Subgroups — #16019690

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127982	C₂	1	2	yes	maximal, complemented
2	#13126641	C₂	1	1	yes	maximal, characteristic, complemented
3	#13126635	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117091	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116965	C₂	1	2	yes	maximal, complemented
6	#13116918	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116810	C₂	1	2	yes	maximal, complemented
8	#13116797	C₂	1	2	yes	maximal, complemented
9	#13116715	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116672	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2213	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719476	C₂²	1	2	yes	complemented
13	#11719462	—	2	1	no	—
14	#11719462	C₂²	1	2	yes	—
15	#11719458	—	2	1	no	—
16	C₁₃×#509	C₂²	1	2	yes	complemented
17	C₁₃×#493	C₂²	1	1	yes	characteristic
18	C₁₃×#443	C₂²	1	1	yes	characteristic, complemented
19	#11718855	C₂²	1	1	yes	characteristic
20	#11718821	—	2	1	no	—
21	#11718821	C₂²	1	2	yes	—
22	C₂×#88067	C₂²	1	2	yes	complemented
23	C₂×#88067	C₂²	1	2	yes	complemented
24	#11718804	—	2	1	no	—
25	#11718738	—	2	2	no	—
26	#11718576	—	2	2	no	—
27	D₁₃×SD₃₂	—	2	2	no	—
28	#11718573	—	2	2	no	—
29	#11718572	—	2	2	no	—
30	#11718570	C₂²	1	2	yes	—
31	#11718568	C₂²	1	2	yes	complemented
32	#11718557	C₂²	1	2	yes	complemented
33	#11718487	—	2	2	no	—
34	#11718361	C₂²	1	1	yes	characteristic
35	#11718361	C₂²	1	1	yes	characteristic
36	#11718360	C₂²	1	1	yes	characteristic, complemented
37	C₁₃×#364	C₂²	1	1	yes	characteristic, complemented
38	C₁₃×#361	C₂²	1	2	yes	complemented
39	#11718159	C₂²	1	1	yes	characteristic, complemented
40	#11718109	C₂²	1	2	yes	complemented
41	#11718107	C₂²	1	2	yes	complemented
42	#11718089	C₂²	1	1	yes	characteristic
43	#11718085	C₂²	1	1	yes	characteristic, complemented
44	C₄×#6090	C₂²	1	1	yes	characteristic, complemented
45	#88257	—	2	2	no	—
46	#88251	—	2	2	no	—
47	C₁₃×#135	C₂³	1	1	yes	characteristic
48	C₂×C₁₃×D₈	—	2	1	no	—
49	C₂×C₁₃×D₈	C₂³	1	2	yes	—
50	C₁₃×#131	—	2	1	no	—
51	C₁₃×#127	—	2	1	no	—
52	#88205	—	2	2	no	—
53	#88186	—	2	2	no	—
54	#88186	—	2	2	no	—
55	Q₁₆×D₁₃	—	2	2	no	—
56	#88169	—	4	2	no	—
57	#88166	—	2	2	no	—
58	#88166	—	2	2	no	—
59	#88166	—	4	1	no	—
60	#88165	—	4	2	no	—
61	#88165	—	4	2	no	—
62	D₈×D₁₃	—	2	2	no	—
63	D₈×D₁₃	—	4	1	no	—
64	#88163	—	2	2	no	—
65	#88163	—	4	2	no	—
66	C₂×Dic₅₂	—	2	1	no	—
67	C₂×Dic₅₂	—	4	1	no	—
68	C₂×Dic₅₂	C₂³	1	2	yes	—
69	#88158	—	4	1	no	—
70	#88158	D₄	1	1	yes	characteristic
71	#88158	C₂³	1	1	yes	characteristic
72	#88155	—	2	1	no	—
73	#88155	C₂³	1	2	yes	—
74	#88123	—	2	1	no	—
75	C₁₃×SD₃₂	—	2	2	no	—
76	C₁₃×D₁₆	—	2	2	no	—
