Subgroups — #137×#10587739 #31797545

Order 17536.

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×#10587739	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#10587739	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578443	C₂	1	2	yes	maximal, complemented
4	#22578442	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#10587739	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#10587739	C₂	1	2	yes	maximal, complemented
7	#22578414	C₂	1	1	yes	maximal, characteristic, complemented
8	#22578396	C₂	1	1	yes	maximal, characteristic, complemented
9	#22578394	C₂	1	1	yes	maximal, characteristic
10	M₁₆×#10587739	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₃₇	C₂	1	1	yes	maximal, characteristic
12	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
13	Q₈×#10587739	—	2	1	no	—
14	Q₈×#10587739	—	2	1	no	—
15	Q₈×#10587739	C₂²	1	2	yes	—
16	Q₈×#10587739	C₂²	1	1	yes	characteristic
17	Q₈×#10587739	C₂²	1	2	yes	—
18	Q₈×#10587739	C₂²	1	1	yes	characteristic
19	D₄×#10587739	—	2	1	no	—
20	D₄×#10587739	—	2	1	no	—
21	D₄×#10587739	C₂²	1	1	yes	characteristic
22	D₄×#10587739	C₂²	1	1	yes	characteristic
23	#17984167	—	2	1	no	—
24	#17984167	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#10587739	—	2	1	no	—
26	C₂×C₄⋊#10587739	C₂²	1	1	yes	characteristic
27	C₂×C₄×#10587739	—	2	1	no	—
28	C₂×C₄×#10587739	—	2	1	no	—
29	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
30	#17984157	C₂²	1	2	yes	—
31	#17984156	C₂²	1	2	yes	—
32	#17984151	C₂²	1	2	yes	—
33	#17984151	C₂²	1	2	yes	—
34	#17984151	C₂²	1	2	yes	—
35	#17984150	C₂²	1	2	yes	—
36	#17984142	C₂²	1	2	yes	—
37	C₈×#10587739	C₂²	1	2	yes	—
38	#41×D₁₃₇	C₂²	1	1	yes	characteristic
39	C₂×Q₈×D₁₃₇	C₂²	1	1	yes	characteristic
40	#17984121	C₄	1	1	yes	characteristic, complemented
41	#17984120	C₄	1	2	yes	complemented
42	#17984119	C₄	1	1	yes	characteristic, complemented
43	#17984118	C₄	1	1	yes	characteristic, complemented
44	#17984117	C₄	1	2	yes	complemented
45	Q₁₆×D₁₃₇	C₂²	1	2	yes	—
46	SD₁₆×D₁₃₇	C₂²	1	2	yes	—
47	M₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
48	#137×C₁₃₇	C₄	1	1	yes	characteristic, complemented
49	C₂²×#10587739	—	2	1	no	—
50	C₂²×#10587739	—	2	1	no	—
51	C₂²×#10587739	—	2	1	no	—
52	C₂²×#10587739	D₄	1	1	yes	characteristic
53	C₂²×#10587739	D₄	1	1	yes	characteristic
54	#14555520	—	2	1	no	—
55	#14555519	—	2	1	no	—
56	#14555518	C₂³	1	2	yes	—
57	C₄⋊#10587739	—	2	2	no	—
58	C₄⋊#10587739	—	2	2	no	—
59	C₄⋊#10587739	—	2	1	no	—
60	C₄⋊#10587739	—	2	1	no	—
61	C₄⋊#10587739	—	2	1	no	—
62	C₄⋊#10587739	C₂³	1	2	yes	—
63	C₄⋊#10587739	C₂³	1	1	yes	characteristic
64	C₄⋊#10587739	C₂³	1	1	yes	characteristic
