Subgroups — #16855855

Order 3968.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521594	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521374	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521370	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521332	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521327	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521247	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521242	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521194	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521161	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521158	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521152	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521150	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#449	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×#92497	—	2	1	no	—
18	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#121	C₂²	1	1	yes	characteristic
21	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
22	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
23	C₂×#92481	C₂²	1	1	yes	characteristic
24	#11758517	C₂²	1	1	yes	characteristic
25	#11758516	—	2	1	no	—
26	D₄×Dic₃₁	—	2	1	no	—
27	D₄×Dic₃₁	—	2	1	no	—
28	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
31	C₂×#92479	C₂²	1	1	yes	characteristic
32	#11758507	C₂²	1	1	yes	characteristic
33	#11758503	—	2	1	no	—
34	#11758503	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₃₁	—	2	1	no	—
36	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
37	#11758470	C₂²	1	1	yes	characteristic
38	#11758465	C₂²	1	1	yes	characteristic
39	#11758462	—	2	1	no	—
40	(C₂²⋊D₃₁)⋊C₄	—	2	1	no	—
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#11758423	C₂²	1	1	yes	characteristic
45	#11758420	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758420	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
49	#11758409	C₂²	1	1	yes	characteristic
50	#11758409	C₂²	1	1	yes	characteristic
51	#11758406	C₂²	1	1	yes	characteristic
52	#11758405	C₂²	1	1	yes	characteristic
53	#11758403	C₂²	1	1	yes	characteristic
54	#11758403	C₂²	1	1	yes	characteristic
55	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
56	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	#41×C₃₁	—	2	2	no	—
60	#41×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₃₁	—	2	1	no	—
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₃₁	—	2	1	no	—
69	C₂×C₂²⋊D₃₁	—	2	1	no	—
70	C₂×C₂²⋊D₃₁	—	2	1	no	—
71	C₂²×Dic₃₁	—	2	1	no	—
72	C₂²×Dic₃₁	—	2	1	no	—
73	C₂²×Dic₃₁	—	2	1	no	—
74	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
75	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
76	#92497	—	4	4	no	—
77	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₃₁	—	2	1	no	—
79	C₂×Dic₆₂	—	2	1	no	—
80	SD₁₆×C₃₁	—	2	2	no	—
81	D₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂²⋊Dic₃₁	—	2	1	no	—
87	C₂²⋊Dic₃₁	—	2	1	no	—
88	C₂²⋊Dic₃₁	—	2	1	no	—
89	C₂²⋊Dic₃₁	—	2	1	no	—
90	#92481	—	2	2	no	—
91	#92479	—	2	2	no	—
92	(C₂×D₃₁)⋊C₄	—	2	1	no	—
93	(C₂×D₃₁)⋊C₄	—	2	1	no	—
94	C₄⋊Dic₃₁	—	2	1	no	—
95	C₄⋊Dic₃₁	—	2	1	no	—
96	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
104	C₄×Dic₃₁	D₄	1	1	yes	characteristic
105	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
106	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
109	C₂³×C₃₁	—	2	1	no	—
110	C₂³×C₃₁	—	2	1	no	—