77	C₁₃×M₃₂	C₂³	1	2	yes	—
78	C₁₃×#108	C₂³	1	1	yes	characteristic
79	C₁₃×#104	—	2	1	no	—
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	#88073	—	2	2	no	—
82	#88069	D₄	1	1	yes	characteristic, complemented
83	#88068	D₄	1	1	yes	characteristic, complemented
84	#88067	—	2	2	no	—
85	#88067	—	2	2	no	—
86	#88067	—	4	2	no	—
87	#88067	D₄	1	1	yes	characteristic, complemented
88	#88066	D₄	1	1	yes	characteristic, complemented
89	#88064	—	2	2	no	—
90	#88059	D₄	1	1	yes	characteristic
91	#88059	C₂³	1	1	yes	characteristic
92	C₂×#6090	C₂³	1	1	yes	characteristic
93	C₂×#6090	C₂³	1	1	yes	characteristic
94	#88045	—	2	1	no	—
95	#88045	D₄	1	2	yes	—
96	Dic₁₀₄	—	2	2	no	—
97	#88040	—	2	2	no	—
98	#88038	—	2	2	no	—
99	C₁₆×D₁₃	—	2	2	no	—
100	#33856	—	13	1	no	maximal, Sylow
101	C₁₃×#41	—	2	1	no	—
102	C₂×D₄×C₁₃	—	2	2	no	—
103	C₂×D₄×C₁₃	—	8	1	no	—
104	C₂×C₂²⋊D₁₃	—	8	2	no	—
105	Q₈×D₁₃	—	4	2	no	—
106	#6129	—	4	2	no	—
107	#6129	—	4	2	no	—
108	#6129	—	4	2	no	—
109	#6129	—	8	2	no	—
110	D₄×D₁₃	—	4	2	no	—
111	D₄×D₁₃	—	4	2	no	—
112	#6127	—	2	1	no	—
113	#6127	—	4	2	no	—
114	#6127	C₂×D₄	1	2	yes	—
115	C₂×Dic₂₆	—	2	2	no	—
116	C₂×Dic₂₆	—	4	2	no	—
117	C₁₃×Q₁₆	C₂×D₄	1	1	yes	characteristic
118	C₁₃×SD₁₆	—	2	1	no	—
119	C₁₃×D₈	—	2	2	no	—
120	C₁₃×D₈	—	2	2	no	—
121	C₁₃×D₈	—	4	1	no	—
122	C₁₃×D₈	—	4	1	no	—
123	C₁₃×D₈	C₂×D₄	1	1	yes	characteristic
124	C₁₃×M₁₆	—	2	1	no	—
125	C₁₃×M₁₆	—	4	1	no	—
126	C₂×C₈×C₁₃	—	4	1	no	—
127	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
129	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
130	#6107	—	4	2	no	—
131	#6105	—	4	2	no	—
132	#6105	—	4	2	no	—
133	#6105	—	4	2	no	—
134	#6104	—	4	2	no	—
135	#6104	—	4	2	no	—
136	C₄⋊Dic₁₃	—	8	1	no	—
137	#6099	—	2	1	no	—
138	#6099	C₂×D₄	1	2	yes	—
139	Dic₅₂	—	2	2	no	—
140	Dic₅₂	—	2	2	no	—
141	Dic₅₂	—	4	2	no	—
142	Dic₅₂	—	4	1	no	—
143	Dic₅₂	—	4	1	no	—
144	Dic₅₂	D₈	1	1	yes	characteristic
145	Dic₅₂	C₂×D₄	1	1	yes	characteristic
146	D₁₀₄	D₈	1	1	yes	characteristic
147	D₁₀₄	C₂×D₄	1	1	yes	characteristic
148	#6095	—	2	1	no	—
149	#6095	—	4	2	no	—
150	#6095	D₈	1	2	yes	—
151	#6094	—	2	2	no	—
152	#6094	—	4	1	no	—
153	C₈×D₁₃	—	2	2	no	—
154	C₈×D₁₃	—	4	1	no	—
155	C₁₃×C₁₆	—	2	2	no	—
156	#6090	—	2	1	no	—
157	#6090	C₂×D₄	1	1	yes	characteristic
158	#6090	C₂×D₄	1	1	yes	characteristic
159	#3416	—	13	2	no	—
160	#3293	—	13	1	no	—
161	#3292	—	13	1	no	—
162	#2248	—	13	1	no	—
163	#2223	—	13	2	no	—
164	#2214	—	13	1	no	—
165	#2213	D₁₃	1	1	yes	characteristic, complemented
166	#2194	—	13	2	no	—
167	#2193	—	13	2	no	—
168	#2178	—	13	1	no	—
169	#2170	—	13	1	no	—
170	C₂³×C₁₃	—	4	2	no	—
171	C₂²×D₁₃	—	8	2	no	—
172	Q₈×C₁₃	—	2	1	no	—
173	D₄×C₁₃	—	2	1	no	—
174	D₄×C₁₃	—	4	1	no	—
