Subgroups — #137×#261 #13131592

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#261	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#261	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₃×#137	C₂	1	1	yes	maximal, characteristic
4	#11719164	C₂	1	1	yes	maximal, characteristic
5	#11719153	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719110	C₂	1	2	yes	maximal, complemented
7	Q₁₆×#261	C₂	1	2	yes	maximal, complemented
8	#11719106	C₂	1	2	yes	maximal, complemented
9	SD₁₆×#261	C₂	1	2	yes	maximal, complemented
10	#11719098	C₂	1	1	yes	maximal, characteristic, complemented
11	M₁₆×#261	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
14	Q₈×#261	—	2	1	no	—
15	Q₈×#261	—	2	1	no	—
16	Q₈×#261	C₂²	1	2	yes	—
17	Q₈×#261	C₂²	1	2	yes	—
18	Q₈×#261	C₂²	1	1	yes	characteristic
19	Q₈×#261	C₂²	1	1	yes	characteristic
20	D₄×#261	—	2	1	no	—
21	D₄×#261	—	2	1	no	—
22	D₄×#261	C₂²	1	1	yes	characteristic
23	D₄×#261	C₂²	1	1	yes	characteristic
24	#88237	—	2	1	no	—
25	#88237	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#261	—	2	1	no	—
27	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
28	C₂×C₄×#261	—	2	1	no	—
29	C₂×C₄×#261	—	2	1	no	—
30	C₂×C₄×#261	C₂²	1	1	yes	characteristic
31	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
32	C₁₃×#137	C₄	1	1	yes	characteristic, complemented
33	#88205	C₄	1	1	yes	characteristic, complemented
34	#88196	C₄	1	1	yes	characteristic, complemented
35	#88172	C₄	1	2	yes	complemented
36	Q₁₆×D₁₃	C₂²	1	2	yes	—
37	#88169	C₄	1	2	yes	complemented
38	SD₁₆×D₁₃	C₂²	1	2	yes	—
39	#88163	C₄	1	1	yes	characteristic, complemented
40	M₁₆×D₁₃	C₂²	1	1	yes	characteristic
41	#88117	C₂²	1	2	yes	—
42	#88117	C₂²	1	2	yes	—
43	#88117	C₂²	1	2	yes	—
44	#88115	C₂²	1	2	yes	—
45	#88102	C₂²	1	2	yes	—
46	#88101	C₂²	1	2	yes	—
47	#88100	C₂²	1	2	yes	—
48	C₈×#261	C₂²	1	2	yes	—
49	C₂²×#261	—	2	1	no	—
50	C₂²×#261	—	2	1	no	—
51	C₂²×#261	—	2	1	no	—
52	C₂²×#261	D₄	1	1	yes	characteristic
53	C₂²×#261	D₄	1	1	yes	characteristic
54	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
56	#6131	C₂×C₄	1	1	yes	characteristic
57	Q₈×D₁₃	—	2	1	no	—
58	Q₈×D₁₃	—	2	1	no	—
59	Q₈×D₁₃	C₂×C₄	1	2	yes	complemented
60	Q₈×D₁₃	C₂³	1	2	yes	—
61	Q₈×D₁₃	C₂³	1	1	yes	characteristic
62	#6129	—	2	1	no	—
63	#6129	C₂×C₄	1	1	yes	characteristic
64	D₄×D₁₃	—	2	1	no	—
65	D₄×D₁₃	C₂³	1	1	yes	characteristic
66	#6127	—	2	1	no	—
67	#6127	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
71	C₂×Dic₂₆	—	2	1	no	—
72	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂²⋊#261	—	2	1	no	—
74	C₂²⋊#261	—	2	1	no	—
75	C₂²⋊#261	—	2	1	no	—
76	#6122	—	2	1	no	—
77	C₄⋊#261	—	2	2	no	—
78	C₄⋊#261	—	2	2	no	—
79	C₄⋊#261	—	2	1	no	—
80	C₄⋊#261	—	2	1	no	—
81	C₄⋊#261	—	2	1	no	—
82	C₄⋊#261	C₂³	1	1	yes	characteristic
83	C₄⋊#261	C₂³	1	2	yes	—
84	C₄⋊#261	C₂³	1	1	yes	characteristic
85	C₄×#261	—	2	1	no	—
86	C₄×#261	—	2	1	no	—