111	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₃₁	—	2	1	no	—
113	Q₈×C₃₁	—	2	1	no	—
114	Q₈×C₃₁	#41	1	2	yes	complemented
115	D₄×C₃₁	—	2	4	no	—
116	D₄×C₃₁	—	2	1	no	—
117	D₄×C₃₁	—	2	1	no	—
118	D₄×C₃₁	—	2	1	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	#41	1	2	yes	complemented
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₃₁	—	4	2	no	—
129	C₂²⋊D₃₁	—	4	2	no	—
130	C₂²⋊D₃₁	—	4	2	no	—
131	C₂×Dic₃₁	—	2	2	no	—
132	C₂×Dic₃₁	—	2	1	no	—
133	C₂×Dic₃₁	—	2	1	no	—
134	C₂×Dic₃₁	—	2	1	no	—
135	C₂×Dic₃₁	—	4	1	no	—
136	C₂×Dic₃₁	—	4	2	no	—
137	C₂×Dic₃₁	—	4	2	no	—
138	C₂×Dic₃₁	—	4	2	no	—
139	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
143	D₁₂₄	—	2	1	no	—
144	D₁₂₄	—	2	1	no	—
145	C₄×D₃₁	—	4	2	no	—
146	Dic₆₂	—	4	2	no	—
147	C₈×C₃₁	—	2	1	no	—
148	C₈×C₃₁	—	2	1	no	—
149	C₃₁⋊C₈	—	2	1	no	—
150	C₃₁⋊C₈	—	2	1	no	—
151	#3115	—	31	1	no	maximal, Sylow
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	—	4	1	no	—
158	C₂²×C₃₁	—	4	1	no	—
159	C₂²×C₃₁	—	4	1	no	—
160	C₂²×C₃₁	—	4	1	no	—
161	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₃₁	—	4	1	no	—
163	C₂×D₃₁	—	4	1	no	—
164	C₂×D₃₁	—	4	1	no	—
165	C₄×C₃₁	—	2	1	no	—
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	4	1	no	—
168	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
169	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	Dic₃₁	—	2	1	no	—
171	Dic₃₁	—	2	1	no	—
172	Dic₃₁	—	2	1	no	—
173	Dic₃₁	—	4	1	no	—
174	Dic₃₁	—	4	1	no	—
175	Dic₃₁	—	4	1	no	—
176	Dic₃₁	AGL(1,ℤ/8)	1	2	yes	—
177	#532	—	31	1	no	—
178	C₄×#41	—	31	1	no	—
179	#482	—	31	1	no	—
180	#482	—	31	1	no	—
181	#475	—	31	1	no	—
182	#474	—	31	1	no	—
183	#460	—	31	1	no	—
184	#459	—	31	1	no	—
185	#449	—	31	1	no	—
186	#449	D₃₁	1	1	yes	characteristic, complemented
187	#442	—	31	1	no	—
188	#442	—	31	1	no	—
189	#440	—	31	1	no	—
190	#440	—	31	1	no	—
191	#432	—	31	1	no	—
192	C₂×C₃₁	—	4	1	no	—
193	C₂×C₃₁	—	4	2	no	—
194	C₂×C₃₁	—	8	1	no	—
195	C₂×C₃₁	#525	1	1	yes	characteristic
196	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂×C₃₁	#575	1	1	yes	characteristic
198	D₃₁	—	8	1	no	—
199	C₂×#41	—	62	1	no	—
200	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
201	C₂×SD₁₆	—	31	1	no	—
202	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
203	C₂×D₈	—	31	1	no	—
204	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
205	C₄⋊Q₈	—	31	1	no	—
206	#127	—	31	1	no	—
207	#124	—	62	1	no	—
208	#121	—	31	1	no	—
209	#121	—	62	1	no	—
210	#121	C₂×D₃₁	1	1	yes	characteristic
211	C₄×Q₈	—	31	1	no	—
212	C₄×Q₈	—	31	1	no	—
213	C₄×D₄	—	31	1	no	—
214	C₄×D₄	—	31	1	no	—
215	C₄×D₄	—	31	1	no	—
216	C₄×D₄	—	31	1	no	—
217	C₄×D₄	—	62	1	no	—
218	C₄×D₄	—	62	1	no	—
219	C₄×D₄	—	62	1	no	—
220	#117	—	31	1	no	—
221	#117	—	62	1	no	—
222	C₂×C₄²	—	31	1	no	—
223	C₂×C₄²	—	62	1	no	—
224	#107	—	31	1	no	—
225	#107	—	31	1	no	—
226	#106	—	31	1	no	—
227	#106	—	31	1	no	—
228	C₄⋊C₈	—	31	1	no	—
229	C₄⋊C₈	—	31	1	no	—
230	#103	—	31	1	no	—
231	#103	C₂×D₃₁	1	1	yes	characteristic
232	#102	—	31	1	no	—
233	#102	—	31	1	no	—
234	#102	—	31	1	no	—
235	#102	—	31	1	no	—
236	#102	—	31	1	no	—
237	#102	C₂×D₃₁	1	1	yes	characteristic
238	C₂²⋊C₈	—	31	1	no	—
239	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
240	#97	—	31	1	no	—
241	#97	—	31	1	no	—
242	C₃₁	#3115	1	1	yes	characteristic, complemented, Sylow
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	124	4	no	—
246	C₂×Q₈	—	62	1	no	—
