Subgroups — #16854554

Order 3968.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758519	C₂	1	1	yes	maximal, characteristic
2	#14521529	C₂	1	1	yes	maximal, characteristic
3	#14521154	C₂	1	2	yes	maximal, complemented
4	#14521147	C₂	1	2	yes	maximal, complemented
5	C₄×#92482	C₂	1	2	yes	maximal, complemented
6	C₄×#92481	C₂	1	2	yes	maximal, complemented
7	#14521092	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521043	C₂	1	1	yes	maximal, characteristic
9	C₂×#11758423	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92495	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	—	2	1	no	—
15	C₄×Q₈×C₃₁	—	2	1	no	—
16	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
17	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#117	—	2	2	no	—
19	C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×#92482	C₂²	1	2	yes	—
22	#11758519	C₄	1	8	yes	complemented
23	C₂×#92481	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	C₂×#92474	C₂²	1	1	yes	characteristic
26	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
27	#11758469	—	2	1	no	—
28	C₄⋊C₄×D₃₁	—	2	1	no	—
29	(C₂²⋊D₃₁)⋊C₄	—	2	2	no	—
30	C₄×D₁₂₄	C₂²	1	2	yes	—
31	C₄×Dic₆₂	C₂²	1	2	yes	—
32	#11758423	C₂²	1	2	yes	—
33	#11758423	C₂²	1	2	yes	—
34	#11758398	C₂²	1	1	yes	characteristic
35	#11758398	C₂²	1	1	yes	characteristic
36	#11758397	C₂²	1	1	yes	characteristic
37	#11758397	C₂²	1	1	yes	characteristic
38	#11758396	C₂²	1	2	yes	—
39	#11758395	C₂²	1	2	yes	—
40	#11758391	C₂²	1	2	yes	—
41	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	—	2	2	no	—
44	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₃₁	—	2	1	no	—
49	C₂²×Dic₃₁	—	2	1	no	—
50	#92495	—	2	2	no	—
51	#92495	C₂×C₄	1	4	yes	—
52	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₃₁	—	2	1	no	—
54	C₂×C₄×D₃₁	—	2	2	no	—
55	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₃₁	—	2	1	no	—
63	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₄²×C₃₁	—	2	2	no	—
65	C₄²×C₃₁	—	2	2	no	—
66	C₄²×C₃₁	—	2	1	no	—
67	C₄²×C₃₁	C₂³	1	2	yes	—
68	#92482	C₂×C₄	1	8	yes	complemented
69	#92481	C₂×C₄	1	8	yes	complemented
70	(C₂×D₃₁)⋊C₄	—	2	2	no	—
71	C₄⋊Dic₃₁	—	2	1	no	—
72	C₄⋊Dic₃₁	C₂³	1	2	yes	—
73	Dic₃₁⋊C₄	—	2	2	no	—
74	C₄×Dic₃₁	—	2	2	no	—
75	#92474	C₂×C₄	1	4	yes	complemented
76	C₂×C₃₁⋊C₈	—	2	2	no	—
77	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
80	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₃₁	—	2	1	no	—
82	Q₈×C₃₁	—	2	4	no	—
83	Q₈×C₃₁	—	2	2	no	—
84	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#41	1	2	yes	—
94	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₃₁	—	2	4	no	—
96	C₂×Dic₃₁	—	2	1	no	—
97	C₂×Dic₃₁	—	2	2	no	—
98	C₂×Dic₃₁	—	8	1	no	—
99	D₁₂₄	—	2	1	no	—
100	D₁₂₄	C₂²×C₄	1	2	yes	—
101	C₄×D₃₁	—	2	4	no	—
102	C₄×D₃₁	—	8	2	no	—
103	Dic₆₂	—	2	1	no	—
104	Dic₆₂	C₂²×C₄	1	2	yes	—
105	C₃₁⋊C₈	—	2	1	no	—
106	C₃₁⋊C₈	—	2	1	no	—
107	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
108	#2947	—	31	1	no	maximal, Sylow
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	C₄×D₄	1	2	yes	—
112	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₃₁	—	2	2	no	—
114	C₂×D₃₁	—	4	1	no	—
115	C₂×D₃₁	—	4	1	no	—
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	2	2	no	—
118	C₄×C₃₁	—	2	2	no	—
119	C₄×C₃₁	—	4	1	no	—
120	C₄×C₃₁	—	4	1	no	—
121	C₄×C₃₁	C₄×D₄	1	2	yes	—
122	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
124	Dic₃₁	—	2	2	no	—
125	Dic₃₁	—	4	2	no	—
126	C₂×#137	—	31	1	no	—
127	#520	—	31	1	no	—
128	#519	D₃₁	1	1	yes	characteristic, complemented
129	#441	—	31	2	no	—
130	#440	—	31	2	no	—
131	C₄×Q₁₆	—	31	2	no	—
132	C₄×SD₁₆	—	31	2	no	—
133	#428	—	31	1	no	—
134	#416	—	31	1	no	—
135	C₂×#103	—	31	1	no	—
136	#405	—	31	1	no	—
137	C₂×C₃₁	—	2	2	no	—