175	D₄×C₁₃	—	4	2	no	—
176	D₄×C₁₃	—	4	2	no	—
177	D₄×C₁₃	—	16	2	no	—
178	C₂×C₄×C₁₃	—	2	1	no	—
179	C₂×C₄×C₁₃	—	4	1	no	—
180	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
181	C₂²⋊D₁₃	—	2	2	no	—
182	C₂²⋊D₁₃	—	4	1	no	—
183	C₂²⋊D₁₃	—	8	2	no	—
184	C₂²⋊D₁₃	—	8	2	no	—
185	C₂²⋊D₁₃	—	8	2	no	—
186	C₂×Dic₁₃	—	4	2	no	—
187	C₂×Dic₁₃	—	8	2	no	—
188	C₂×Dic₁₃	—	8	2	no	—
189	D₅₂	—	2	1	no	—
190	D₅₂	C₂×D₈	1	2	yes	—
191	C₄×D₁₃	—	2	2	no	—
192	C₄×D₁₃	—	4	1	no	—
193	C₄×D₁₃	—	8	2	no	—
194	Dic₂₆	—	2	1	no	—
195	Dic₂₆	—	4	2	no	—
196	Dic₂₆	—	4	2	no	—
197	Dic₂₆	—	8	2	no	—
198	Dic₂₆	C₂×D₈	1	2	yes	—
199	C₈×C₁₃	—	4	1	no	—
200	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
201	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
202	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
203	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
204	#1055	—	2	2	no	—
205	#1055	—	4	1	no	—
206	#578	—	13	2	no	—
207	#578	—	26	1	no	—
208	#576	—	13	2	no	—
209	#576	—	26	1	no	—
210	#510	—	13	2	no	—
211	#510	—	26	2	no	—
212	#509	—	26	2	no	—
213	#509	C₂×D₁₃	1	2	yes	complemented
214	#508	—	13	1	no	—
215	#508	—	26	2	no	—
216	C₂×SD₃₂	—	13	2	no	—
217	C₂×SD₃₂	—	13	2	no	—
218	C₂×SD₃₂	—	26	2	no	—
219	#504	—	13	2	no	—
220	#494	—	13	1	no	—
221	#493	C₂×D₁₃	1	1	yes	characteristic
222	#473	—	13	2	no	—
223	#473	—	26	1	no	—
224	#472	—	13	2	no	—
225	#472	—	26	1	no	—
226	#456	—	26	2	no	—
227	#453	—	26	2	no	—
228	#443	—	13	1	no	—
229	#443	—	13	1	no	—
230	#443	C₂×D₁₃	1	1	yes	characteristic, complemented
231	#368	—	13	1	no	—
232	#364	C₂×D₁₃	1	1	yes	characteristic, complemented
233	#362	—	13	2	no	—
234	#361	C₂×D₁₃	1	2	yes	complemented
235	#359	—	13	1	no	—
236	#359	—	13	1	no	—
237	#350	—	13	2	no	—
238	C₄×C₁₆	—	13	1	no	—
239	C₂²×C₁₃	—	4	1	no	—
240	C₂²×C₁₃	—	8	2	no	—
241	C₂²×C₁₃	—	16	2	no	—
242	C₂²×C₁₃	#576	1	1	yes	characteristic
243	C₂×D₁₃	—	2	2	no	—
244	C₂×D₁₃	—	4	1	no	—
245	C₂×D₁₃	—	32	2	no	—
246	C₄×C₁₃	—	2	2	no	—
247	C₄×C₁₃	—	4	1	no	—
248	C₄×C₁₃	D₄²	1	1	yes	characteristic
249	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
250	Dic₁₃	—	2	2	no	—
251	Dic₁₃	—	4	1	no	—
252	Dic₁₃	—	8	2	no	—
253	2₋¹⁺⁴	—	26	2	no	—
254	H(5,𝔽₂)	—	26	2	no	—
255	#137	—	26	2	no	—
256	#137	—	26	2	no	—
257	#137	—	52	2	no	—
258	#137	—	52	2	no	—
259	AGL(1,ℤ/8)	—	26	2	no	—
260	AGL(1,ℤ/8)	—	26	2	no	—
261	AGL(1,ℤ/8)	—	52	2	no	—
262	AGL(1,ℤ/8)	—	52	2	no	—
263	#135	—	13	1	no	—
264	#135	—	13	1	no	—
265	#135	—	26	2	no	—
266	#135	—	26	2	no	—
267	#135	—	52	1	no	—
268	#135	—	52	1	no	—
269	#135	C₂²×D₁₃	1	1	yes	characteristic
270	C₂×Q₁₆	—	13	2	no	—
271	C₂×Q₁₆	—	26	2	no	—
272	C₂×Q₁₆	—	26	1	no	—
273	C₂×Q₁₆	—	52	1	no	—
274	C₂×D₈	—	2	1	no	—
