Subgroups — #16854624

Order 3968.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758527	C₂	1	1	yes	maximal, characteristic
2	#14521816	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521815	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521168	C₂	1	4	yes	maximal, complemented
5	#14521163	C₂	1	4	yes	maximal, complemented
6	C₂×#11758410	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521067	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#421	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×#92495	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#117	C₂²	1	1	yes	characteristic
14	#11758527	C₄	1	2	yes	complemented
15	#11758527	C₄	1	2	yes	complemented
16	#11758527	C₂²	1	2	yes	—
17	#11758527	C₂²	1	2	yes	—
18	#11758525	—	2	2	no	—
19	#11758525	C₂²	1	4	yes	complemented
20	C₂×#92482	C₄	1	1	yes	characteristic, complemented
21	C₂×#92481	C₄	1	1	yes	characteristic, complemented
22	C₂×#92480	C₄	1	1	yes	characteristic, complemented
23	C₂×#92479	C₄	1	1	yes	characteristic, complemented
24	C₂×#92474	—	2	2	no	—
25	C₂×#92474	C₂²	1	1	yes	characteristic
26	C₂×#92474	C₂²	1	1	yes	characteristic
27	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	#11758484	C₂²	1	4	yes	complemented
29	M₁₆×D₃₁	—	2	2	no	—
30	#11758478	—	2	2	no	—
31	C₃₁×#104	C₂²	1	4	yes	complemented
32	#11758410	C₂²	1	2	yes	complemented
33	#11758410	C₂²	1	2	yes	complemented
34	#11758393	C₂²	1	4	yes	complemented
35	#11758391	C₂²	1	2	yes	complemented
36	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
37	#41×C₃₁	—	2	1	no	—
38	#41×C₃₁	—	2	1	no	—
39	#41×C₃₁	C₂×C₄	1	2	yes	—
40	#41×C₃₁	C₂³	1	2	yes	—
41	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₃₁	—	2	1	no	—
43	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₃₁	—	2	1	no	—
47	#92495	—	2	1	no	—
48	#92495	—	2	1	no	—
49	#92495	C₂×C₄	1	2	yes	—
50	#92495	C₂³	1	2	yes	—
51	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₁	—	2	1	no	—
53	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₃₁	—	2	1	no	—
55	M₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂×C₈×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂²⋊C₄×C₃₁	—	2	1	no	—
59	C₄²×C₃₁	C₂³	1	2	yes	—
60	#92482	C₂×C₄	1	2	yes	—
61	#92482	C₂×C₄	1	2	yes	—
62	#92481	C₂×C₄	1	2	yes	—
63	#92481	C₂×C₄	1	2	yes	—
64	#92480	C₂×C₄	1	2	yes	—
65	#92480	C₂×C₄	1	2	yes	—
66	#92479	C₂×C₄	1	2	yes	—
67	#92479	C₂×C₄	1	2	yes	—
68	#92474	—	2	1	no	—
69	#92474	—	2	4	no	—
70	#92474	D₄	1	2	yes	—
71	#92474	D₄	1	2	yes	—
72	#92474	C₂³	1	2	yes	—
73	C₂×C₃₁⋊C₈	—	2	1	no	—
74	C₂×C₃₁⋊C₈	—	2	4	no	—
75	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
79	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
82	#92469	—	2	4	no	—
83	C₈×D₃₁	—	2	4	no	—
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	#41	1	1	yes	characteristic
86	C₂²×D₃₁	—	2	1	no	—
87	Q₈×C₃₁	—	2	1	no	—
88	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
89	D₄×C₃₁	—	2	2	no	—
90	D₄×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	2	1	no	—
92	D₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₃₁	—	2	2	no	—
104	C₂²⋊D₃₁	—	2	2	no	—
105	C₂×Dic₃₁	—	2	1	no	—
106	D₁₂₄	—	2	1	no	—
107	D₁₂₄	C₂²×C₄	1	2	yes	—
108	C₄×D₃₁	—	2	2	no	—
109	C₄×D₃₁	—	2	2	no	—
110	Dic₆₂	—	2	1	no	—
111	Dic₆₂	C₂²×C₄	1	2	yes	—
112	C₈×C₃₁	—	2	2	no	—
113	C₃₁⋊C₈	—	2	2	no	—
114	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
118	C₃₁⋊C₈	C₂×D₄	1	4	yes	—
119	#2957	—	31	1	no	maximal, Sylow
120	C₂²×C₃₁	—	2	2	no	—
121	C₂²×C₃₁	—	4	1	no	—
122	C₂²×C₃₁	—	8	1	no	—
123	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
126	C₂×D₃₁	—	2	2	no	—
127	C₂×D₃₁	—	8	1	no	—
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	4	2	no	—
130	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	Dic₃₁	—	2	2	no	—
135	C₂×#135	—	31	1	no	—
136	#568	—	31	1	no	—
137	#568	—	31	1	no	—
138	#444	—	31	4	no	—
139	#443	—	31	4	no	—
140	C₂×#108	—	31	1	no	—
141	#421	—	31	1	no	—
142	#421	D₃₁	1	1	yes	characteristic, complemented
143	#405	—	31	1	no	—
144	C₂×C₃₁	—	2	1	no	—
145	C₂×C₃₁	—	2	1	no	—
146	C₂×C₃₁	—	2	1	no	—
147	C₂×C₃₁	—	4	2	no	—
148	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
149	D₃₁	—	4	2	no	—
150	C₂×#41	—	31	1	no	—
151	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