138	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₃₁	#577	1	1	yes	characteristic
140	C₂×C₃₁	#578	1	1	yes	characteristic
141	D₃₁	—	4	2	no	—
142	C₂×#41	—	31	1	no	—
143	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
144	#137	—	31	8	no	—
145	C₂×Q₁₆	—	31	2	no	—
146	C₂×SD₁₆	—	31	2	no	—
147	C₂×M₁₆	—	31	1	no	—
148	C₂²×C₈	—	31	1	no	—
149	C₄×Q₈	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	31	2	no	—
152	C₄×Q₈	C₂×D₃₁	1	2	yes	—
153	C₄×Q₈	C₂×D₃₁	1	2	yes	—
154	C₄×D₄	—	31	2	no	—
155	C₄×D₄	—	62	2	no	—
156	#117	—	2	2	no	—
157	#117	—	62	1	no	—
158	#117	C₂×D₃₁	1	1	yes	characteristic
159	#117	C₂×D₃₁	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	31	1	no	—
161	C₂×C₄⋊C₄	—	62	1	no	—
162	#107	—	31	2	no	—
163	#106	—	31	2	no	—
164	#103	—	31	2	no	—
165	#103	—	31	2	no	—
166	#103	—	31	1	no	—
167	#103	—	31	1	no	—
168	#102	—	31	1	no	—
169	#102	—	31	1	no	—
170	#97	—	31	2	no	—
171	C₄×C₈	—	31	2	no	—
172	C₃₁	#2947	1	1	yes	characteristic, complemented, Sylow
173	#41	—	31	4	no	—
174	#41	—	62	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	31	1	no	—
178	C₂×Q₈	C₄×D₃₁	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
180	C₂×D₄	—	31	1	no	—
181	C₂×D₄	—	62	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	62	1	no	—
184	C₂²×C₄	—	62	1	no	—
185	C₂²×C₄	—	62	2	no	—
186	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
187	Q₁₆	—	31	8	no	—
188	SD₁₆	—	31	8	no	—
189	M₁₆	—	31	4	no	—
190	C₂×C₈	—	31	2	no	—
191	C₂×C₈	—	31	1	no	—
192	C₂×C₈	—	31	1	no	—
193	C₂×C₈	—	62	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	31	2	no	—
197	C₄⋊C₄	—	62	2	no	—
198	C₄⋊C₄	—	62	1	no	—
199	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	62	2	no	—
205	C₂²⋊C₄	C₂²⋊D₃₁	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	62	2	no	—
210	C₄²	C₂²×D₃₁	1	2	yes	—
211	C₂³	—	62	1	no	—
212	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	31	2	no	—
216	Q₈	—	62	1	no	—
217	Q₈	C₂×C₄×D₃₁	1	4	yes	—
218	D₄	—	31	2	no	—
219	D₄	—	62	4	no	—
220	D₄	—	62	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	62	1	no	—
226	C₂×C₄	—	62	4	no	—
227	C₂×C₄	—	62	2	no	—
228	C₂×C₄	—	248	2	no	—
229	C₂×C₄	—	248	1	no	—
230	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
231	C₂×C₄	#92495	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
236	C₈	—	31	4	no	—
237	C₈	—	62	1	no	—
238	C₈	—	62	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	62	2	no	—
242	C₂²	—	124	1	no	—
243	C₂²	—	124	1	no	—
244	C₂²	C₄×C₂²⋊D₃₁	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	4	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	62	2	no	—
252	C₄	—	124	2	no	—
253	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
254	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
255	C₄	C₂×#92495	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	124	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₃₁	1	1	yes	characteristic
259	C₂	#14521857	1	1	yes	characteristic
260	C₂	#14521864	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#519	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#519	D₃₁
hypocenter	C₃₁	#2947
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #14521154	C₂	semidirect
4. #14521147	C₂	semidirect
5. C₄×#92482	C₂	semidirect
6. C₄×#92481	C₂	semidirect
7. #14521092	C₂	semidirect
9. C₂×#11758423	C₂	semidirect
10. #14521005	C₂	semidirect
11. C₃₁×#519	C₂	semidirect
22. #11758519	C₄	semidirect
44. C₂×Q₈×C₃₁	C₂×C₄	semidirect
68. #92482	C₂×C₄	semidirect
69. #92481	C₂×C₄	semidirect
75. #92474	C₂×C₄	semidirect
128. #519	D₃₁	semidirect
172. C₃₁	#2947	semidirect
178. C₂×Q₈	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×#11758519	2
#14521529	2
#14521154	2
#14521147	2
C₄×#92482	2
C₄×#92481	2
#14521092	2
#14521043	2
C₂×#11758423	2
#14521005	2
C₃₁×#519	2
#2947	31