Subgroups — #16019725

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127989	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127977	C₂	1	1	yes	maximal, characteristic, complemented
3	#13126623	C₂	1	2	yes	maximal, complemented
4	#13117097	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116961	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116956	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116906	C₂	1	2	yes	maximal, complemented
8	#13116809	C₂	1	2	yes	maximal, complemented
9	#13116802	C₂	1	2	yes	maximal, complemented
10	#13116718	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719475	—	2	2	no	—
13	#11719458	C₂²	1	2	yes	complemented
14	#11719457	C₂²	1	2	yes	—
15	C₁₃×#504	C₂²	1	1	yes	characteristic
16	C₁₃×#504	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#443	C₂²	1	1	yes	characteristic
18	#11718804	C₂²	1	2	yes	—
19	#11718803	C₂²	1	2	yes	complemented
20	#11718570	—	2	2	no	—
21	#11718570	C₂²	1	2	yes	complemented
22	#11718569	—	2	2	no	—
23	#11718569	C₂²	1	2	yes	—
24	C₂×Dic₁₀₄	—	2	1	no	—
25	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
26	#11718565	—	2	1	no	—
27	#11718565	—	2	1	no	—
28	#11718565	C₂²	1	1	yes	characteristic
29	C₂×#88040	C₂²	1	2	yes	complemented
30	C₂×D₂₀₈	—	2	1	no	—
31	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
32	#11718559	—	2	2	no	—
33	#11718488	—	2	2	no	—
34	#11718486	—	2	2	no	—
35	#11718361	C₂²	1	2	yes	complemented
36	#11718357	C₂²	1	2	yes	complemented
37	C₁₃×#364	C₂²	1	1	yes	characteristic, complemented
38	C₁₃×#350	C₂²	1	1	yes	characteristic
39	C₁₃×#350	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
41	#11718109	C₂²	1	2	yes	complemented
42	#11718108	C₂²	1	2	yes	complemented
43	#11718102	C₂²	1	2	yes	complemented
44	#11718098	C₂²	1	1	yes	characteristic, complemented
45	#88257	—	2	2	no	—
46	#88256	—	2	2	no	—
47	C₁₃×#135	D₄	1	1	yes	characteristic
48	C₁₃×#131	C₂³	1	1	yes	characteristic
49	C₁₃×#131	C₂³	1	1	yes	characteristic
50	#88205	—	2	2	no	—
51	#88203	—	2	2	no	—
52	#88172	—	4	2	no	—
53	#88170	—	4	2	no	—
54	SD₁₆×D₁₃	—	4	2	no	—
55	#88165	—	4	2	no	—
56	#88163	—	2	2	no	—
57	#88163	—	4	2	no	—
58	#88162	—	2	2	no	—
59	#88162	—	4	2	no	—
60	C₂×Dic₅₂	—	2	2	no	—
61	C₂×Dic₅₂	C₂³	1	2	yes	—
62	#88158	—	2	2	no	—
63	#88158	—	2	2	no	—
64	#88158	C₂³	1	2	yes	—
65	C₂×D₁₀₄	—	2	2	no	—
66	C₂×D₁₀₄	C₂³	1	2	yes	—
67	C₂×#6095	—	2	2	no	—
68	#88155	—	2	2	no	—
69	#88129	—	2	2	no	—
70	C₁₃×M₃₂	—	2	1	no	—
71	C₁₃×M₃₂	—	2	1	no	—
72	C₁₃×M₃₂	C₂³	1	1	yes	characteristic
73	C₁₃×M₃₂	C₂³	1	1	yes	characteristic
74	C₂×C₁₃×C₁₆	—	2	1	no	—
75	C₂×C₁₃×C₁₆	—	2	1	no	—
