Subgroups — #16855573

Order 3968.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521677	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521356	C₂	1	2	yes	maximal, complemented
3	#14521349	C₂	1	2	yes	maximal, complemented
4	#14521339	C₂	1	2	yes	maximal, complemented
5	#14521334	C₂	1	2	yes	maximal, complemented
6	#14521089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11758396	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521076	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521050	C₂	1	1	yes	maximal, characteristic
10	#14521045	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#520	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₄×D₄×C₃₁	—	2	2	no	—
15	C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#117	—	2	1	no	—
17	C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	#11758503	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
22	C₂×#92474	C₂²	1	1	yes	characteristic
23	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
24	#11758467	—	2	1	no	—
25	#11758465	—	2	1	no	—
26	#11758455	—	2	2	no	—
27	#11758447	C₂²	1	2	yes	—
28	#11758446	C₂²	1	2	yes	—
29	#11758423	C₂²	1	1	yes	characteristic
30	#11758423	C₂²	1	2	yes	—
31	#11758423	C₂²	1	1	yes	characteristic
32	#11758420	C₂²	1	1	yes	characteristic
33	#11758420	C₂²	1	2	yes	—
34	#11758420	C₂²	1	1	yes	characteristic
35	#11758396	C₂²	1	2	yes	—
36	#11758396	C₂²	1	2	yes	—
37	#11758396	C₂²	1	2	yes	—
38	#11758395	C₂²	1	2	yes	—
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
41	#41×C₃₁	—	2	2	no	—
42	#41×C₃₁	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₃₁	—	2	1	no	—
45	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Dic₃₁	—	2	1	no	—
51	C₂×Dic₆₂	—	2	2	no	—
52	C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
59	C₄²×C₃₁	—	2	2	no	—
60	C₄²×C₃₁	C₂³	1	2	yes	—
61	C₂²⋊Dic₃₁	—	2	1	no	—
62	C₄⋊Dic₃₁	—	2	2	no	—
63	C₄⋊Dic₃₁	—	2	1	no	—
64	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₃₁	C₂³	1	2	yes	—
67	Dic₃₁⋊C₄	—	2	2	no	—
68	Dic₃₁⋊C₄	—	2	2	no	—
69	C₄×Dic₃₁	—	2	1	no	—
70	#92474	—	2	2	no	—
71	C₂×C₃₁⋊C₈	—	2	2	no	—
72	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
73	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₂³×C₃₁	—	2	1	no	—
76	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₃₁	—	2	1	no	—
78	Q₈×C₃₁	C₂×Q₈	1	2	yes	—
79	D₄×C₃₁	—	2	4	no	—
80	D₄×C₃₁	—	2	1	no	—
81	D₄×C₃₁	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	8	2	no	—
86	C₂×C₄×C₃₁	—	8	1	no	—
87	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₃₁	#41	1	2	yes	—
92	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₃₁	—	2	1	no	—
94	C₂×Dic₃₁	—	2	1	no	—
95	C₂×Dic₃₁	—	2	2	no	—
96	C₂×Dic₃₁	—	8	1	no	—
97	Dic₆₂	—	8	2	no	—
98	C₃₁⋊C₈	—	2	1	no	—
99	C₃₁⋊C₈	—	2	1	no	—
100	C₃₁⋊C₈	—	2	2	no	—
101	#3084	—	31	1	no	maximal, Sylow
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	4	2	no	—
113	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
114	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
115	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
116	Dic₃₁	—	4	1	no	—
117	Dic₃₁	—	4	2	no	—
118	Dic₃₁	—	4	1	no	—
119	#544	—	31	1	no	—
120	#520	D₃₁	1	1	yes	characteristic
121	#479	—	31	2	no	—
122	#478	—	31	2	no	—
123	#476	—	31	2	no	—
124	#475	—	31	2	no	—
125	#428	—	31	1	no	—
126	C₂×#106	—	31	1	no	—
127	#424	—	31	1	no	—
128	#417	—	31	1	no	—
129	#416	—	31	1	no	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	—	4	2	no	—
