Subgroups — #16854592

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 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	#14521529	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521159	C₂	1	2	yes	maximal, complemented
4	#14521154	C₂	1	2	yes	maximal, complemented
5	C₄×#92481	C₂	1	2	yes	maximal, complemented
6	C₄×#92480	C₂	1	2	yes	maximal, complemented
7	C₂×#11758396	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521050	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521048	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92495	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₄×D₄×C₃₁	—	2	2	no	—
16	C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#117	—	2	1	no	—
18	C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	#11758527	C₄	1	8	yes	complemented
22	C₂×#92482	C₂²	1	1	yes	characteristic
23	C₂×#92481	C₂²	1	1	yes	characteristic
24	C₂×#92480	C₂²	1	1	yes	characteristic
25	C₂×#92479	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
27	C₂×#92474	C₂²	1	1	yes	characteristic
28	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
29	#11758469	—	2	1	no	—
30	C₄⋊C₄×D₃₁	—	2	1	no	—
31	(C₂²⋊D₃₁)⋊C₄	—	2	2	no	—
32	C₄×D₁₂₄	C₂²	1	2	yes	—
33	C₄×Dic₆₂	C₂²	1	2	yes	—
34	#11758423	C₂²	1	2	yes	—
35	#11758420	C₂²	1	2	yes	—
36	#11758398	C₂²	1	2	yes	—
37	#11758397	C₂²	1	2	yes	—
38	#11758396	C₂²	1	2	yes	—
39	#11758396	C₂²	1	2	yes	—
40	#11758391	C₂²	1	2	yes	—
41	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
42	#41×C₃₁	—	2	2	no	—
43	#41×C₃₁	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	—	2	1	no	—
46	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₃₁	—	2	1	no	—
52	C₂²×Dic₃₁	—	2	1	no	—
53	#92495	—	2	2	no	—
54	#92495	C₂×C₄	1	4	yes	complemented
55	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₃₁	—	2	1	no	—
57	C₂×C₄×D₃₁	—	2	2	no	—
58	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	C₂³	1	2	yes	—
68	#92482	C₂×C₄	1	4	yes	complemented
69	#92481	C₂×C₄	1	4	yes	complemented
70	#92480	C₂×C₄	1	4	yes	complemented
71	#92479	C₂×C₄	1	4	yes	complemented
72	(C₂×D₃₁)⋊C₄	—	2	2	no	—
73	C₄⋊Dic₃₁	—	2	1	no	—
74	C₄⋊Dic₃₁	C₂³	1	2	yes	—
75	Dic₃₁⋊C₄	—	2	2	no	—
76	C₄×Dic₃₁	—	2	2	no	—
77	#92474	—	2	2	no	—
78	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	complemented
79	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
80	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₃₁	—	2	1	no	—
85	Q₈×C₃₁	—	2	1	no	—
86	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
87	D₄×C₃₁	—	2	4	no	—
88	D₄×C₃₁	—	2	1	no	—
89	D₄×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	8	2	no	—
94	C₂×C₄×C₃₁	—	8	1	no	—
95	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	#41	1	2	yes	—
100	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₃₁	—	2	4	no	—
102	C₂×Dic₃₁	—	2	1	no	—
103	C₂×Dic₃₁	—	2	2	no	—
104	C₂×Dic₃₁	—	8	1	no	—
105	D₁₂₄	—	2	1	no	—
106	D₁₂₄	C₂²×C₄	1	2	yes	—
107	C₄×D₃₁	—	2	4	no	—
108	C₄×D₃₁	—	8	2	no	—
109	Dic₆₂	—	2	1	no	—
110	Dic₆₂	C₂²×C₄	1	2	yes	—
111	C₃₁⋊C₈	—	2	2	no	—
112	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
113	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
114	#2952	—	31	1	no	maximal, Sylow
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	4	1	no	—
119	C₂²×C₃₁	—	4	1	no	—
120	C₂²×C₃₁	C₄×D₄	1	2	yes	—
121	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×D₃₁	—	2	2	no	—
123	C₂×D₃₁	—	4	1	no	—
124	C₂×D₃₁	—	4	1	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	4	2	no	—
129	C₄×C₃₁	C₄×D₄	1	2	yes	—
130	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	Dic₃₁	—	2	2	no	—
133	Dic₃₁	—	4	2	no	—
134	C₂×#135	—	31	1	no	—
135	#520	—	31	1	no	—
136	#520	D₃₁	1	1	yes	characteristic, complemented
137	#442	—	31	2	no	—
138	#441	—	31	2	no	—
139	C₄×SD₁₆	—	31	2	no	—
140	C₄×SD₁₆	—	31	2	no	—
141	C₂×#106	—	31	1	no	—
142	#417	—	31	1	no	—
143	#417	—	31	1	no	—
144	#405	—	31	1	no	—
145	C₂×C₃₁	—	2	2	no	—
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₃₁	#577	1	1	yes	characteristic