152	#135	—	31	2	no	—
153	#135	—	31	2	no	—
154	#135	—	31	2	no	—
155	#135	—	31	2	no	—
156	C₂×Q₁₆	—	31	1	no	—
157	C₂×SD₁₆	—	31	1	no	—
158	C₂×SD₁₆	—	31	1	no	—
159	C₂×D₈	—	31	1	no	—
160	#131	—	31	4	no	—
161	#131	—	31	4	no	—
162	#131	—	62	2	no	—
163	#131	—	62	2	no	—
164	C₂×M₁₆	—	31	1	no	—
165	C₂×M₁₆	—	31	1	no	—
166	C₂×M₁₆	—	62	2	no	—
167	C₂×M₁₆	—	62	2	no	—
168	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
169	C₂²×C₈	—	31	1	no	—
170	#117	C₂×D₃₁	1	1	yes	characteristic
171	#108	—	31	2	no	—
172	#108	—	31	2	no	—
173	#104	—	31	4	no	—
174	#104	C₂×D₃₁	1	4	yes	complemented
175	#97	—	31	2	no	—
176	C₄×C₈	—	31	2	no	—
177	C₃₁	#2957	1	1	yes	characteristic, complemented, Sylow
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	31	2	no	—
181	#41	—	31	2	no	—
182	#41	—	62	1	no	—
183	#41	—	62	1	no	—
184	#41	C₄×D₃₁	1	2	yes	—
185	#41	C₂²×D₃₁	1	2	yes	—
186	C₂×Q₈	—	31	1	no	—
187	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	31	1	no	—
190	C₂×D₄	—	62	1	no	—
191	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	62	1	no	—
194	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
195	Q₁₆	—	31	2	no	—
196	Q₁₆	—	31	2	no	—
197	SD₁₆	—	31	2	no	—
198	SD₁₆	—	31	2	no	—
199	SD₁₆	—	31	2	no	—
200	SD₁₆	—	31	2	no	—
201	D₈	—	31	2	no	—
202	D₈	—	31	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	31	2	no	—
205	M₁₆	—	31	2	no	—
206	M₁₆	—	31	2	no	—
207	M₁₆	—	62	4	no	—
208	M₁₆	—	62	4	no	—
209	M₁₆	—	62	1	no	—
210	M₁₆	C₂²×D₃₁	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	31	1	no	—
213	C₂×C₈	—	31	1	no	—
214	C₂×C₈	—	31	1	no	—
215	C₂×C₈	—	31	1	no	—
216	C₂×C₈	—	31	1	no	—
217	C₂×C₈	—	31	2	no	—
218	C₂×C₈	—	31	1	no	—
219	C₂×C₈	—	62	4	no	—
220	C₂×C₈	—	62	4	no	—
221	C₂×C₈	—	62	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	C₂²×D₃₁	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	62	1	no	—
227	C₂³	#92497	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	31	2	no	—
230	Q₈	—	62	1	no	—
231	Q₈	C₂×C₄×D₃₁	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	31	2	no	—
236	D₄	—	62	2	no	—
237	D₄	—	62	2	no	—
238	D₄	—	62	1	no	—
239	D₄	C₂×C₄×D₃₁	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	62	1	no	—
245	C₂×C₄	—	62	2	no	—
246	C₂×C₄	—	62	2	no	—
247	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
249	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
250	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
251	C₂×C₄	#92497	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	31	1	no	—
255	C₈	—	31	1	no	—
256	C₈	—	31	1	no	—
257	C₈	—	31	1	no	—
258	C₈	—	31	4	no	—
259	C₈	—	62	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	62	2	no	—
264	C₂²	—	248	1	no	—
265	C₂²	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
266	C₂²	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
267	C₂²	C₂×#92497	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	62	2	no	—
271	C₄	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
272	C₄	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
274	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	124	2	no	—
280	C₂	C₂×(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#421	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	C₂×#11758527	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#421	D₃₁
hypocenter	C₃₁	#2957
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #14521816	C₂	semidirect
3. #14521815	C₂	semidirect
4. #14521168	C₂	semidirect
5. #14521163	C₂	semidirect
6. C₂×#11758410	C₂	semidirect
7. #14521067	C₂	semidirect
8. #14521005	C₂	semidirect
9. C₃₁×#421	C₂	semidirect
14. #11758527	C₄	semidirect
15. #11758527	C₄	semidirect
19. #11758525	C₂²	semidirect
20. C₂×#92482	C₄	semidirect
21. C₂×#92481	C₄	semidirect
22. C₂×#92480	C₄	semidirect
23. C₂×#92479	C₄	semidirect
28. #11758484	C₂²	semidirect
31. C₃₁×#104	C₂²	semidirect
32. #11758410	C₂²	semidirect
33. #11758410	C₂²	semidirect
34. #11758393	C₂²	semidirect
35. #11758391	C₂²	semidirect
36. C₄×C₃₁⋊C₈	C₂²	semidirect
142. #421	D₃₁	semidirect
174. #104	C₂×D₃₁	semidirect
177. C₃₁	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758527	2
#14521816	2
#14521815	2
#14521168	2
#14521163	2
C₂×#11758410	2
#14521067	2
#14521005	2
C₃₁×#421	2
#2957	31