C₄⋊C₄	—	26	1	no	—
253	C₂×C₄⋊C₄	—	52	1	no	—
254	C₂×C₂²⋊C₄	—	52	2	no	—
255	SD₃₂	—	13	8	no	—
256	D₁₆	—	13	8	no	—
257	M₃₂	—	13	4	no	—
258	C₂×C₁₆	—	2	2	no	—
259	C₂×C₁₆	—	13	2	no	—
260	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
262	#107	—	13	2	no	—
263	#107	—	26	1	no	—
264	#107	C₂²×D₁₃	1	1	yes	characteristic
265	#107	C₂²×D₁₃	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#103	—	26	2	no	—
268	#102	—	26	2	no	—
269	#102	—	26	2	no	—
270	#102	—	26	2	no	—
271	#97	—	13	2	no	—
272	C₄×C₈	—	13	2	no	—
273	C₂×C₁₃	—	2	2	no	—
274	C₂×C₁₃	—	8	2	no	—
275	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
276	C₂×C₁₃	#3415	1	1	yes	characteristic
277	C₂×C₁₃	#3416	1	1	yes	characteristic
278	D₁₃	—	8	4	no	—
279	C₂⁴	—	52	1	no	—
280	#41	—	2	4	no	—
281	#41	—	4	2	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	26	4	no	—
286	C₂×D₄	—	26	2	no	—
287	C₂×D₄	—	52	1	no	—
288	C₂×D₄	—	52	2	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	52	2	no	—
291	C₂²×C₄	—	52	2	no	—
292	C₂²×C₄	—	52	2	no	—
293	C₂²×C₄	—	52	1	no	—
294	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
295	Q₁₆	—	2	1	no	—
296	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
297	SD₁₆	—	2	4	no	—
298	D₈	—	2	1	no	—
299	D₈	—	13	4	no	—
300	D₈	—	26	4	no	—
301	D₈	—	26	2	no	—
302	D₈	C₂×C₄×D₁₃	1	2	yes	—
303	M₁₆	—	26	2	no	—
304	C₂×C₈	—	13	2	no	—
305	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
306	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
307	C₂×C₈	#6129	1	2	yes	—
308	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	13	1	no	—
312	C₄⋊C₄	—	13	1	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	—	52	2	no	—
315	C₄⋊C₄	—	52	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	13	2	no	—
318	C₂²⋊C₄	—	52	2	no	—
319	C₂²⋊C₄	—	52	2	no	—
320	C₂²⋊C₄	—	52	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	—	13	2	no	—
323	C₄²	—	52	2	no	—
324	C₁₆	—	2	1	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	13	4	no	—
327	C₁₃	#33454	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	4	1	no	—
329	C₂³	—	52	2	no	—
330	C₂³	—	52	4	no	—
331	C₂³	—	208	1	no	—
332	C₂³	—	208	1	no	—
333	C₂³	D₈×D₁₃	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	D₄	—	2	2	no	—
337	D₄	—	4	4	no	—
338	D₄	—	4	1	no	—
339	D₄	—	26	4	no	—
340	D₄	—	52	4	no	—
341	D₄	—	52	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	13	2	no	—
347	C₂×C₄	—	13	2	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	208	2	no	—
350	C₂×C₄	—	208	2	no	—
351	C₂×C₄	—	208	2	no	—
352	C₂×C₄	—	208	1	no	—
353	C₂×C₄	#88135	1	2	yes	—
354	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
355	C₂×C₄	#88170	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
357	C₈	—	26	2	no	—
358	C₈	#88135	1	2	yes	—
359	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
360	C₈	C₂×#6129	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	52	4	no	—
367	C₂²	—	104	2	no	—
368	C₂²	—	104	2	no	—
369	C₂²	—	208	4	no	—
370	C₂²	#11718417	1	2	yes	—
371	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
372	C₄	—	4	2	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	26	2	no	—
376	C₄	—	26	2	no	—
377	C₄	—	104	2	no	—
378	C₄	#11718417	1	2	yes	—
379	C₄	C₂×#88135	1	1	yes	characteristic
380	C₄	C₂×#88170	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	8	2	no	—
383	C₂	—	104	4	no	—
384	C₂	C₂×#11718417	1	1	yes	characteristic
385	C₂	#13127974	1	1	yes	characteristic
386	C₂	#13127979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2138	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718854	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2138	D₁₃
hypocenter	C₁₃	#33454
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
2. #13123712	C₂	semidirect
3. #13123687	C₂	semidirect
4. #13116709	C₂	semidirect
5. #13116703	C₂	semidirect
6. #13116685	C₂	semidirect
7. #13116680	C₂	semidirect
8. #13116625	C₂	semidirect
9. C₂×#11718101	C₂	semidirect
10. #13116432	C₂	semidirect
11. C₁₃×#2138	C₂	semidirect
19. #11718854	C₄	semidirect
20. Q₁₆×Dic₁₃	C₂²	semidirect
23. D₈×Dic₁₃	C₂²	semidirect
32. C₁₃×#358	C₂²	semidirect
33. C₁₃×#358	C₂²	semidirect
34. C₁₃×#357	C₂²	semidirect
35. C₁₃×#357	C₂²	semidirect
38. #11718094	C₂²	semidirect
39. C₁₆×Dic₁₃	C₂²	semidirect
40. #11718082	C₂²	semidirect
41. #11718081	C₂²	semidirect
45. C₁₃×#135	C₂×C₄	semidirect
61. C₂×D₁₀₄	C₂×C₄	semidirect
76. #88068	C₂×C₄	semidirect
77. #88066	C₂×C₄	semidirect
82. #88052	C₂×C₄	semidirect
143. #2138	D₁₃	semidirect
201. #358	C₂×D₁₃	semidirect
202. #358	C₂×D₁₃	semidirect
205. #357	C₂×D₁₃	semidirect
206. #357	C₂×D₁₃	semidirect
232. #135	C₄×D₁₃	semidirect
327. C₁₃	#33454	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
#13123712	2
#13123687	2
#13116709	2
#13116703	2
#13116685	2
#13116680	2
#13116625	2
C₂×#11718101	2
#13116432	2
C₁₃×#2138	2
#33454	13