76	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
78	C₁₃×#108	D₄	1	1	yes	characteristic
79	C₁₃×#104	D₄	1	1	yes	characteristic
80	C₁₃×#104	D₄	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	#88064	—	2	2	no	—
83	#88063	—	2	2	no	—
84	#88059	C₂³	1	2	yes	—
85	#88045	—	2	2	no	—
86	Dic₁₀₄	—	2	2	no	—
87	Dic₁₀₄	—	2	1	no	—
88	Dic₁₀₄	—	2	1	no	—
89	Dic₁₀₄	D₄	1	1	yes	characteristic
90	#88040	—	2	2	no	—
91	#88040	—	2	2	no	—
92	#88040	—	4	2	no	—
93	#88040	D₄	1	2	yes	—
94	D₂₀₈	—	2	2	no	—
95	D₂₀₈	—	2	1	no	—
96	D₂₀₈	—	2	1	no	—
97	D₂₀₈	D₄	1	1	yes	characteristic
98	#33859	—	13	1	no	maximal, Sylow
99	C₁₃×#41	C₂×D₄	1	1	yes	characteristic
100	C₁₃×#41	C₂×D₄	1	1	yes	characteristic
101	#6131	—	4	2	no	—
102	Q₈×D₁₃	—	4	2	no	—
103	#6129	—	4	2	no	—
104	#6129	—	8	2	no	—
105	D₄×D₁₃	—	4	2	no	—
106	D₄×D₁₃	—	8	2	no	—
107	#6127	—	2	2	no	—
108	#6127	—	4	2	no	—
109	#6127	—	4	2	no	—
110	C₂×D₅₂	—	2	2	no	—
111	C₂×D₅₂	—	4	2	no	—
112	C₂×Dic₂₆	—	2	2	no	—
113	C₂×Dic₂₆	—	4	2	no	—
114	C₁₃×Q₁₆	D₈	1	1	yes	characteristic
115	C₁₃×SD₁₆	D₈	1	1	yes	characteristic
116	C₁₃×SD₁₆	D₈	1	1	yes	characteristic
117	C₁₃×D₈	D₈	1	1	yes	characteristic
118	C₁₃×M₁₆	—	2	1	no	—
119	C₁₃×M₁₆	—	2	1	no	—
120	C₁₃×M₁₆	C₂×D₄	1	1	yes	characteristic
121	C₁₃×M₁₆	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₁₃	—	2	1	no	—
123	C₂×C₈×C₁₃	—	2	1	no	—
124	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
126	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
127	#6107	—	4	2	no	—
128	#6106	—	4	2	no	—
129	#6105	—	4	2	no	—
130	#6104	—	4	2	no	—
131	#6103	—	8	2	no	—
132	#6099	—	2	2	no	—
133	Dic₅₂	—	2	2	no	—
134	Dic₅₂	—	2	2	no	—
135	Dic₅₂	—	4	2	no	—
136	Dic₅₂	C₂×D₄	1	2	yes	—
137	D₁₀₄	—	2	2	no	—
138	D₁₀₄	—	2	2	no	—
139	D₁₀₄	—	4	2	no	—
140	D₁₀₄	C₂×D₄	1	2	yes	—
141	#6095	—	2	2	no	—
142	#6095	—	4	2	no	—
143	#6095	—	4	2	no	—
144	#6095	—	4	2	no	—
145	#6095	—	4	2	no	—
146	#6094	—	4	2	no	—
147	C₈×D₁₃	—	4	2	no	—
148	C₁₃×C₁₆	—	2	1	no	—
149	C₁₃×C₁₆	—	2	1	no	—
150	C₁₃×C₁₆	—	2	1	no	—
151	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
152	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
153	#3417	—	13	1	no	—
154	#3415	—	13	1	no	—
155	#3290	—	13	2	no	—
156	#2249	—	13	1	no	—
157	#2222	—	13	1	no	—
158	#2221	—	13	1	no	—
159	#2212	—	13	2	no	—
160	#2194	—	13	2	no	—
161	#2193	—	13	2	no	—
162	#2178	—	13	1	no	—
163	#2170	D₁₃	1	1	yes	characteristic, complemented
164	C₂²×D₁₃	—	4	2	no	—
165	C₂²×D₁₃	—	8	2	no	—
166	Q₈×C₁₃	C₂×D₈	1	1	yes	characteristic
167	Q₈×C₁₃	C₂×D₈	1	1	yes	characteristic
168	D₄×C₁₃	—	2	1	no	—
169	D₄×C₁₃	—	2	1	no	—
170	D₄×C₁₃	C₂×D₈	1	1	yes	characteristic