65	C₂²⋊#10587739	—	2	1	no	—
66	C₂²⋊#10587739	—	2	1	no	—
67	C₂²⋊#10587739	—	2	1	no	—
68	C₄×#10587739	—	2	1	no	—
69	C₄×#10587739	—	2	1	no	—
70	C₄×#10587739	—	2	2	no	—
71	C₄×#10587739	—	2	2	no	—
72	C₄×#10587739	—	2	1	no	—
73	C₄×#10587739	D₄	1	1	yes	characteristic
74	C₄×#10587739	D₄	1	1	yes	characteristic
75	C₄×#10587739	C₂³	1	1	yes	characteristic
76	C₄×#10587739	C₂³	1	1	yes	characteristic
77	#14555511	C₂×C₄	1	1	yes	characteristic
78	#14555510	—	2	1	no	—
79	#14555510	C₂×C₄	1	1	yes	characteristic
80	#14555509	—	2	1	no	—
81	#14555509	C₂×C₄	1	1	yes	characteristic
82	C₂×Dic₂₇₄	—	2	1	no	—
83	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic, complemented
84	Q₈×D₁₃₇	—	2	1	no	—
85	Q₈×D₁₃₇	—	2	1	no	—
86	Q₈×D₁₃₇	C₂×C₄	1	2	yes	complemented
87	Q₈×D₁₃₇	C₂³	1	2	yes	—
88	Q₈×D₁₃₇	C₂³	1	1	yes	characteristic
89	D₄×D₁₃₇	—	2	1	no	—
90	D₄×D₁₃₇	C₂³	1	1	yes	characteristic
91	C₂×C₄×D₁₃₇	—	2	1	no	—
92	C₂×C₄×D₁₃₇	—	2	1	no	—
93	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
94	Dic₅₄₈	C₂×C₄	1	2	yes	complemented
95	#14555500	C₂×C₄	1	2	yes	complemented
96	#14555500	C₂×C₄	1	2	yes	complemented
97	#14555499	C₂×C₄	1	2	yes	complemented
98	#14555498	C₂×C₄	1	2	yes	complemented
99	#14555497	C₂×C₄	1	2	yes	complemented
100	#14555494	C₂×C₄	1	1	yes	characteristic, complemented
101	#14555493	C₂×C₄	1	2	yes	complemented
102	C₈×D₁₃₇	C₂³	1	2	yes	—
103	#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
104	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
105	Q₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
106	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
107	M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
108	C₂×#10587739	—	2	2	no	—
109	C₂×#10587739	—	2	1	no	—
110	C₂×#10587739	—	2	1	no	—
111	C₂×#10587739	—	2	2	no	—
112	C₂×#10587739	—	4	1	no	—
113	C₂×#10587739	—	4	2	no	—
114	C₂×#10587739	—	4	2	no	—
115	C₂×#10587739	—	4	2	no	—
116	C₂×#10587739	C₂×D₄	1	1	yes	characteristic
117	C₂×#10587739	C₂×D₄	1	1	yes	characteristic
118	C₂×#10587739	C₂×D₄	1	1	yes	characteristic
119	C₂×#10587739	C₂×D₄	1	1	yes	characteristic
120	#11764047	—	2	2	no	—
121	C₂²×D₁₃₇	—	2	1	no	—
122	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
123	Dic₂₇₄	—	2	1	no	—
124	Dic₂₇₄	—	2	2	no	—
125	Dic₂₇₄	—	2	2	no	—
126	Dic₂₇₄	—	2	1	no	—
127	Dic₂₇₄	C₂²×C₄	1	2	yes	—
128	Dic₂₇₄	C₂²×C₄	1	1	yes	characteristic
129	D₅₄₈	—	2	1	no	—
130	D₅₄₈	C₂²×C₄	1	1	yes	characteristic
131	C₂²⋊D₁₃₇	—	2	1	no	—
132	C₂²⋊D₁₃₇	—	2	1	no	—
133	C₂²⋊D₁₃₇	—	2	1	no	—
134	C₂×Dic₁₃₇	—	2	1	no	—
135	C₂×Dic₁₃₇	—	2	1	no	—
136	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
137	C₄×D₁₃₇	—	2	2	no	—
138	C₄×D₁₃₇	—	2	1	no	—