87	C₄×#261	—	2	2	no	—
88	C₄×#261	—	2	2	no	—
89	C₄×#261	—	2	1	no	—
90	C₄×#261	D₄	1	1	yes	characteristic
91	C₄×#261	D₄	1	1	yes	characteristic
92	C₄×#261	C₂³	1	1	yes	characteristic
93	C₄×#261	C₂³	1	1	yes	characteristic
94	#6118	—	2	1	no	—
95	D₁₃⋊C₈	C₂³	1	2	yes	—
96	C₁₃×Q₁₆	C₂×C₄	1	2	yes	complemented
97	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
98	C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic, complemented
99	#6107	C₂×C₄	1	2	yes	complemented
100	#6107	C₂×C₄	1	2	yes	complemented
101	#6106	C₂×C₄	1	2	yes	complemented
102	#6105	C₂×C₄	1	2	yes	complemented
103	#6099	C₂×C₄	1	1	yes	characteristic, complemented
104	Dic₅₂	C₂×C₄	1	2	yes	complemented
105	#6095	C₂×C₄	1	2	yes	complemented
106	#6094	C₂×C₄	1	2	yes	complemented
107	C₈×D₁₃	C₂³	1	2	yes	—
108	C₄×#137	—	13	1	no	maximal, Sylow
109	C₂²×D₁₃	—	2	1	no	—
110	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×#261	—	2	2	no	—
112	C₂×#261	—	2	2	no	—
113	C₂×#261	—	2	1	no	—
114	C₂×#261	—	2	1	no	—
115	C₂×#261	—	4	1	no	—
116	C₂×#261	—	4	2	no	—
117	C₂×#261	—	4	2	no	—
118	C₂×#261	—	4	2	no	—
119	C₂×#261	C₂×D₄	1	1	yes	characteristic
120	C₂×#261	C₂×D₄	1	1	yes	characteristic
121	C₂×#261	C₂×D₄	1	1	yes	characteristic
122	C₂×#261	C₂×D₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	1	no	—
124	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
125	Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
126	D₄×C₁₃	—	2	1	no	—
127	D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²⋊D₁₃	—	2	1	no	—
132	C₂²⋊D₁₃	—	2	1	no	—
133	C₂²⋊D₁₃	—	2	1	no	—
134	C₂×Dic₁₃	—	2	1	no	—
135	C₂×Dic₁₃	—	2	1	no	—
136	C₂×Dic₁₃	#41	1	1	yes	characteristic
137	D₅₂	—	2	1	no	—
138	D₅₂	C₂²×C₄	1	1	yes	characteristic
139	C₄×D₁₃	—	2	2	no	—
140	C₄×D₁₃	—	2	1	no	—
141	C₄×D₁₃	—	2	2	no	—
142	C₄×D₁₃	—	2	1	no	—
143	C₄×D₁₃	—	2	1	no	—
144	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
145	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
146	C₄×D₁₃	#41	1	1	yes	characteristic
147	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
148	Dic₂₆	—	2	1	no	—
149	Dic₂₆	—	2	2	no	—
150	Dic₂₆	—	2	2	no	—
151	Dic₂₆	—	2	1	no	—
152	Dic₂₆	C₂²×C₄	1	2	yes	—
153	Dic₂₆	C₂²×C₄	1	1	yes	characteristic
154	#1057	—	2	2	no	—
155	C₈×C₁₃	C₂²×C₄	1	2	yes	—
156	#1055	C₂²×C₄	1	2	yes	—
157	C₂×#137	—	13	1	no	—
158	C₄×#41	—	13	1	no	—
159	C₂×C₄×Q₈	—	13	1	no	—
160	#441	—	13	2	no	—
161	#440	—	13	2	no	—
162	C₄×Q₁₆	—	13	2	no	—
163	C₄×SD₁₆	—	13	2	no	—
164	#428	—	13	1	no	—
165	#419	—	13	1	no	—
166	#417	—	13	1	no	—
167	C₄×M₁₆	—	13	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₂×D₁₃	—	2	1	no	—
171	C₂×D₁₃	—	2	1	no	—
172	C₂×D₁₃	—	2	1	no	—
173	C₂×D₁₃	—	4	1	no	—
174	C₂×D₁₃	—	4	1	no	—
175	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
176	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
177	#261	—	2	1	no	—
178	#261	—	2	1	no	—
179	#261	—	2	1	no	—
180	#261	—	4	1	no	—