171	D₄×C₁₃	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃	—	2	1	no	—
173	C₂×C₄×C₁₃	—	2	1	no	—
174	C₂×C₄×C₁₃	—	2	1	no	—
175	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
176	C₂²⋊D₁₃	—	4	2	no	—
177	C₂²⋊D₁₃	—	8	2	no	—
178	C₂²⋊D₁₃	—	8	2	no	—
179	C₂×Dic₁₃	—	4	2	no	—
180	C₂×Dic₁₃	—	8	2	no	—
181	D₅₂	—	2	2	no	—
182	D₅₂	—	4	2	no	—
183	D₅₂	—	4	2	no	—
184	D₅₂	—	8	2	no	—
185	C₄×D₁₃	—	4	2	no	—
186	C₄×D₁₃	—	8	2	no	—
187	C₄×D₁₃	—	8	2	no	—
188	Dic₂₆	—	2	2	no	—
189	Dic₂₆	—	4	2	no	—
190	Dic₂₆	—	4	2	no	—
191	Dic₂₆	—	8	2	no	—
192	C₈×C₁₃	—	2	1	no	—
193	C₈×C₁₃	—	2	1	no	—
194	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
195	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
196	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
197	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
198	#1055	—	4	2	no	—
199	#578	—	13	2	no	—
200	#577	—	26	2	no	—
201	#576	—	13	2	no	—
202	#510	—	13	2	no	—
203	#510	—	26	2	no	—
204	#509	—	13	2	no	—
205	#509	—	26	2	no	—
206	#508	—	13	1	no	—
207	#508	—	26	1	no	—
208	#508	—	26	1	no	—
209	C₂×Q₃₂	—	13	1	no	—
210	C₂×Q₃₂	—	26	1	no	—
211	C₂×SD₃₂	—	13	2	no	—
212	C₂×D₁₆	—	13	1	no	—
213	C₂×D₁₆	—	26	1	no	—
214	#504	C₂×D₁₃	1	1	yes	characteristic
215	#504	C₂×D₁₃	1	1	yes	characteristic, complemented
216	#495	—	13	2	no	—
217	#473	—	13	2	no	—
218	#472	—	13	2	no	—
219	#471	—	26	2	no	—
220	#455	—	26	2	no	—
221	#454	—	26	2	no	—
222	#443	—	13	2	no	—
223	#443	C₂×D₁₃	1	1	yes	characteristic
224	#368	—	13	1	no	—
225	#364	C₂×D₁₃	1	1	yes	characteristic, complemented
226	#362	—	13	2	no	—
227	#361	—	13	2	no	—
228	#359	—	13	2	no	—
229	#350	C₂×D₁₃	1	1	yes	characteristic
230	#350	C₂×D₁₃	1	1	yes	characteristic, complemented
231	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
232	C₂²×C₁₃	—	2	1	no	—
233	C₂²×C₁₃	—	2	1	no	—
234	C₂²×C₁₃	#576	1	1	yes	characteristic
235	C₂×D₁₃	—	4	2	no	—
236	C₂×D₁₃	—	8	2	no	—
237	C₂×D₁₃	—	16	2	no	—
238	C₂×D₁₃	—	32	2	no	—
239	C₄×C₁₃	—	2	1	no	—
240	C₄×C₁₃	—	2	1	no	—
241	C₄×C₁₃	—	4	1	no	—
242	C₄×C₁₃	D₄²	1	1	yes	characteristic
243	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
244	Dic₁₃	—	4	2	no	—
245	Dic₁₃	—	8	2	no	—
246	2₋¹⁺⁴	—	26	2	no	—
247	H(5,𝔽₂)	—	26	2	no	—
248	#137	—	26	2	no	—
249	#137	—	26	2	no	—
250	#137	—	52	2	no	—
251	#137	—	52	2	no	—
252	AGL(1,ℤ/8)	—	26	2	no	—
253	AGL(1,ℤ/8)	—	26	2	no	—
254	AGL(1,ℤ/8)	—	52	2	no	—
255	AGL(1,ℤ/8)	—	52	2	no	—
256	#135	—	13	2	no	—
257	#135	—	26	2	no	—
258	#135	—	26	2	no	—
259	#135	—	52	2	no	—
260	#135	D₅₂	1	1	yes	characteristic
261	C₂×Q₁₆	—	13	2	no	—
262	C₂×Q₁₆	—	26	2	no	—
263	C₂×SD₁₆	—	26	2	no	—
