Subgroups — #16854623

Order 3968.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521814	C₂	1	2	yes	maximal, complemented
3	#14521169	C₂	1	2	yes	maximal, complemented
4	#14521168	C₂	1	2	yes	maximal, complemented
5	#14521167	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521166	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521165	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521064	C₂	1	2	yes	maximal, complemented
10	#14521003	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₃₁	C₂²	1	2	yes	—
13	C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₃₁×#131	C₂²	1	2	yes	complemented
15	#11758527	C₂²	1	2	yes	—
16	#11758525	C₂²	1	2	yes	complemented
17	#11758496	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₃₁	C₄	1	1	yes	characteristic, complemented
19	#11758493	C₄	1	2	yes	complemented
20	SD₁₆×D₃₁	C₄	1	2	yes	complemented
21	#11758489	C₄	1	1	yes	characteristic, complemented
22	D₈×D₃₁	C₄	1	1	yes	characteristic, complemented
23	#11758484	—	2	2	no	—
24	#11758484	C₂²	1	2	yes	complemented
25	M₁₆×D₃₁	—	2	2	no	—
26	M₁₆×D₃₁	C₂²	1	2	yes	—
27	#11758481	C₂²	1	1	yes	characteristic
28	#11758478	—	2	2	no	—
29	#11758478	C₂²	1	2	yes	complemented
30	C₂×#92469	—	2	2	no	—
31	C₂×#92469	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
33	#11758449	C₂²	1	1	yes	characteristic
34	C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#104	C₂²	1	2	yes	complemented
36	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
37	#11758425	C₂²	1	2	yes	complemented
38	#11758413	C₂²	1	2	yes	complemented
39	#11758410	C₂²	1	2	yes	complemented
40	#11758407	C₂²	1	1	yes	characteristic, complemented
41	#11758403	C₂²	1	1	yes	characteristic, complemented
42	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
43	#11758393	C₂²	1	2	yes	complemented
44	#11758391	C₂²	1	1	yes	characteristic, complemented
45	#41×C₃₁	C₂³	1	2	yes	—
46	#92499	C₂×C₄	1	2	yes	—
47	Q₈×D₃₁	C₂×C₄	1	2	yes	—
48	#92497	—	2	2	no	—
49	#92497	C₂×C₄	1	2	yes	—
50	D₄×D₃₁	—	2	2	no	—
51	D₄×D₃₁	C₂×C₄	1	2	yes	—
52	#92495	—	2	2	no	—
53	#92495	C₂³	1	2	yes	—
54	C₂×C₄×D₃₁	—	2	2	no	—
55	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
56	Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
57	SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
58	D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
59	M₁₆×C₃₁	—	2	2	no	—
60	M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	#92482	C₂×C₄	1	2	yes	—
66	#92481	C₂×C₄	1	2	yes	—
67	#92480	C₂×C₄	1	2	yes	—
68	#92479	C₂×C₄	1	2	yes	—
69	(C₂×D₃₁)⋊C₄	—	2	1	no	—
70	Dic₃₁⋊C₄	—	2	1	no	—
71	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
72	#92474	—	2	2	no	—
73	#92474	C₂³	1	2	yes	—
74	C₂×C₃₁⋊C₈	—	2	2	no	—
75	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
77	Dic₁₂₄	C₂×C₄	1	1	yes	characteristic
78	D₂₄₈	C₂×C₄	1	1	yes	characteristic
79	#92470	C₂×C₄	1	2	yes	—
80	#92469	—	2	2	no	—
81	#92469	—	2	2	no	—
82	#92469	—	2	2	no	—
83	#92469	D₄	1	1	yes	characteristic
84	#92469	D₄	1	1	yes	characteristic
85	#92469	D₄	1	1	yes	characteristic
86	#92469	D₄	1	1	yes	characteristic
87	C₈×D₃₁	—	2	2	no	—
88	C₈×D₃₁	—	2	2	no	—
89	C₈×D₃₁	—	2	2	no	—
90	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
91	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
92	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