181	#261	—	4	2	no	—
182	#261	—	4	1	no	—
183	#261	#137	1	2	yes	direct product factor
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
188	Dic₁₃	—	2	2	no	—
189	Dic₁₃	—	2	1	no	—
190	Dic₁₃	C₄×D₄	1	1	yes	characteristic
191	Dic₁₃	C₂×#41	1	1	yes	characteristic
192	C₂×#41	—	13	1	no	—
193	C₂²×Q₈	—	13	1	no	—
194	#137	—	13	1	no	—
195	#137	—	13	1	no	—
196	#137	—	13	2	no	—
197	#137	—	13	2	no	—
198	#137	—	13	1	no	—
199	#137	#261	1	1	yes	characteristic, direct product factor
200	C₂×Q₁₆	—	13	2	no	—
201	C₂×SD₁₆	—	13	2	no	—
202	C₂×M₁₆	—	13	1	no	—
203	C₂×M₁₆	—	13	1	no	—
204	C₄×Q₈	—	13	2	no	—
205	C₄×Q₈	—	13	2	no	—
206	C₄×Q₈	—	13	1	no	—
207	C₄×Q₈	—	13	1	no	—
208	C₄×Q₈	—	26	1	no	—
209	C₄×Q₈	—	26	1	no	—
210	C₄×D₄	—	13	1	no	—
211	C₄×D₄	—	13	1	no	—
212	C₄×D₄	—	26	1	no	—
213	C₄×D₄	—	26	1	no	—
214	#117	—	13	1	no	—
215	#117	—	26	1	no	—
216	C₂×C₄⋊C₄	—	13	1	no	—
217	C₂×C₄⋊C₄	—	26	1	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	26	1	no	—
220	C₂×C₄²	—	26	1	no	—
221	#107	—	13	2	no	—
222	#106	—	13	2	no	—
223	#103	—	13	2	no	—
224	#103	—	13	2	no	—
225	#103	—	13	2	no	—
226	#102	—	13	2	no	—
227	#97	—	13	2	no	—
228	C₄×C₈	—	13	2	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	4	1	no	—
231	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
232	D₁₃	—	2	1	no	—
233	D₁₃	—	4	1	no	—
234	D₁₃	C₂×#137	1	1	yes	characteristic
235	D₁₃	#575	1	1	yes	characteristic
236	#41	—	13	1	no	—
237	#41	—	13	1	no	—
238	#41	—	13	1	no	—
239	#41	—	26	1	no	—
240	#41	—	26	1	no	—
241	#41	C₂×#261	1	1	yes	characteristic
242	C₂×Q₈	—	13	2	no	—
243	C₂×Q₈	—	13	1	no	—
244	C₂×Q₈	—	13	2	no	—
245	C₂×Q₈	—	13	1	no	—
246	C₂×Q₈	—	26	1	no	—
247	C₂×Q₈	—	26	1	no	—
248	C₂×Q₈	—	26	1	no	—
249	C₂×Q₈	C₂×#261	1	1	yes	characteristic, complemented
250	C₂×D₄	—	13	1	no	—
251	C₂×D₄	—	26	1	no	—
252	C₂²×C₄	—	13	1	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	26	1	no	—
256	C₂²×C₄	—	26	1	no	—
257	C₂²×C₄	—	26	1	no	—
258	C₂²×C₄	—	26	1	no	—
259	C₂²×C₄	—	26	1	no	—
260	Q₁₆	—	13	2	no	—
261	Q₁₆	—	13	2	no	—
262	Q₁₆	—	13	2	no	—
263	Q₁₆	C₂×#261	1	2	yes	complemented
264	SD₁₆	—	13	2	no	—
265	SD₁₆	—	13	2	no	—
266	SD₁₆	—	13	2	no	—
267	SD₁₆	C₂×#261	1	2	yes	complemented
268	M₁₆	—	13	1	no	—
269	M₁₆	—	13	2	no	—
270	M₁₆	—	26	1	no	—
271	M₁₆	—	26	1	no	—
272	M₁₆	C₂×#261	1	1	yes	characteristic, complemented
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	13	2	no	—
275	C₄⋊C₄	—	13	2	no	—
276	C₄⋊C₄	—	13	1	no	—
277	C₄⋊C₄	—	13	1	no	—
278	C₄⋊C₄	—	26	1	no	—
279	C₄⋊C₄	—	26	2	no	—
280	C₄⋊C₄	—	26	2	no	—
281	C₄⋊C₄	—	26	1	no	—
282	C₄⋊C₄	—	26	1	no	—
283	C₂²⋊C₄	—	26	1	no	—
284	C₂²⋊C₄	—	26	1	no	—
285	C₂²⋊C₄	—	26	1	no	—
286	C₄²	—	13	1	no	—
287	C₄²	—	13	1	no	—
288	C₄²	—	13	1	no	—
289	C₄²	—	13	1	no	—
290	C₄²	—	26	1	no	—
291	C₄²	—	26	1	no	—
292	C₄²	—	26	1	no	—
293	C₄²	—	26	2	no	—
294	C₄²	—	26	2	no	—
295	C₁₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	13	1	no	—