264	C₂×SD₁₆	—	52	2	no	—
265	C₂×D₈	—	13	2	no	—
266	C₂×D₈	—	26	2	no	—
267	#131	—	26	2	no	—
268	#131	C₂²×D₁₃	1	1	yes	characteristic
269	#131	C₂²×D₁₃	1	1	yes	characteristic
270	#124	—	26	2	no	—
271	Q₃₂	—	13	1	no	—
272	Q₃₂	—	26	2	no	—
273	Q₃₂	—	26	1	no	—
274	Q₃₂	—	26	1	no	—
275	SD₃₂	—	13	2	no	—
276	SD₃₂	—	26	2	no	—
277	SD₃₂	—	26	2	no	—
278	SD₃₂	—	52	2	no	—
279	D₁₆	—	13	1	no	—
280	D₁₆	—	26	2	no	—
281	D₁₆	—	26	1	no	—
282	D₁₆	—	26	1	no	—
283	M₃₂	—	2	1	no	—
284	M₃₂	—	2	1	no	—
285	M₃₂	C₂²×D₁₃	1	1	yes	characteristic
286	M₃₂	C₂²×D₁₃	1	1	yes	characteristic
287	C₂×C₁₆	—	2	1	no	—
288	C₂×C₁₆	—	2	1	no	—
289	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
290	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
291	#108	—	13	2	no	—
292	#108	D₅₂	1	1	yes	characteristic
293	#104	—	26	2	no	—
294	#104	D₅₂	1	1	yes	characteristic
295	#104	D₅₂	1	1	yes	characteristic
296	#101	—	26	2	no	—
297	#100	—	26	2	no	—
298	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
299	C₂×C₁₃	—	2	1	no	—
300	C₂×C₁₃	—	4	1	no	—
301	C₂×C₁₃	—	4	1	no	—
302	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
303	D₁₃	—	8	2	no	—
304	D₁₃	—	16	2	no	—
305	#41	—	26	2	no	—
306	#41	—	52	2	no	—
307	#41	—	52	2	no	—
308	#41	—	52	2	no	—
309	#41	—	52	2	no	—
310	#41	—	104	2	no	—
311	#41	C₂×D₅₂	1	1	yes	characteristic
312	#41	C₂×D₅₂	1	1	yes	characteristic
313	C₂×Q₈	—	26	2	no	—
314	C₂×Q₈	—	52	2	no	—
315	C₂×Q₈	—	52	2	no	—
316	C₂×D₄	—	26	2	no	—
317	C₂×D₄	—	52	2	no	—
318	C₂×D₄	—	52	2	no	—
319	C₂×D₄	—	104	2	no	—
320	Q₁₆	—	13	2	no	—
321	Q₁₆	—	26	2	no	—
322	Q₁₆	—	26	2	no	—
323	Q₁₆	—	52	2	no	—
324	Q₁₆	—	52	2	no	—
325	Q₁₆	D₁₀₄	1	1	yes	characteristic
326	SD₁₆	—	26	2	no	—
327	SD₁₆	—	52	2	no	—
328	SD₁₆	—	52	2	no	—
329	SD₁₆	—	52	2	no	—
330	SD₁₆	—	52	2	no	—
331	SD₁₆	—	52	2	no	—
332	SD₁₆	—	52	2	no	—
333	SD₁₆	D₁₀₄	1	1	yes	characteristic
334	SD₁₆	D₁₀₄	1	1	yes	characteristic
335	D₈	—	13	2	no	—
336	D₈	—	26	2	no	—
337	D₈	—	26	2	no	—
338	D₈	—	52	2	no	—
339	D₈	—	52	2	no	—
340	D₈	D₁₀₄	1	1	yes	characteristic
341	M₁₆	—	2	1	no	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	26	2	no	—
344	M₁₆	—	52	2	no	—
345	M₁₆	C₂×D₅₂	1	1	yes	characteristic
346	M₁₆	C₂×D₅₂	1	1	yes	characteristic
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	52	2	no	—
350	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
351	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
352	C₂²⋊C₄	—	104	2	no	—
353	C₄²	C₂×D₅₂	1	1	yes	characteristic
354	C₁₆	—	2	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	2	1	no	—
357	C₁₆	D₄×D₁₃	1	1	yes	characteristic
358	C₁₆	D₄×D₁₃	1	1	yes	characteristic