93	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₃₁	—	2	2	no	—
95	C₂²×D₃₁	#41	1	1	yes	characteristic
96	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
97	D₄×C₃₁	—	2	2	no	—
98	D₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₃₁	—	2	2	no	—
103	C₂²⋊D₃₁	—	2	2	no	—
104	C₂²⋊D₃₁	—	2	2	no	—
105	C₂×Dic₃₁	—	2	2	no	—
106	C₂×Dic₃₁	—	2	1	no	—
107	C₂×Dic₃₁	#41	1	1	yes	characteristic
108	D₁₂₄	—	2	2	no	—
109	D₁₂₄	C₂²×C₄	1	2	yes	—
110	C₄×D₃₁	—	2	2	no	—
111	C₄×D₃₁	—	2	2	no	—
112	C₄×D₃₁	—	2	2	no	—
113	C₄×D₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄×D₃₁	C₂×D₄	1	1	yes	characteristic
117	Dic₆₂	—	2	2	no	—
118	Dic₆₂	C₂²×C₄	1	2	yes	—
119	C₈×C₃₁	—	2	2	no	—
120	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₃₁⋊C₈	—	2	2	no	—
125	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
128	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
129	#2957	—	31	1	no	maximal, Sylow
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₂×D₃₁	—	2	2	no	—
133	C₂×D₃₁	—	4	1	no	—
134	C₂×D₃₁	—	8	2	no	—
135	C₂×D₃₁	C₄×D₄	1	1	yes	characteristic
136	C₂×D₃₁	C₄×D₄	1	1	yes	characteristic
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	4	1	no	—
139	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	Dic₃₁	—	2	2	no	—
142	Dic₃₁	—	4	1	no	—
143	Dic₃₁	C₄×D₄	1	1	yes	characteristic
144	Dic₃₁	C₄×D₄	1	1	yes	characteristic
145	C₂×#135	—	31	1	no	—
146	#568	—	31	2	no	—
147	#444	—	31	2	no	—
148	#444	—	31	2	no	—
149	#443	—	31	1	no	—
150	#443	—	31	1	no	—
151	#443	—	31	1	no	—
152	#443	D₃₁	1	1	yes	characteristic, complemented
153	C₂×#108	—	31	1	no	—
154	#421	—	31	2	no	—
155	#405	—	31	1	no	—
156	C₂×C₃₁	—	2	1	no	—
157	C₂×C₃₁	—	4	2	no	—
158	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
159	D₃₁	—	2	1	no	—
160	D₃₁	—	2	1	no	—
161	D₃₁	—	4	2	no	—
162	C₂×#41	—	31	2	no	—
163	#135	—	31	2	no	—
164	#135	—	31	1	no	—
165	#135	—	31	2	no	—
166	#135	—	31	1	no	—
167	#135	—	31	1	no	—
168	#135	C₂×D₃₁	1	1	yes	characteristic
169	C₂×Q₁₆	—	31	1	no	—
170	C₂×SD₁₆	—	31	2	no	—
171	C₂×D₈	—	31	1	no	—
172	#131	—	31	2	no	—
173	#131	—	31	2	no	—
174	#131	—	31	2	no	—
175	#131	—	62	2	no	—
176	#131	—	62	2	no	—
177	#131	C₂×D₃₁	1	2	yes	complemented
178	C₂×M₁₆	—	31	1	no	—
179	C₂×M₁₆	—	31	2	no	—
180	C₂×M₁₆	—	62	2	no	—
181	C₂×M₁₆	—	62	2	no	—
182	C₂²×C₈	—	31	1	no	—
183	#117	—	31	1	no	—
184	#108	—	31	2	no	—
185	#108	—	31	1	no	—
186	#108	C₂×D₃₁	1	1	yes	characteristic, complemented
187	#104	—	31	2	no	—
188	#104	—	31	2	no	—
189	#104	—	31	2	no	—
190	#104	C₂×D₃₁	1	2	yes	complemented
191	#97	—	31	1	no	—
192	#97	—	31	1	no	—
193	C₄×C₈	—	31	1	no	—
194	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
195	C₃₁	#2957	1	1	yes	characteristic, complemented, Sylow
196	#41	—	31	2	no	—
197	#41	—	31	2	no	—
198	#41	—	31	2	no	—
199	#41	—	62	2	no	—
200	#41	—	62	2	no	—
201	#41	C₂²×D₃₁	1	2	yes	—
202	C₂×Q₈	—	31	2	no	—
203	C₂×D₄	—	31	2	no	—
204	C₂×D₄	—	62	2	no	—
205	C₂²×C₄	—	31	1	no	—
206	C₂²×C₄	—	62	2	no	—
207	Q₁₆	—	31	2	no	—
208	Q₁₆	—	31	1	no	—
209	Q₁₆	C₄×D₃₁	1	1	yes	characteristic
210	SD₁₆	—	31	2	no	—
211	SD₁₆	—	31	2	no	—
212	SD₁₆	—	31	2	no	—
213	SD₁₆	C₄×D₃₁	1	2	yes	—
214	D₈	—	31	2	no	—
215	D₈	—	31	1	no	—
216	D₈	C₄×D₃₁	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	31	1	no	—
219	M₁₆	—	31	1	no	—
220	M₁₆	—	31	1	no	—
221	M₁₆	—	31	1	no	—
222	M₁₆	—	31	2	no	—
223	M₁₆	—	62	2	no	—
224	M₁₆	—	62	2	no	—