275	C₂×D₈	—	26	2	no	—
276	C₂×D₈	—	52	1	no	—
277	C₂×D₈	C₂²×D₁₃	1	2	yes	—
278	#131	—	2	1	no	—
279	#131	—	13	2	no	—
280	#131	—	26	1	no	—
281	C₄⋊Q₈	—	26	1	no	—
282	#127	—	2	1	no	—
283	Q₃₂	—	13	1	no	—
284	Q₃₂	—	26	2	no	—
285	SD₃₂	—	2	2	no	—
286	SD₃₂	—	13	1	no	—
287	SD₃₂	—	13	1	no	—
288	SD₃₂	—	26	2	no	—
289	SD₃₂	—	26	2	no	—
290	SD₃₂	—	26	2	no	—
291	SD₃₂	—	52	2	no	—
292	D₁₆	—	2	2	no	—
293	D₁₆	—	13	1	no	—
294	M₃₂	—	26	2	no	—
295	M₃₂	C₂²×D₁₃	1	2	yes	—
296	C₂×C₁₆	—	13	1	no	—
297	C₂×C₁₆	—	13	1	no	—
298	C₂×C₁₆	—	26	2	no	—
299	#108	—	13	1	no	—
300	#108	—	13	1	no	—
301	#108	C₂²×D₁₃	1	1	yes	characteristic
302	#104	—	2	1	no	—
303	#104	—	13	2	no	—
304	#104	—	26	1	no	—
305	#101	—	26	2	no	—
306	#100	—	26	2	no	—
307	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
308	C₂×C₁₃	—	2	1	no	—
309	C₂×C₁₃	—	8	1	no	—
310	C₂×C₁₃	—	16	2	no	—
311	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
312	D₁₃	—	4	2	no	—
313	D₁₃	—	8	1	no	—
314	#41	—	2	1	no	—
315	#41	—	13	2	no	—
316	#41	—	26	1	no	—
317	#41	—	52	2	no	—
318	#41	—	52	2	no	—
319	#41	—	52	2	no	—
320	#41	—	52	2	no	—
321	#41	—	104	2	no	—
322	C₂×Q₈	—	26	2	no	—
323	C₂×Q₈	—	52	2	no	—
324	C₂×Q₈	—	52	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	8	1	no	—
327	C₂×D₄	—	52	2	no	—
328	C₂×D₄	—	52	2	no	—
329	C₂×D₄	—	104	2	no	—
330	Q₁₆	—	13	1	no	—
331	Q₁₆	—	13	1	no	—
332	Q₁₆	—	26	2	no	—
333	Q₁₆	—	26	2	no	—
334	Q₁₆	—	52	2	no	—
335	Q₁₆	—	52	2	no	—
336	Q₁₆	—	52	1	no	—
337	Q₁₆	—	52	1	no	—
338	Q₁₆	D₄×D₁₃	1	1	yes	characteristic
339	SD₁₆	—	2	1	no	—
340	SD₁₆	—	13	2	no	—
341	SD₁₆	—	26	1	no	—
342	SD₁₆	—	52	2	no	—
343	SD₁₆	—	52	2	no	—
344	SD₁₆	—	52	2	no	—
345	SD₁₆	—	52	2	no	—
346	D₈	—	2	2	no	—
347	D₈	—	2	2	no	—
348	D₈	—	4	1	no	—
349	D₈	—	4	1	no	—
350	D₈	—	13	1	no	—
351	D₈	—	13	1	no	—
352	D₈	—	52	2	no	—
353	D₈	—	52	2	no	—
354	D₈	D₄×D₁₃	1	1	yes	characteristic
355	M₁₆	—	2	1	no	—
356	M₁₆	—	4	1	no	—
357	M₁₆	—	13	2	no	—
358	M₁₆	—	26	2	no	—
359	M₁₆	—	26	1	no	—
360	M₁₆	—	52	1	no	—
361	C₂×C₈	—	4	1	no	—
362	C₂×C₈	—	26	2	no	—
363	C₂×C₈	—	52	1	no	—
364	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
365	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
366	C₄⋊C₄	—	104	1	no	—
367	C₄²	D₄×D₁₃	1	1	yes	characteristic
368	C₁₆	—	2	2	no	—
369	C₁₆	—	13	1	no	—
370	C₁₆	—	13	1	no	—
371	C₁₆	—	26	1	no	—
372	C₁₃	#33856	1	1	yes	characteristic, complemented, Sylow
373	C₂³	—	4	2	no	—
374	C₂³	—	104	2	no	—
375	Q₈	—	2	1	no	—
376	Q₈	—	13	2	no	—
377	Q₈	—	26	1	no	—
378	Q₈	—	52	2	no	—
379	Q₈	—	52	2	no	—
380	Q₈	—	104	2	no	—
381	D₄	—	2	1	no	—
382	D₄	—	4	1	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	2	no	—
385	D₄	—	13	2	no	—
386	D₄	—	16	2	no	—
387	D₄	—	26	2	no	—
