Subgroups — #16854527

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758495	C₂	1	1	yes	maximal, characteristic
2	C₄×#92499	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521154	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521151	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521147	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521141	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521136	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521134	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521088	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521056	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521049	C₂	1	1	yes	maximal, characteristic
14	#14521001	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#92499	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₃₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
23	C₂×#92482	C₂²	1	1	yes	characteristic
24	C₂×#92481	C₂²	1	1	yes	characteristic
25	#11758495	C₄	1	8	yes	complemented
26	C₂×#92470	C₂²	1	1	yes	characteristic
27	C₂×#92469	C₂²	1	1	yes	characteristic
28	C₂×#92469	C₂²	1	1	yes	characteristic
29	#11758470	—	2	1	no	—
30	#11758470	C₂²	1	1	yes	characteristic
31	#11758469	—	2	1	no	—
32	#11758469	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₃₁	—	2	1	no	—
34	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
35	#11758464	—	2	1	no	—
36	#11758464	C₂²	1	1	yes	characteristic
37	C₄×D₁₂₄	—	2	1	no	—
38	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
39	C₄²×D₃₁	—	2	1	no	—
40	C₄²×D₃₁	—	2	1	no	—
41	C₄²×D₃₁	C₂²	1	1	yes	characteristic
42	C₄×Dic₆₂	—	2	1	no	—
43	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
44	C₃₁×#106	C₂²	1	1	yes	characteristic
45	C₃₁×#103	C₂²	1	1	yes	characteristic
46	C₃₁×#103	C₂²	1	1	yes	characteristic
47	C₃₁×#97	C₂²	1	1	yes	characteristic
48	#11758423	C₂²	1	1	yes	characteristic
49	#11758423	C₂²	1	1	yes	characteristic
50	#11758409	C₂²	1	1	yes	characteristic
51	#11758406	C₂²	1	1	yes	characteristic
52	#11758404	C₂²	1	1	yes	characteristic
53	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
54	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
55	#11758398	C₂²	1	1	yes	characteristic
56	#11758397	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	#11758391	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	#92499	—	2	2	no	—
63	#92499	C₂×C₄	1	4	yes	—
64	Q₈×D₃₁	—	2	2	no	—
65	Q₈×D₃₁	C₂×C₄	1	4	yes	complemented
66	C₂×D₁₂₄	—	2	1	no	—
67	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₃₁	—	2	1	no	—
69	C₂×C₄×D₃₁	—	2	1	no	—
70	C₂×C₄×D₃₁	—	2	1	no	—
71	C₂×C₄×D₃₁	—	2	1	no	—
72	C₂×C₄×D₃₁	—	2	1	no	—
73	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
76	C₂×Dic₆₂	—	2	1	no	—
77	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
78	Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₄²×C₃₁	—	2	1	no	—
86	C₄²×C₃₁	—	2	1	no	—
87	C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	#92482	C₂×C₄	1	4	yes	complemented
89	#92481	C₂×C₄	1	4	yes	complemented
90	(C₂×D₃₁)⋊C₄	—	2	1	no	—
91	(C₂×D₃₁)⋊C₄	—	2	1	no	—
92	(C₂×D₃₁)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₃₁	—	2	1	no	—
94	C₄⋊Dic₃₁	—	2	1	no	—
95	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
97	Dic₃₁⋊C₄	—	2	1	no	—
98	Dic₃₁⋊C₄	—	2	1	no	—
99	Dic₃₁⋊C₄	—	2	1	no	—
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	—	2	1	no	—
102	C₄×Dic₃₁	—	2	1	no	—
103	C₄×Dic₃₁	—	2	1	no	—
104	C₄×Dic₃₁	—	2	1	no	—
105	C₄×Dic₃₁	D₄	1	1	yes	characteristic
106	C₄×Dic₃₁	D₄	1	1	yes	characteristic