247	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	31	1	no	—
251	C₂×D₄	—	62	1	no	—
252	C₂×D₄	—	62	1	no	—
253	C₂×D₄	—	62	1	no	—
254	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	31	1	no	—
258	C₂²×C₄	—	31	1	no	—
259	C₂²×C₄	—	62	1	no	—
260	C₂²×C₄	—	62	1	no	—
261	C₂²×C₄	—	62	1	no	—
262	C₂²×C₄	—	62	1	no	—
263	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	62	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	62	2	no	—
268	C₂×C₈	—	31	1	no	—
269	C₂×C₈	—	31	1	no	—
270	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
272	C₄⋊C₄	—	31	1	no	—
273	C₄⋊C₄	—	31	1	no	—
274	C₄⋊C₄	—	31	1	no	—
275	C₄⋊C₄	—	62	1	no	—
276	C₄⋊C₄	—	62	1	no	—
277	C₄⋊C₄	—	62	1	no	—
278	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	62	1	no	—
281	C₂²⋊C₄	—	62	1	no	—
282	C₂²⋊C₄	—	62	1	no	—
283	C₂²⋊C₄	—	62	1	no	—
284	C₂²⋊C₄	—	62	1	no	—
285	C₂²⋊C₄	—	62	1	no	—
286	C₄²	—	31	1	no	—
287	C₄²	—	31	1	no	—
288	C₄²	—	31	1	no	—
289	C₄²	—	31	1	no	—
290	C₄²	—	62	1	no	—
291	C₄²	—	62	1	no	—
292	C₄²	—	62	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	62	1	no	—
296	C₂³	D₄×D₃₁	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	124	2	no	—
299	Q₈	#92497	1	2	yes	complemented
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	62	1	no	—
306	D₄	—	62	1	no	—
307	D₄	—	124	2	no	—
308	D₄	—	124	2	no	—
309	D₄	—	124	2	no	—
310	D₄	#92497	1	2	yes	complemented
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	31	1	no	—
317	C₂×C₄	—	31	1	no	—
318	C₂×C₄	—	31	1	no	—
319	C₂×C₄	—	31	1	no	—
320	C₂×C₄	—	62	2	no	—
321	C₂×C₄	—	62	1	no	—
322	C₂×C₄	—	62	1	no	—
323	C₂×C₄	—	62	1	no	—
324	C₂×C₄	—	124	2	no	—
325	C₂×C₄	—	124	2	no	—
326	C₂×C₄	—	124	1	no	—
327	C₂×C₄	—	124	2	no	—
328	C₂×C₄	—	124	2	no	—
329	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	62	1	no	—
334	C₈	—	62	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	124	1	no	—
345	C₂²	—	124	1	no	—
346	C₂²	—	124	1	no	—
347	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	31	2	no	—
352	C₄	—	62	1	no	—
353	C₄	—	62	1	no	—
354	C₄	—	62	1	no	—
355	C₄	—	124	1	no	—
356	C₄	—	124	1	no	—
357	C₄	—	124	1	no	—
358	C₄	C₂×#92497	1	1	yes	characteristic
359	C₄	C₂×#92497	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	248	1	no	—
364	C₂	#14521555	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₃₁	1	1	yes	characteristic
366	C₂	#14521846	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#449	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	#449	D₃₁
hypocenter	C₃₁	#3115
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #14521594	C₂	semidirect
2. #41×Dic₃₁	C₂	semidirect
3. #14521374	C₂	semidirect
4. #14521370	C₂	semidirect
5. #14521332	C₂	semidirect
6. #14521327	C₂	semidirect
7. #14521247	C₂	semidirect
8. #14521242	C₂	semidirect
9. #14521194	C₂	semidirect
10. #14521161	C₂	semidirect
11. #14521158	C₂	semidirect
12. #14521152	C₂	semidirect
13. #14521150	C₂	semidirect
14. #14521103	C₂	semidirect
15. C₃₁×#449	C₂	semidirect
114. Q₈×C₃₁	#41	semidirect
120. D₄×C₃₁	#41	semidirect
186. #449	D₃₁	semidirect
242. C₃₁	#3115	semidirect
299. Q₈	#92497	semidirect
310. D₄	#92497	semidirect

Maximal subgroups

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

subgroup	index
#14521594	2
#41×Dic₃₁	2
#14521374	2
#14521370	2
#14521332	2
#14521327	2
#14521247	2
#14521242	2
#14521194	2
#14521161	2
#14521158	2
#14521152	2
#14521150	2
#14521103	2
C₃₁×#449	2
#3115	31