139	C₄×D₁₃₇	—	2	2	no	—
140	C₄×D₁₃₇	—	2	1	no	—
141	C₄×D₁₃₇	—	2	1	no	—
142	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
143	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
144	C₄×D₁₃₇	#41	1	1	yes	characteristic
145	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
146	#11764040	C₂²×C₄	1	2	yes	—
147	Q₈×C₁₃₇	—	2	1	no	—
148	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
149	Q₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
150	D₄×C₁₃₇	—	2	1	no	—
151	D₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃₇	—	2	1	no	—
153	C₂×C₄×C₁₃₇	—	2	1	no	—
154	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
155	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
156	C₂²×C₁₃₇	—	2	1	no	—
157	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
158	C₂×D₁₃₇	—	2	1	no	—
159	C₂×D₁₃₇	—	2	1	no	—
160	C₂×D₁₃₇	—	2	1	no	—
161	C₂×D₁₃₇	—	4	1	no	—
162	C₂×D₁₃₇	—	4	1	no	—
163	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
165	#10587739	—	2	1	no	—
166	#10587739	—	2	1	no	—
167	#10587739	—	2	1	no	—
168	#10587739	—	4	1	no	—
169	#10587739	—	4	1	no	—
170	#10587739	—	4	2	no	—
171	#10587739	#137	1	2	yes	direct product factor
172	C₄×C₁₃₇	—	2	1	no	—
173	C₄×C₁₃₇	—	2	2	no	—
174	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
176	Dic₁₃₇	—	2	1	no	—
177	Dic₁₃₇	—	2	2	no	—
178	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
179	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₁₃₇	—	2	1	no	—
181	C₂×C₁₃₇	—	4	1	no	—
182	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
183	D₁₃₇	—	2	1	no	—
184	D₁₃₇	—	4	1	no	—
185	D₁₃₇	C₂×#137	1	1	yes	characteristic
186	D₁₃₇	#575	1	1	yes	characteristic
187	C₁₃₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	C₄×#137	—	137	1	no	maximal, Sylow
189	C₂×#137	—	137	1	no	—
190	C₄×#41	—	137	1	no	—
191	C₂×C₄×Q₈	—	137	1	no	—
192	#441	—	137	2	no	—
193	#440	—	137	2	no	—
194	C₄×Q₁₆	—	137	2	no	—
195	C₄×SD₁₆	—	137	2	no	—
196	#428	—	137	1	no	—
197	#419	—	137	1	no	—
198	#417	—	137	1	no	—
199	C₄×M₁₆	—	137	1	no	—
200	C₂×#41	—	137	1	no	—
201	C₂²×Q₈	—	137	1	no	—
202	#137	—	137	1	no	—
203	#137	—	137	1	no	—
204	#137	—	137	2	no	—
205	#137	—	137	2	no	—
206	#137	—	137	1	no	—
207	#137	#10587739	1	1	yes	characteristic, direct product factor
208	C₂×Q₁₆	—	137	2	no	—
209	C₂×SD₁₆	—	137	2	no	—
210	C₂×M₁₆	—	137	1	no	—
211	C₂×M₁₆	—	137	1	no	—
212	C₄×Q₈	—	274	1	no	—
213	C₄×Q₈	—	274	1	no	—
214	C₄×Q₈	—	137	2	no	—
215	C₄×Q₈	—	137	2	no	—
216	C₄×Q₈	—	137	1	no	—
217	C₄×Q₈	—	137	1	no	—
218	C₄×D₄	—	274	1	no	—
219	C₄×D₄	—	274	1	no	—
220	C₄×D₄	—	137	1	no	—
221	C₄×D₄	—	137	1	no	—
222	#117	—	274	1	no	—
223	#117	—	137	1	no	—