359	C₁₃	#33859	1	1	yes	characteristic, complemented, Sylow
360	C₂³	—	52	2	no	—
361	C₂³	—	104	2	no	—
362	Q₈	—	26	2	no	—
363	Q₈	—	52	2	no	—
364	Q₈	—	52	2	no	—
365	Q₈	—	104	2	no	—
366	Q₈	C₂×D₁₀₄	1	1	yes	characteristic
367	Q₈	C₂×D₁₀₄	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	1	no	—
370	D₄	—	26	2	no	—
371	D₄	—	52	2	no	—
372	D₄	—	52	2	no	—
373	D₄	—	52	2	no	—
374	D₄	—	104	2	no	—
375	D₄	—	104	2	no	—
376	D₄	—	104	2	no	—
377	D₄	C₂×D₁₀₄	1	1	yes	characteristic
378	D₄	C₂×D₁₀₄	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	52	2	no	—
383	C₂×C₄	—	52	2	no	—
384	C₂×C₄	—	104	2	no	—
385	C₂×C₄	—	104	2	no	—
386	C₂×C₄	—	104	2	no	—
387	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	—	52	2	no	—
391	C₈	C₂×D₁₀₄	1	1	yes	characteristic
392	C₈	C₂×D₁₀₄	1	1	yes	characteristic
393	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
394	C₈	#88256	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	52	2	no	—
398	C₂²	—	104	2	no	—
399	C₂²	—	416	2	no	—
400	C₂²	—	208	2	no	—
401	C₂²	#11719457	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	52	2	no	—
406	C₄	—	104	2	no	—
407	C₄	D₄×D₅₂	1	1	yes	characteristic
408	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	4	1	no	—
412	C₂	—	104	2	no	—
413	C₂	—	208	2	no	—
414	C₂	D₄×D₁₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2170	C₂
socle	C₂×C₁₃	D₄×D₈
hypercenter	#2170	D₁₃
hypocenter	C₁₃	#33859
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #13127989	C₂	semidirect
2. #13127977	C₂	semidirect
3. #13126623	C₂	semidirect
4. #13117097	C₂	semidirect
5. #13116961	C₂	semidirect
6. #13116956	C₂	semidirect
7. #13116906	C₂	semidirect
8. #13116809	C₂	semidirect
9. #13116802	C₂	semidirect
10. #13116718	C₂	semidirect
11. C₁₃×#2170	C₂	semidirect
13. #11719458	C₂²	semidirect
16. C₁₃×#504	C₂²	semidirect
19. #11718803	C₂²	semidirect
21. #11718570	C₂²	semidirect
25. C₂×Dic₁₀₄	C₂²	semidirect
29. C₂×#88040	C₂²	semidirect
35. #11718361	C₂²	semidirect
36. #11718357	C₂²	semidirect
37. C₁₃×#364	C₂²	semidirect
39. C₁₃×#350	C₂²	semidirect
40. C₄×C₁₃×C₁₆	C₂²	semidirect
41. #11718109	C₂²	semidirect
42. #11718108	C₂²	semidirect
43. #11718102	C₂²	semidirect
44. #11718098	C₂²	semidirect
163. #2170	D₁₃	semidirect
215. #504	C₂×D₁₃	semidirect
225. #364	C₂×D₁₃	semidirect
230. #350	C₂×D₁₃	semidirect
231. C₄×C₁₆	C₂×D₁₃	semidirect
359. C₁₃	#33859	semidirect

Maximal subgroups

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

subgroup	index
#13127989	2
#13127977	2
#13126623	2
#13117097	2
#13116961	2
#13116956	2
#13116906	2
#13116809	2
#13116802	2
#13116718	2
C₁₃×#2170	2
#33859	13