Subgroups — #16854581

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758489	C₂	1	1	yes	maximal, characteristic
2	#14521528	C₂	1	2	yes	maximal, complemented
3	#14521149	C₂	1	2	yes	maximal, complemented
4	#14521146	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521143	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521047	C₂	1	2	yes	maximal, complemented
10	#14521003	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#92497	C₂²	1	2	yes	—
13	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	D₄×Dic₃₁	C₂²	1	2	yes	—
16	C₂×#92480	C₂²	1	2	yes	—
17	C₄×C₂²⋊D₃₁	—	2	2	no	—
18	C₂×Dic₃₁⋊C₄	—	2	2	no	—
19	#11758489	C₄	1	8	yes	complemented
20	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
21	C₂×#92469	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
24	#11758464	C₂²	1	2	yes	—
25	(C₂²⋊D₃₁)⋊C₄	—	2	2	no	—
26	#11758454	—	2	2	no	—
27	#11758449	C₂²	1	1	yes	characteristic
28	C₄×Dic₆₂	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#102	C₂²	1	2	yes	—
31	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	#11758420	C₂²	1	2	yes	—
33	#11758406	C₂²	1	1	yes	characteristic
34	#11758404	C₂²	1	2	yes	—
35	#11758403	C₂²	1	1	yes	characteristic
36	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
37	#11758398	C₂²	1	2	yes	—
38	#11758395	C₂²	1	2	yes	—
39	#11758391	C₂²	1	1	yes	characteristic
40	C₂×D₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂×C₂²⋊D₃₁	—	2	2	no	—
43	C₂²×Dic₃₁	—	2	2	no	—
44	C₂²×Dic₃₁	—	2	2	no	—
45	#92497	—	2	4	no	—
46	#92497	C₂×C₄	1	8	yes	complemented
47	C₂×C₄×D₃₁	—	2	2	no	—
48	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
49	C₂×Dic₆₂	C₂³	1	2	yes	—
50	D₈×C₃₁	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₃₁	—	2	2	no	—
55	C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₃₁	—	2	2	no	—
57	#92480	C₂×C₄	1	8	yes	complemented
58	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
59	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
60	C₄⋊Dic₃₁	C₂³	1	2	yes	—
61	Dic₃₁⋊C₄	—	2	2	no	—
62	Dic₃₁⋊C₄	—	2	2	no	—
63	Dic₃₁⋊C₄	—	2	2	no	—
64	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
65	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
66	C₄×Dic₃₁	—	2	2	no	—
67	C₄×Dic₃₁	—	2	2	no	—
68	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	Dic₁₂₄	C₂×C₄	1	4	yes	complemented
72	#92469	—	2	2	no	—
73	C₈×D₃₁	C₂×C₄	1	4	yes	complemented
74	C₂³×C₃₁	—	2	2	no	—
75	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
76	D₄×C₃₁	—	2	2	no	—
77	D₄×C₃₁	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	8	2	no	—
80	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₃₁	—	2	8	no	—
84	C₂×Dic₃₁	—	2	2	no	—
85	C₂×Dic₃₁	—	2	8	no	—
86	C₂×Dic₃₁	—	2	2	no	—
87	C₂×Dic₃₁	—	8	2	no	—
88	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
91	C₄×D₃₁	—	8	2	no	—
92	C₄×D₃₁	C₂²×C₄	1	2	yes	—
93	C₄×D₃₁	#41	1	2	yes	—
94	Dic₆₂	—	2	2	no	—
95	Dic₆₂	C₂²×C₄	1	4	yes	—
96	C₈×C₃₁	—	2	1	no	—
97	C₈×C₃₁	C₂²×C₄	1	2	yes	—
98	C₃₁⋊C₈	—	2	1	no	—
99	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
100	#2951	—	31	1	no	maximal, Sylow
101	C₂²×C₃₁	—	2	4	no	—
102	C₂²×C₃₁	—	4	2	no	—
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₃₁	—	2	1	no	—
106	C₂×D₃₁	—	2	1	no	—
107	C₂×D₃₁	C₄×D₄	1	2	yes	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	4	2	no	—
111	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
113	Dic₃₁	—	2	4	no	—
114	Dic₃₁	—	2	1	no	—
115	Dic₃₁	—	2	1	no	—
116	Dic₃₁	—	4	2	no	—
117	Dic₃₁	C₄×D₄	1	2	yes	—
118	C₂×#135	—	31	1	no	—
119	#520	—	31	2	no	—
120	#440	—	31	2	no	—
121	#440	—	31	2	no	—
122	C₄×Q₁₆	—	31	1	no	—
123	C₄×Q₁₆	—	31	1	no	—
124	C₄×D₈	—	31	1	no	—
125	C₄×D₈	D₃₁	1	1	yes	characteristic, complemented
126	C₂×#107	—	31	1	no	—
127	#417	—	31	2	no	—
128	#405	—	31	1	no	—
129	C₂×C₃₁	—	4	4	no	—