224	C₂×C₄⋊C₄	—	274	1	no	—
225	C₂×C₄⋊C₄	—	137	1	no	—
226	C₂×C₄²	—	274	1	no	—
227	C₂×C₄²	—	274	1	no	—
228	C₂×C₄²	—	137	1	no	—
229	#107	—	137	2	no	—
230	#106	—	137	2	no	—
231	#103	—	137	2	no	—
232	#103	—	137	2	no	—
233	#103	—	137	2	no	—
234	#102	—	137	2	no	—
235	#97	—	137	2	no	—
236	C₄×C₈	—	137	2	no	—
237	#41	—	274	1	no	—
238	#41	—	274	1	no	—
239	#41	—	137	1	no	—
240	#41	—	137	1	no	—
241	#41	—	137	1	no	—
242	#41	C₂×#10587739	1	1	yes	characteristic
243	C₂×Q₈	—	274	1	no	—
244	C₂×Q₈	—	274	1	no	—
245	C₂×Q₈	—	274	1	no	—
246	C₂×Q₈	—	137	2	no	—
247	C₂×Q₈	—	137	1	no	—
248	C₂×Q₈	—	137	2	no	—
249	C₂×Q₈	—	137	1	no	—
250	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic, complemented
251	C₂×D₄	—	274	1	no	—
252	C₂×D₄	—	137	1	no	—
253	C₂²×C₄	—	274	1	no	—
254	C₂²×C₄	—	274	1	no	—
255	C₂²×C₄	—	274	1	no	—
256	C₂²×C₄	—	274	1	no	—
257	C₂²×C₄	—	274	1	no	—
258	C₂²×C₄	—	137	1	no	—
259	C₂²×C₄	—	137	1	no	—
260	C₂²×C₄	—	137	1	no	—
261	Q₁₆	—	137	2	no	—
262	Q₁₆	—	137	2	no	—
263	Q₁₆	—	137	2	no	—
264	Q₁₆	C₂×#10587739	1	2	yes	complemented
265	SD₁₆	—	137	2	no	—
266	SD₁₆	—	137	2	no	—
267	SD₁₆	—	137	2	no	—
268	SD₁₆	C₂×#10587739	1	2	yes	complemented
269	M₁₆	—	274	1	no	—
270	M₁₆	—	274	1	no	—
271	M₁₆	—	137	1	no	—
272	M₁₆	—	137	2	no	—
273	M₁₆	C₂×#10587739	1	1	yes	characteristic, complemented
274	C₂×C₈	—	137	2	no	—
275	C₂×C₈	—	137	2	no	—
276	C₄⋊C₄	—	274	2	no	—
277	C₄⋊C₄	—	274	2	no	—
278	C₄⋊C₄	—	274	1	no	—
279	C₄⋊C₄	—	274	1	no	—
280	C₄⋊C₄	—	274	1	no	—
281	C₄⋊C₄	—	137	2	no	—
282	C₄⋊C₄	—	137	1	no	—
283	C₄⋊C₄	—	137	1	no	—
284	C₂²⋊C₄	—	274	1	no	—
285	C₂²⋊C₄	—	274	1	no	—
286	C₂²⋊C₄	—	274	1	no	—
287	C₄²	—	274	1	no	—
288	C₄²	—	274	1	no	—
289	C₄²	—	274	1	no	—
290	C₄²	—	274	2	no	—
291	C₄²	—	274	2	no	—
292	C₄²	—	137	1	no	—
293	C₄²	—	137	1	no	—
294	C₄²	—	137	1	no	—
295	C₄²	—	137	1	no	—
296	C₂³	—	274	1	no	—
297	C₂³	—	137	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	274	2	no	—
300	Q₈	—	274	2	no	—
301	Q₈	—	274	1	no	—
302	Q₈	—	274	1	no	—
303	Q₈	—	137	2	no	—
304	Q₈	—	137	1	no	—
305	Q₈	C₂²×#10587739	1	2	yes	—
306	Q₈	C₂²×#10587739	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	274	1	no	—
309	D₄	—	274	1	no	—
310	D₄	—	274	1	no	—
311	D₄	—	274	1	no	—
312	D₄	—	137	1	no	—
313	D₄	C₂²×#10587739	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	274	1	no	—
317	C₂×C₄	—	274	2	no	—
318	C₂×C₄	—	274	2	no	—
319	C₂×C₄	—	274	1	no	—
320	C₂×C₄	—	274	1	no	—
321	C₂×C₄	—	274	1	no	—
322	C₂×C₄	—	274	2	no	—
323	C₂×C₄	—	274	1	no	—
324	C₂×C₄	—	274	1	no	—
325	C₂×C₄	—	274	1	no	—
326	C₂×C₄	—	274	2	no	—
327	C₂×C₄	—	548	1	no	—
328	C₂×C₄	—	548	2	no	—
329	C₂×C₄	—	548	2	no	—
330	C₂×C₄	—	548	2	no	—