107	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
110	#92470	C₂×C₄	1	4	yes	complemented
111	#92469	—	2	2	no	—
112	#92469	C₂×C₄	1	4	yes	complemented
113	C₂²×D₃₁	—	2	1	no	—
114	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₃₁	—	2	1	no	—
116	Q₈×C₃₁	—	2	1	no	—
117	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
118	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₃₁	—	2	1	no	—
127	C₂×Dic₃₁	—	2	1	no	—
128	C₂×Dic₃₁	—	2	1	no	—
129	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
132	D₁₂₄	—	2	4	no	—
133	D₁₂₄	—	2	1	no	—
134	D₁₂₄	C₂²×C₄	1	2	yes	—
135	C₄×D₃₁	—	2	4	no	—
136	C₄×D₃₁	—	2	4	no	—
137	C₄×D₃₁	—	2	2	no	—
138	C₄×D₃₁	—	4	1	no	—
139	C₄×D₃₁	—	4	2	no	—
140	C₄×D₃₁	—	4	2	no	—
141	C₄×D₃₁	—	4	2	no	—
142	C₄×D₃₁	C₂²×C₄	1	2	yes	—
143	C₄×D₃₁	#41	1	2	yes	—
144	Dic₆₂	—	2	1	no	—
145	Dic₆₂	—	2	4	no	—
146	Dic₆₂	C₂²×C₄	1	2	yes	—
147	C₈×C₃₁	—	2	1	no	—
148	C₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₃₁⋊C₈	—	2	1	no	—
150	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
151	C₄×#137	—	31	1	no	maximal, Sylow
152	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₃₁	—	2	1	no	—
154	C₂×D₃₁	—	2	1	no	—
155	C₂×D₃₁	—	2	2	no	—
156	C₂×D₃₁	—	4	1	no	—
157	C₂×D₃₁	—	4	1	no	—
158	C₂×D₃₁	C₄×D₄	1	2	yes	—
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	4	1	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
166	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
167	Dic₃₁	—	2	2	no	—
168	Dic₃₁	—	2	1	no	—
169	Dic₃₁	—	4	1	no	—
170	Dic₃₁	—	4	1	no	—
171	Dic₃₁	C₄×D₄	1	2	yes	—
172	Dic₃₁	#137	1	2	yes	complemented
173	C₂×#137	—	31	1	no	—
174	C₄×#41	—	31	1	no	—
175	C₂×C₄×Q₈	—	31	1	no	—
176	#441	—	31	1	no	—
177	#441	D₃₁	1	1	yes	characteristic, complemented
178	#440	—	31	1	no	—
179	#440	—	31	1	no	—
180	C₄×Q₁₆	—	31	1	no	—
181	C₄×Q₁₆	—	31	1	no	—
182	C₄×SD₁₆	—	31	1	no	—
183	C₄×SD₁₆	—	31	1	no	—
184	#428	—	31	1	no	—
185	#419	—	31	1	no	—
186	#417	—	31	1	no	—
187	C₄×M₁₆	—	31	1	no	—
188	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
190	C₂×C₃₁	#575	1	1	yes	characteristic
191	D₃₁	—	2	2	no	—
192	D₃₁	—	4	2	no	—
193	C₂×#41	—	31	1	no	—
194	C₂²×Q₈	—	31	1	no	—
195	#137	—	31	8	no	—
196	C₂×Q₁₆	—	31	1	no	—
197	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
198	C₂×SD₁₆	—	31	1	no	—
199	C₂×SD₁₆	—	31	1	no	—
200	C₂×M₁₆	—	31	1	no	—
201	C₂×M₁₆	—	31	1	no	—
202	C₄×Q₈	—	31	1	no	—
203	C₄×Q₈	—	31	1	no	—
204	C₄×Q₈	—	31	1	no	—
205	C₄×Q₈	—	31	1	no	—
206	C₄×Q₈	—	62	1	no	—
207	C₄×Q₈	—	62	1	no	—
208	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
209	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
210	C₄×D₄	—	31	1	no	—
211	C₄×D₄	—	31	1	no	—
212	C₄×D₄	—	62	1	no	—
213	C₄×D₄	—	62	1	no	—
214	#117	—	31	1	no	—
215	#117	—	62	1	no	—
216	C₂×C₄⋊C₄	—	31	1	no	—
217	C₂×C₄⋊C₄	—	62	1	no	—
218	C₂×C₄²	—	31	1	no	—
219	C₂×C₄²	—	62	1	no	—
220	C₂×C₄²	—	62	1	no	—
221	#107	—	31	1	no	—
222	#107	—	31	1	no	—
223	#106	—	31	1	no	—
224	#106	C₂×D₃₁	1	1	yes	characteristic
225	#103	—	31	1	no	—
226	#103	—	31	1	no	—
227	#103	—	31	1	no	—
228	#103	—	31	1	no	—
229	#103	C₂×D₃₁	1	1	yes	characteristic
230	#103	C₂×D₃₁	1	1	yes	characteristic
231	#102	—	31	1	no	—
232	#102	—	31	1	no	—
233	#97	—	31	1	no	—
234	#97	C₂×D₃₁	1	1	yes	characteristic
235	C₄×C₈	—	31	1	no	—
236	C₄×C₈	—	31	1	no	—
237	C₃₁	C₄×#137	1	1	yes	characteristic, complemented, Sylow
238	#41	—	31	4	no	—
239	#41	—	62	2	no	—
240	C₂×Q₈	—	31	4	no	—
241	C₂×Q₈	—	31	1	no	—
242	C₂×Q₈	—	62	1	no	—
243	C₂×Q₈	—	62	2	no	—
244	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