225	M₁₆	—	62	2	no	—
226	M₁₆	—	62	2	no	—
227	M₁₆	C₂²×D₃₁	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	31	1	no	—
230	C₂×C₈	—	31	1	no	—
231	C₂×C₈	—	31	1	no	—
232	C₂×C₈	—	31	1	no	—
233	C₂×C₈	—	31	1	no	—
234	C₂×C₈	—	31	1	no	—
235	C₂×C₈	—	62	2	no	—
236	C₂×C₈	—	62	2	no	—
237	C₂×C₈	—	62	2	no	—
238	C₂×C₈	—	62	2	no	—
239	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
241	C₄⋊C₄	—	62	1	no	—
242	C₂²⋊C₄	—	62	1	no	—
243	C₄²	—	31	1	no	—
244	C₄²	C₂²×D₃₁	1	1	yes	characteristic
245	C₂³	—	31	1	no	—
246	C₂³	—	62	2	no	—
247	Q₈	—	31	2	no	—
248	Q₈	—	62	2	no	—
249	Q₈	C₂×C₄×D₃₁	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	31	2	no	—
252	D₄	—	62	2	no	—
253	D₄	—	62	2	no	—
254	D₄	—	62	2	no	—
255	D₄	—	62	2	no	—
256	D₄	C₂×C₄×D₃₁	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	31	1	no	—
260	C₂×C₄	—	31	1	no	—
261	C₂×C₄	—	31	1	no	—
262	C₂×C₄	—	31	1	no	—
263	C₂×C₄	—	31	1	no	—
264	C₂×C₄	—	62	2	no	—
265	C₂×C₄	—	62	2	no	—
266	C₂×C₄	—	62	2	no	—
267	C₂×C₄	—	62	2	no	—
268	C₂×C₄	—	62	1	no	—
269	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	31	1	no	—
272	C₈	—	31	1	no	—
273	C₈	—	31	1	no	—
274	C₈	—	31	1	no	—
275	C₈	—	62	2	no	—
276	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
277	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
278	C₈	D₄×D₃₁	1	1	yes	characteristic
279	C₈	D₄×D₃₁	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	31	1	no	—
282	C₂²	—	31	1	no	—
283	C₂²	—	62	2	no	—
284	C₂²	—	124	1	no	—
285	C₂²	—	248	2	no	—
286	C₂²	#41×D₃₁	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	31	1	no	—
290	C₄	—	31	1	no	—
291	C₄	—	62	2	no	—
292	C₄	—	124	1	no	—
293	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
294	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	62	1	no	—
298	C₂	—	62	1	no	—
299	C₂	—	124	2	no	—
300	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#443	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	#135×D₃₁	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#443	D₃₁
hypocenter	C₃₁	#2957
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #14521814	C₂	semidirect
3. #14521169	C₂	semidirect
4. #14521168	C₂	semidirect
5. #14521167	C₂	semidirect
6. #14521166	C₂	semidirect
7. #14521165	C₂	semidirect
8. #108×D₃₁	C₂	semidirect
9. #14521064	C₂	semidirect
10. #14521003	C₂	semidirect
11. C₃₁×#443	C₂	semidirect
14. C₃₁×#131	C₂²	semidirect
16. #11758525	C₂²	semidirect
17. #11758496	C₄	semidirect
18. Q₁₆×D₃₁	C₄	semidirect
19. #11758493	C₄	semidirect
20. SD₁₆×D₃₁	C₄	semidirect
21. #11758489	C₄	semidirect
22. D₈×D₃₁	C₄	semidirect
24. #11758484	C₂²	semidirect
29. #11758478	C₂²	semidirect
34. C₃₁×#108	C₂²	semidirect
35. C₃₁×#104	C₂²	semidirect
36. C₄×C₈×C₃₁	C₂²	semidirect
37. #11758425	C₂²	semidirect
38. #11758413	C₂²	semidirect
39. #11758410	C₂²	semidirect
40. #11758407	C₂²	semidirect
41. #11758403	C₂²	semidirect
42. C₈×Dic₃₁	C₂²	semidirect
43. #11758393	C₂²	semidirect
44. #11758391	C₂²	semidirect
152. #443	D₃₁	semidirect
177. #131	C₂×D₃₁	semidirect
186. #108	C₂×D₃₁	semidirect
190. #104	C₂×D₃₁	semidirect
194. C₄×C₈	C₂×D₃₁	semidirect
195. C₃₁	#2957	semidirect

Maximal subgroups

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

subgroup	index
#135×D₃₁	2
#14521814	2
#14521169	2
#14521168	2
#14521167	2
#14521166	2
#14521165	2
#108×D₃₁	2
#14521064	2
#14521003	2
C₃₁×#443	2
#2957	31