331	C₂×C₄	—	137	1	no	—
332	C₂×C₄	—	137	1	no	—
333	C₂×C₄	—	137	1	no	—
334	C₂×C₄	—	137	1	no	—
335	C₂×C₄	—	137	1	no	—
336	C₂×C₄	—	137	1	no	—
337	C₂×C₄	—	137	1	no	—
338	C₂×C₄	—	137	1	no	—
339	C₂×C₄	—	137	1	no	—
340	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
341	C₈	—	274	2	no	—
342	C₈	—	137	2	no	—
343	C₈	C₂²×#10587739	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	274	1	no	—
346	C₂²	—	274	1	no	—
347	C₂²	—	274	1	no	—
348	C₂²	—	548	1	no	—
349	C₂²	—	548	1	no	—
350	C₂²	—	137	1	no	—
351	C₂²	—	137	1	no	—
352	C₂²	D₄×#10587739	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	274	1	no	—
356	C₄	—	274	1	no	—
357	C₄	—	274	1	no	—
358	C₄	—	274	1	no	—
359	C₄	—	274	2	no	—
360	C₄	—	548	1	no	—
361	C₄	—	548	2	no	—
362	C₄	—	548	1	no	—
363	C₄	—	137	2	no	—
364	C₄	—	137	1	no	—
365	C₄	—	137	1	no	—
366	C₄	D₄×#10587739	1	1	yes	characteristic
367	C₄	C₂³×#10587739	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	274	1	no	—
371	C₂	—	548	1	no	—
372	C₂	—	137	1	no	—
373	C₂	—	137	1	no	—
374	C₂	C₂×D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	#137×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#41×D₁₃₇	C₂²
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	#137	#10587739
hypocenter	C₁₃₇	C₄×#137
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
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×#10587739	C₂	semidirect
3. #22578443	C₂	semidirect
4. #22578442	C₂	semidirect
5. Q₁₆×#10587739	C₂	semidirect
6. SD₁₆×#10587739	C₂	semidirect
7. #22578414	C₂	semidirect
8. #22578396	C₂	semidirect
10. M₁₆×#10587739	C₂	semidirect
40. #17984121	C₄	semidirect
41. #17984120	C₄	semidirect
42. #17984119	C₄	semidirect
43. #17984118	C₄	semidirect
44. #17984117	C₄	semidirect
48. #137×C₁₃₇	C₄	semidirect
83. C₂×Dic₂₇₄	C₂×C₄	semidirect
86. Q₈×D₁₃₇	C₂×C₄	semidirect
94. Dic₅₄₈	C₂×C₄	semidirect
95. #14555500	C₂×C₄	semidirect
96. #14555500	C₂×C₄	semidirect
97. #14555499	C₂×C₄	semidirect
98. #14555498	C₂×C₄	semidirect
99. #14555497	C₂×C₄	semidirect
100. #14555494	C₂×C₄	semidirect
101. #14555493	C₂×C₄	semidirect
104. C₂×Q₈×C₁₃₇	C₂×C₄	semidirect
105. Q₁₆×C₁₃₇	C₂×C₄	semidirect
106. SD₁₆×C₁₃₇	C₂×C₄	semidirect
107. M₁₆×C₁₃₇	C₂×C₄	semidirect
171. #10587739	#137	direct (N × H)
187. C₁₃₇	C₄×#137	semidirect
207. #137	#10587739	direct (N × H)
250. C₂×Q₈	C₂×#10587739	semidirect
264. Q₁₆	C₂×#10587739	semidirect
268. SD₁₆	C₂×#10587739	semidirect
273. M₁₆	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
#41×#10587739	2
C₂×Q₈×#10587739	2
#22578443	2
#22578442	2
Q₁₆×#10587739	2
SD₁₆×#10587739	2
#22578414	2
#22578396	2
#22578394	2
M₁₆×#10587739	2
#137×D₁₃₇	2
C₄×#137	137