246	C₂×D₄	—	31	1	no	—
247	C₂×D₄	—	62	1	no	—
248	C₂²×C₄	—	31	1	no	—
249	C₂²×C₄	—	31	1	no	—
250	C₂²×C₄	—	31	1	no	—
251	C₂²×C₄	—	62	1	no	—
252	C₂²×C₄	—	62	1	no	—
253	C₂²×C₄	—	62	1	no	—
254	C₂²×C₄	—	62	1	no	—
255	C₂²×C₄	—	62	1	no	—
256	Q₁₆	—	31	4	no	—
257	Q₁₆	C₄×D₃₁	1	4	yes	complemented
258	SD₁₆	—	31	4	no	—
259	SD₁₆	—	31	4	no	—
260	M₁₆	—	31	4	no	—
261	M₁₆	—	62	2	no	—
262	C₂×C₈	—	31	1	no	—
263	C₂×C₈	—	31	1	no	—
264	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	31	1	no	—
269	C₄⋊C₄	—	31	1	no	—
270	C₄⋊C₄	—	62	1	no	—
271	C₄⋊C₄	—	62	1	no	—
272	C₄⋊C₄	—	62	1	no	—
273	C₄⋊C₄	—	62	1	no	—
274	C₄⋊C₄	—	62	1	no	—
275	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
277	C₂²⋊C₄	—	62	1	no	—
278	C₂²⋊C₄	—	62	1	no	—
279	C₂²⋊C₄	—	62	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	31	1	no	—
283	C₄²	—	31	1	no	—
284	C₄²	—	31	1	no	—
285	C₄²	—	62	1	no	—
286	C₄²	—	62	1	no	—
287	C₄²	—	62	1	no	—
288	C₄²	—	62	1	no	—
289	C₄²	—	62	1	no	—
290	C₄²	C₂²×D₃₁	1	1	yes	characteristic
291	C₂³	—	31	1	no	—
292	C₂³	—	62	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	31	2	no	—
296	Q₈	—	62	4	no	—
297	Q₈	—	62	1	no	—
298	Q₈	C₂×C₄×D₃₁	1	2	yes	—
299	Q₈	C₂×C₄×D₃₁	1	2	yes	—
300	D₄	—	31	2	no	—
301	D₄	—	62	4	no	—
302	D₄	—	62	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	31	1	no	—
308	C₂×C₄	—	31	1	no	—
309	C₂×C₄	—	31	2	no	—
310	C₂×C₄	—	31	2	no	—
311	C₂×C₄	—	31	1	no	—
312	C₂×C₄	—	62	2	no	—
313	C₂×C₄	—	62	4	no	—
314	C₂×C₄	—	62	4	no	—
315	C₂×C₄	—	62	1	no	—
316	C₂×C₄	—	62	1	no	—
317	C₂×C₄	—	62	1	no	—
318	C₂×C₄	—	124	1	no	—
319	C₂×C₄	—	124	2	no	—
320	C₂×C₄	—	124	2	no	—
321	C₂×C₄	—	124	2	no	—
322	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
323	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	31	2	no	—
327	C₈	—	62	1	no	—
328	C₈	C₂×C₄×D₃₁	1	2	yes	—
329	C₂²	—	31	2	no	—
330	C₂²	—	62	1	no	—
331	C₂²	—	62	1	no	—
332	C₂²	—	62	2	no	—
333	C₂²	—	124	1	no	—
334	C₂²	—	124	1	no	—
335	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	31	2	no	—
343	C₄	—	31	2	no	—
344	C₄	—	62	2	no	—
345	C₄	—	62	1	no	—
346	C₄	—	124	1	no	—
347	C₄	—	124	1	no	—
348	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
349	C₄	#41×D₃₁	1	1	yes	characteristic
350	C₂	—	62	2	no	—
351	C₂	—	124	2	no	—
352	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic
353	C₂	#137×D₃₁	1	1	yes	characteristic
354	C₂	#14521848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#441	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#92499	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#441	D₃₁
hypocenter	C₃₁	C₄×#137
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₃₁	C₂	semidirect
4. #14521154	C₂	semidirect
5. #14521151	C₂	semidirect
6. #14521147	C₂	semidirect
7. Q₁₆×Dic₃₁	C₂	semidirect
8. #14521141	C₂	semidirect
9. #14521136	C₂	semidirect
10. #14521134	C₂	semidirect
11. #14521088	C₂	semidirect
12. #14521056	C₂	semidirect
14. #14521001	C₂	semidirect
15. C₃₁×#441	C₂	semidirect
25. #11758495	C₄	semidirect
65. Q₈×D₃₁	C₂×C₄	semidirect
78. Q₁₆×C₃₁	C₂×C₄	semidirect
88. #92482	C₂×C₄	semidirect
89. #92481	C₂×C₄	semidirect
110. #92470	C₂×C₄	semidirect
112. #92469	C₂×C₄	semidirect
172. Dic₃₁	#137	semidirect
177. #441	D₃₁	semidirect
237. C₃₁	C₄×#137	semidirect
257. Q₁₆	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758495	2
C₄×#92499	2
C₄×Q₈×D₃₁	2
#14521154	2
#14521151	2
#14521147	2
Q₁₆×Dic₃₁	2
#14521141	2
#14521136	2
#14521134	2
#14521088	2
#14521056	2
#14521049	2
#14521001	2
C₃₁×#441	2
C₄×#137	31