388	D₄	—	26	1	no	—
389	D₄	—	52	1	no	—
390	D₄	—	104	2	no	—
391	D₄	—	104	2	no	—
392	D₄	—	104	2	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	26	2	no	—
396	C₂×C₄	—	52	1	no	—
397	C₂×C₄	—	52	2	no	—
398	C₂×C₄	—	104	2	no	—
399	C₂×C₄	—	104	2	no	—
400	C₂×C₄	—	104	2	no	—
401	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
402	C₈	—	4	1	no	—
403	C₈	—	26	2	no	—
404	C₈	—	52	1	no	—
405	C₈	D₈×D₁₃	1	1	yes	characteristic
406	C₈	D₈×D₁₃	1	1	yes	characteristic
407	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
408	C₈	#88256	1	1	yes	characteristic
409	C₂²	—	4	1	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	16	2	no	—
412	C₂²	—	26	2	no	—
413	C₂²	—	52	1	no	—
414	C₂²	—	416	2	no	—
415	C₂²	#11719479	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	4	1	no	—
418	C₄	—	26	2	no	—
419	C₄	—	52	1	no	—
420	C₄	—	104	2	no	—
421	C₄	#11719314	1	1	yes	characteristic
422	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
423	C₂	—	2	1	no	—
424	C₂	—	8	1	no	—
425	C₂	—	16	2	no	—
426	C₂	—	52	2	no	—
427	C₂	—	104	1	no	—
428	C₂	#13126508	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13126508
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2213	C₂
socle	C₂×C₁₃	D₄×D₈
hypercenter	#2213	D₁₃
hypocenter	C₁₃	#33856
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2213

Complements

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

N	H ≅ G/N	type
1. #13127982	C₂	semidirect
2. #13126641	C₂	semidirect
3. #13126635	C₂	semidirect
4. #13117091	C₂	semidirect
5. #13116965	C₂	semidirect
6. #13116918	C₂	semidirect
7. #13116810	C₂	semidirect
8. #13116797	C₂	semidirect
9. #13116715	C₂	semidirect
10. #13116672	C₂	semidirect
11. C₁₃×#2213	C₂	semidirect
12. #11719476	C₂²	semidirect
16. C₁₃×#509	C₂²	semidirect
18. C₁₃×#443	C₂²	semidirect
22. C₂×#88067	C₂²	semidirect
23. C₂×#88067	C₂²	semidirect
31. #11718568	C₂²	semidirect
32. #11718557	C₂²	semidirect
36. #11718360	C₂²	semidirect
37. C₁₃×#364	C₂²	semidirect
38. C₁₃×#361	C₂²	semidirect
39. #11718159	C₂²	semidirect
40. #11718109	C₂²	semidirect
41. #11718107	C₂²	semidirect
43. #11718085	C₂²	semidirect
44. C₄×#6090	C₂²	semidirect
82. #88069	D₄	semidirect
83. #88068	D₄	semidirect
87. #88067	D₄	semidirect
88. #88066	D₄	semidirect
165. #2213	D₁₃	semidirect
213. #509	C₂×D₁₃	semidirect
230. #443	C₂×D₁₃	semidirect
232. #364	C₂×D₁₃	semidirect
234. #361	C₂×D₁₃	semidirect
372. C₁₃	#33856	semidirect

Maximal subgroups

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

subgroup	index
#13127982	2
#13126641	2
#13126635	2
#13117091	2
#13116965	2
#13116918	2
#13116810	2
#13116797	2
#13116715	2
#13116672	2
C₁₃×#2213	2
#33856	13