130	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂×C₃₁	#576	1	1	yes	characteristic
132	C₂×C₃₁	#578	1	1	yes	characteristic
133	D₃₁	—	2	2	no	—
134	C₂×#41	—	31	2	no	—
135	#135	—	31	8	no	—
136	C₂×Q₁₆	—	31	1	no	—
137	C₂×SD₁₆	—	31	2	no	—
138	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
139	C₂×M₁₆	—	31	1	no	—
140	C₂²×C₈	—	31	1	no	—
141	C₄×Q₈	—	31	2	no	—
142	C₄×Q₈	—	31	2	no	—
143	C₄×D₄	—	31	2	no	—
144	C₄×D₄	—	62	2	no	—
145	C₄×D₄	—	62	2	no	—
146	C₄×D₄	C₂×D₃₁	1	2	yes	—
147	#117	—	31	1	no	—
148	#117	—	62	2	no	—
149	C₂×C₄⋊C₄	—	31	2	no	—
150	C₂×C₄⋊C₄	—	62	2	no	—
151	#107	—	31	1	no	—
152	#107	—	31	2	no	—
153	#107	C₂×D₃₁	1	1	yes	characteristic
154	#103	—	31	2	no	—
155	#103	—	31	2	no	—
156	#102	—	31	2	no	—
157	#102	C₂×D₃₁	1	2	yes	—
158	#97	—	31	1	no	—
159	#97	—	31	1	no	—
160	C₄×C₈	—	31	1	no	—
161	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
162	C₃₁	#2951	1	1	yes	characteristic, complemented, Sylow
163	#41	—	31	8	no	—
164	#41	—	62	4	no	—
165	C₂×Q₈	—	31	2	no	—
166	C₂×D₄	—	62	2	no	—
167	C₂×D₄	C₂²×D₃₁	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	31	1	no	—
170	C₂²×C₄	—	62	2	no	—
171	C₂²×C₄	—	62	2	no	—
172	C₂²×C₄	—	62	2	no	—
173	Q₁₆	—	31	4	no	—
174	SD₁₆	—	31	8	no	—
175	D₈	C₄×D₃₁	1	4	yes	complemented
176	M₁₆	—	62	2	no	—
177	C₂×C₈	—	31	1	no	—
178	C₂×C₈	—	31	4	no	—
179	C₂×C₈	—	31	1	no	—
180	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
182	C₄⋊C₄	—	31	1	no	—
183	C₄⋊C₄	—	31	1	no	—
184	C₄⋊C₄	—	31	2	no	—
185	C₄⋊C₄	—	62	2	no	—
186	C₄⋊C₄	—	62	2	no	—
187	C₄⋊C₄	—	62	2	no	—
188	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	31	1	no	—
191	C₂²⋊C₄	—	31	1	no	—
192	C₂²⋊C₄	—	62	2	no	—
193	C₄²	—	31	1	no	—
194	C₄²	—	62	2	no	—
195	C₄²	—	62	2	no	—
196	C₄²	C₂²×D₃₁	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	31	1	no	—
199	Q₈	—	31	4	no	—
200	Q₈	—	62	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	62	8	no	—
203	D₄	C₂×C₄×D₃₁	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	31	1	no	—
207	C₂×C₄	—	31	2	no	—
208	C₂×C₄	—	31	2	no	—
209	C₂×C₄	—	31	1	no	—
210	C₂×C₄	—	31	1	no	—
211	C₂×C₄	—	62	8	no	—
212	C₂×C₄	—	62	2	no	—
213	C₂×C₄	—	62	2	no	—
214	C₂×C₄	—	248	2	no	—
215	C₂×C₄	—	248	2	no	—
216	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
217	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	31	2	no	—
221	C₈	—	62	1	no	—
222	C₈	C₂×C₄×D₃₁	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	31	2	no	—
227	C₂²	—	62	1	no	—
228	C₂²	—	62	1	no	—
229	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	31	2	no	—
234	C₄	—	62	1	no	—
235	C₄	—	62	4	no	—
236	C₄	—	62	1	no	—
237	C₄	—	124	2	no	—
238	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
239	C₄	#41×D₃₁	1	1	yes	characteristic
240	C₂	—	4	4	no	—
241	C₂	—	62	2	no	—
242	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic
243	C₂	#14521850	1	1	yes	characteristic
244	C₂	#14521863	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×D₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#11758489	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×D₈	D₃₁
hypocenter	C₃₁	#2951
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #14521528	C₂	semidirect
3. #14521149	C₂	semidirect
4. #14521146	C₂	semidirect
5. C₄×Dic₁₂₄	C₂	semidirect
6. #14521143	C₂	semidirect
7. D₈×Dic₃₁	C₂	semidirect
8. #107×D₃₁	C₂	semidirect
9. #14521047	C₂	semidirect
10. #14521003	C₂	semidirect
11. C₄×D₈×C₃₁	C₂	semidirect
19. #11758489	C₄	semidirect
46. #92497	C₂×C₄	semidirect
50. D₈×C₃₁	C₂×C₄	semidirect
57. #92480	C₂×C₄	semidirect
71. Dic₁₂₄	C₂×C₄	semidirect
73. C₈×D₃₁	C₂×C₄	semidirect
125. C₄×D₈	D₃₁	semidirect
162. C₃₁	#2951	semidirect
175. D₈	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×#11758489	2
#14521528	2
#14521149	2
#14521146	2
C₄×Dic₁₂₄	2
#14521143	2
D₈×Dic₃₁	2
#107×D₃₁	2
#14521047	2
#14521003	2
C₄×D₈×C₃₁	2
#2951	31