132	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₃₁	#577	1	1	yes	characteristic
134	C₂×C₃₁	#577	1	1	yes	characteristic
135	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
136	C₂×M₁₆	—	31	1	no	—
137	C₂²×C₈	—	31	1	no	—
138	C₄⋊Q₈	—	31	2	no	—
139	C₄⋊Q₈	—	62	1	no	—
140	#125	—	31	2	no	—
141	#125	—	62	1	no	—
142	C₂²⋊Q₈	—	62	2	no	—
143	C₄×Q₈	C₂×D₃₁	1	2	yes	—
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	C₂×D₃₁	1	2	yes	—
146	#117	—	2	1	no	—
147	#117	—	31	1	no	—
148	#117	C₂×D₃₁	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	31	1	no	—
151	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
152	#107	—	31	2	no	—
153	#106	—	31	2	no	—
154	#106	—	31	2	no	—
155	#106	—	31	2	no	—
156	C₄⋊C₈	—	31	2	no	—
157	C₄⋊C₈	—	31	2	no	—
158	#103	—	31	1	no	—
159	#103	—	31	1	no	—
160	#103	—	31	2	no	—
161	#102	—	31	1	no	—
162	#102	—	31	1	no	—
163	#102	—	31	2	no	—
164	C₃₁	#3084	1	1	yes	characteristic, complemented, Sylow
165	#41	—	2	2	no	—
166	#41	Dic₆₂	1	4	yes	complemented
167	C₂×Q₈	—	62	2	no	—
168	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	62	1	no	—
175	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
176	M₁₆	—	62	2	no	—
177	C₂×C₈	—	31	1	no	—
178	C₂×C₈	—	31	1	no	—
179	C₂×C₈	—	31	2	no	—
180	C₂×C₈	—	62	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	31	2	no	—
184	C₄⋊C₄	—	31	1	no	—
185	C₄⋊C₄	—	31	1	no	—
186	C₄⋊C₄	—	62	2	no	—
187	C₄⋊C₄	—	62	2	no	—
188	C₄⋊C₄	—	62	1	no	—
189	C₄⋊C₄	—	62	2	no	—
190	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
191	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	62	1	no	—
195	C₂²⋊C₄	C₂²⋊D₃₁	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	62	1	no	—
198	C₄²	C₂²×D₃₁	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	248	2	no	—
203	Q₈	C₂×Dic₆₂	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₆₂	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	62	1	no	—
213	C₂×C₄	—	62	1	no	—
214	C₂×C₄	—	62	2	no	—
215	C₂×C₄	—	248	1	no	—
216	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
217	C₂×C₄	#92495	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
221	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
222	C₈	—	62	1	no	—
223	C₈	—	62	1	no	—
224	C₈	—	62	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#11758501	1	2	yes	—
231	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	124	2	no	—
237	C₄	—	124	1	no	—
238	C₄	—	124	1	no	—
239	C₄	#11758501	1	2	yes	—
240	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
241	C₄	C₂×#92495	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#11758501	1	1	yes	characteristic
245	C₂	#14521857	1	1	yes	characteristic
246	C₂	#14521860	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#520	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#41×C₃₁	C₂²
hypercenter	#520	D₃₁
hypocenter	C₃₁	#3084
LCS	G ▷ C₂×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
1. #14521677	C₂	semidirect
2. #14521356	C₂	semidirect
3. #14521349	C₂	semidirect
4. #14521339	C₂	semidirect
5. #14521334	C₂	semidirect
6. #14521089	C₂	semidirect
7. C₂×#11758396	C₂	semidirect
8. #14521076	C₂	semidirect
42. #41×C₃₁	Q₈	semidirect
164. C₃₁	#3084	semidirect
166. #41	Dic₆₂	semidirect

Maximal subgroups

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

subgroup	index
#14521677	2
#14521356	2
#14521349	2
#14521339	2
#14521334	2
#14521089	2
C₂×#11758396	2
#14521076	2
#14521050	2
#14521045	2
C₃₁×#520	2
#3084	31