149	C₂×C₃₁	#577	1	1	yes	characteristic
150	D₃₁	—	4	2	no	—
151	C₂×#41	—	31	1	no	—
152	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
153	#135	—	31	8	no	—
154	C₂×Q₁₆	—	31	1	no	—
155	C₂×SD₁₆	—	31	1	no	—
156	C₂×SD₁₆	—	31	1	no	—
157	C₂×D₈	—	31	1	no	—
158	C₂×M₁₆	—	31	1	no	—
159	C₂²×C₈	—	31	1	no	—
160	C₄×Q₈	—	31	2	no	—
161	C₄×Q₈	C₂×D₃₁	1	2	yes	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	31	2	no	—
164	C₄×D₄	—	62	2	no	—
165	C₄×D₄	C₂×D₃₁	1	2	yes	—
166	#117	—	2	1	no	—
167	#117	—	62	1	no	—
168	#117	C₂×D₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	31	1	no	—
171	C₂×C₄⋊C₄	—	62	1	no	—
172	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
173	#106	—	31	2	no	—
174	#106	—	31	2	no	—
175	#103	—	31	2	no	—
176	#103	—	31	2	no	—
177	#102	—	31	2	no	—
178	#102	—	31	2	no	—
179	#97	—	31	2	no	—
180	C₄×C₈	—	31	2	no	—
181	C₃₁	#2952	1	1	yes	characteristic, complemented, Sylow
182	#41	—	2	2	no	—
183	#41	—	31	4	no	—
184	#41	—	62	2	no	—
185	#41	C₄×D₃₁	1	4	yes	complemented
186	C₂×Q₈	—	31	1	no	—
187	C₂×Q₈	C₂²×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	C₂²×D₃₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	62	1	no	—
196	C₂²×C₄	—	62	1	no	—
197	C₂²×C₄	—	62	2	no	—
198	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
199	Q₁₆	—	31	4	no	—
200	SD₁₆	—	31	4	no	—
201	SD₁₆	—	31	4	no	—
202	D₈	—	31	4	no	—
203	M₁₆	—	62	2	no	—
204	C₂×C₈	—	31	4	no	—
205	C₂×C₈	—	31	1	no	—
206	C₂×C₈	—	31	1	no	—
207	C₂×C₈	—	31	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	31	2	no	—
211	C₄⋊C₄	—	62	2	no	—
212	C₄⋊C₄	—	62	1	no	—
213	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	62	2	no	—
218	C₂²⋊C₄	C₂²⋊D₃₁	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	62	2	no	—
221	C₄²	C₂²×D₃₁	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	62	1	no	—
224	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	31	2	no	—
227	Q₈	—	62	1	no	—
228	Q₈	C₂×C₄×D₃₁	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	31	2	no	—
232	D₄	—	62	4	no	—
233	D₄	—	62	1	no	—
234	D₄	C₂×C₄×D₃₁	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	62	1	no	—
241	C₂×C₄	—	62	4	no	—
242	C₂×C₄	—	62	2	no	—
243	C₂×C₄	—	248	2	no	—
244	C₂×C₄	—	248	1	no	—
245	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
246	C₂×C₄	#92495	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
251	C₈	—	31	2	no	—
252	C₈	—	31	2	no	—
253	C₈	—	62	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	62	2	no	—
260	C₂²	—	124	1	no	—
261	C₂²	—	124	1	no	—
262	C₂²	C₄×C₂²⋊D₃₁	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	62	2	no	—
269	C₄	—	124	2	no	—
270	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
271	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
272	C₄	C₂×#92495	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	124	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₃₁	1	1	yes	characteristic
277	C₂	#14521857	1	1	yes	characteristic
278	C₂	#14521860	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#520	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#11758527	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#520	D₃₁
hypocenter	C₃₁	#2952
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14521529	C₂	semidirect
3. #14521159	C₂	semidirect
4. #14521154	C₂	semidirect
5. C₄×#92481	C₂	semidirect
6. C₄×#92480	C₂	semidirect
7. C₂×#11758396	C₂	semidirect
8. #14521050	C₂	semidirect
9. #14521048	C₂	semidirect
10. #14521005	C₂	semidirect
11. C₃₁×#520	C₂	semidirect
21. #11758527	C₄	semidirect
43. #41×C₃₁	C₂×C₄	semidirect
54. #92495	C₂×C₄	semidirect
68. #92482	C₂×C₄	semidirect
69. #92481	C₂×C₄	semidirect
70. #92480	C₂×C₄	semidirect
71. #92479	C₂×C₄	semidirect
78. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
136. #520	D₃₁	semidirect
181. C₃₁	#2952	semidirect
185. #41	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×#11758527	2
#14521529	2
#14521159	2
#14521154	2
C₄×#92481	2
C₄×#92480	2
C₂×#11758396	2
#14521050	2
#14521048	2
#14521005	2
C₃₁×#520	2
#2952	31