297	C₂³	—	26	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	13	2	no	—
300	Q₈	—	13	1	no	—
301	Q₈	—	26	2	no	—
302	Q₈	—	26	2	no	—
303	Q₈	—	26	1	no	—
304	Q₈	—	26	1	no	—
305	Q₈	C₂²×#261	1	2	yes	—
306	Q₈	C₂²×#261	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	13	1	no	—
309	D₄	—	26	1	no	—
310	D₄	—	26	1	no	—
311	D₄	—	26	1	no	—
312	D₄	—	26	1	no	—
313	D₄	C₂²×#261	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	26	1	no	—
326	C₂×C₄	—	26	1	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	1	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	1	no	—
332	C₂×C₄	—	26	1	no	—
333	C₂×C₄	—	26	1	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	C₂²×#261	1	1	yes	characteristic
341	C₈	—	13	2	no	—
342	C₈	—	26	2	no	—
343	C₈	C₂²×#261	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	13	1	no	—
346	C₂²	—	13	1	no	—
347	C₂²	—	26	1	no	—
348	C₂²	—	26	1	no	—
349	C₂²	—	26	1	no	—
350	C₂²	—	52	1	no	—
351	C₂²	—	52	1	no	—
352	C₂²	D₄×#261	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	13	2	no	—
356	C₄	—	13	1	no	—
357	C₄	—	13	1	no	—
358	C₄	—	26	1	no	—
359	C₄	—	26	1	no	—
360	C₄	—	26	1	no	—
361	C₄	—	26	1	no	—
362	C₄	—	26	2	no	—
363	C₄	—	52	1	no	—
364	C₄	—	52	1	no	—
365	C₄	—	52	2	no	—
366	C₄	D₄×#261	1	1	yes	characteristic
367	C₄	C₂³×#261	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	13	1	no	—
371	C₂	—	13	1	no	—
372	C₂	—	26	1	no	—
373	C₂	—	52	1	no	—
374	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₁₃×#137	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#41×D₁₃	C₂²
cosocle	C₄×D₁₃	C₂⁴
hypercenter	#137	#261
hypocenter	C₁₃	C₄×#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×#261	C₂	semidirect
5. #11719153	C₂	semidirect
6. #11719110	C₂	semidirect
7. Q₁₆×#261	C₂	semidirect
8. #11719106	C₂	semidirect
9. SD₁₆×#261	C₂	semidirect
10. #11719098	C₂	semidirect
11. M₁₆×#261	C₂	semidirect
32. C₁₃×#137	C₄	semidirect
33. #88205	C₄	semidirect
34. #88196	C₄	semidirect
35. #88172	C₄	semidirect
37. #88169	C₄	semidirect
39. #88163	C₄	semidirect
55. C₂×Q₈×C₁₃	C₂×C₄	semidirect
59. Q₈×D₁₃	C₂×C₄	semidirect
72. C₂×Dic₂₆	C₂×C₄	semidirect
96. C₁₃×Q₁₆	C₂×C₄	semidirect
97. C₁₃×SD₁₆	C₂×C₄	semidirect
98. C₁₃×M₁₆	C₂×C₄	semidirect
99. #6107	C₂×C₄	semidirect
100. #6107	C₂×C₄	semidirect
101. #6106	C₂×C₄	semidirect
102. #6105	C₂×C₄	semidirect
103. #6099	C₂×C₄	semidirect
104. Dic₅₂	C₂×C₄	semidirect
105. #6095	C₂×C₄	semidirect
106. #6094	C₂×C₄	semidirect
183. #261	#137	direct (N × H)
199. #137	#261	direct (N × H)
249. C₂×Q₈	C₂×#261	semidirect
263. Q₁₆	C₂×#261	semidirect
267. SD₁₆	C₂×#261	semidirect
272. M₁₆	C₂×#261	semidirect
295. C₁₃	C₄×#137	semidirect

Maximal subgroups

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

subgroup	index
#41×#261	2
C₂×Q₈×#261	2
D₁₃×#137	2
#11719164	2
#11719153	2
#11719110	2
Q₁₆×#261	2
#11719106	2
SD₁₆×#261	2
#11719098	2
M₁₆×#261	2
C₄×#137	13