Subgroups — #16855873

Order 3968.

386 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521606	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521390	C₂	1	2	yes	maximal, complemented
3	#14521344	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521339	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521259	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521254	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521211	C₂	1	2	yes	maximal
8	#14521154	C₂	1	2	yes	maximal, complemented
9	#14521149	C₂	1	2	yes	maximal, complemented
10	#14521106	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₃₁	—	2	1	no	—
16	C₄×D₄×C₃₁	—	2	1	no	—
17	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#117	—	2	1	no	—
20	C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₃₁	—	2	1	no	—
22	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×#92482	C₂²	1	2	yes	—
24	C₂×#92480	C₂²	1	2	yes	—
25	#11758503	—	2	1	no	—
26	#11758501	—	2	1	no	—
27	#11758501	C₂²	1	2	yes	—
28	#11758447	—	2	1	no	—
29	#11758446	C₂²	1	1	yes	characteristic
30	#11758446	C₂²	1	1	yes	characteristic
31	C₄×Dic₆₂	—	2	1	no	—
32	C₄×Dic₆₂	C₂²	1	2	yes	—
33	#11758423	C₂²	1	2	yes	—
34	#11758420	C₂²	1	2	yes	—
35	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
36	#11758398	C₂²	1	2	yes	—
37	#11758398	C₂²	1	2	yes	—
38	#11758396	C₂²	1	2	yes	—
39	#11758395	C₂²	1	2	yes	—
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#11758391	C₂²	1	2	yes	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	—	2	2	no	—
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₆₂	—	2	1	no	—
55	C₂×Dic₆₂	C₂³	1	2	yes	—
56	C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₃₁	—	2	1	no	—
61	C₂²⋊C₄×C₃₁	—	2	1	no	—
62	C₂²⋊C₄×C₃₁	—	2	1	no	—
63	C₄²×C₃₁	—	2	1	no	—
64	C₄²×C₃₁	—	2	1	no	—
65	C₄²×C₃₁	—	2	1	no	—
66	C₄²×C₃₁	D₄	1	1	yes	characteristic
67	C₄²×C₃₁	D₄	1	1	yes	characteristic
68	C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₃₁	—	2	2	no	—
71	#92482	—	2	4	no	—
72	#92480	—	2	4	no	—
73	C₄⋊Dic₃₁	—	2	1	no	—
74	C₄⋊Dic₃₁	—	2	1	no	—
75	C₄⋊Dic₃₁	—	2	1	no	—
76	C₄⋊Dic₃₁	C₂³	1	2	yes	—
77	Dic₃₁⋊C₄	—	2	2	no	—
78	Dic₃₁⋊C₄	—	2	2	no	—
79	C₄×Dic₃₁	—	2	2	no	—
80	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
81	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₃₁	—	2	1	no	—
85	Q₈×C₃₁	#41	1	2	yes	—
86	D₄×C₃₁	—	2	4	no	—
87	D₄×C₃₁	—	2	1	no	—
88	D₄×C₃₁	#41	1	2	yes	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	4	1	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₃₁	—	2	2	no	—
106	C₂×Dic₃₁	—	2	2	no	—
107	Dic₆₂	—	2	2	no	—
108	Dic₆₂	—	2	2	no	—
109	Dic₆₂	—	4	2	no	—
110	C₃₁⋊C₈	—	2	2	no	—
111	C₃₁⋊C₈	—	2	2	no	—
112	#3116	—	31	1	no	maximal, Sylow
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	4	1	no	—
118	C₂²×C₃₁	—	4	1	no	—
119	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₃₁	—	2	1	no	—
121	C₄×C₃₁	—	2	2	no	—
122	C₄×C₃₁	—	2	1	no	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	4	1	no	—
126	C₄×C₃₁	—	4	1	no	—
127	C₄×C₃₁	#137	1	2	yes	—
128	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
129	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
130	Dic₃₁	—	4	2	no	—
131	Dic₃₁	—	4	2	no	—
132	#533	—	31	1	no	—
133	C₄×#41	D₃₁	1	1	yes	characteristic
134	#485	—	31	2	no	—
135	#477	—	31	1	no	—
136	#476	—	31	1	no	—
137	#462	—	31	1	no	—
138	#461	—	31	1	no	—
139	#452	—	31	2	no	—
140	#441	—	31	2	no	—
141	#440	—	31	2	no	—
142	#432	—	31	1	no	—
143	C₂×C₃₁	—	4	1	no	—
144	C₂×C₃₁	—	4	2	no	—
145	C₂×C₃₁	#525	1	1	yes	characteristic
146	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
147	C₂×C₃₁	#575	1	1	yes	characteristic
148	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
149	C₂×Q₁₆	—	31	2	no	—
150	C₂×SD₁₆	—	31	2	no	—
151	C₄⋊Q₈	—	31	1	no	—
152	C₄⋊Q₈	—	31	1	no	—
153	#125	—	62	1	no	—
154	C₂²⋊Q₈	—	31	2	no	—
155	C₂²⋊Q₈	—	62	1	no	—
156	C₄×Q₈	—	31	2	no	—
157	C₄×Q₈	—	62	1	no	—
158	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
159	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
163	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
164	#117	—	2	1	no	—
165	#117	—	62	1	no	—
166	#117	C₂×D₃₁	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
169	#107	—	31	2	no	—
170	#106	—	31	2	no	—
171	C₄⋊C₈	—	31	1	no	—
172	C₄⋊C₈	—	31	1	no	—
173	#103	—	31	2	no	—
174	#103	—	31	2	no	—
175	#103	—	31	2	no	—
176	#102	—	31	2	no	—
177	C₂²⋊C₈	—	31	2	no	—
178	#97	—	31	2	no	—
179	C₃₁	#3116	1	1	yes	characteristic, complemented, Sylow
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	31	2	no	—
183	C₂×Q₈	—	62	1	no	—
184	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
193	Q₁₆	—	62	4	no	—
194	SD₁₆	—	62	4	no	—
195	C₂×C₈	—	31	2	no	—
196	C₂×C₈	—	31	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	31	2	no	—
200	C₄⋊C₄	—	62	2	no	—
201	C₄⋊C₄	—	62	2	no	—
202	C₄⋊C₄	—	62	1	no	—
203	C₄⋊C₄	—	62	1	no	—
204	C₄⋊C₄	—	62	1	no	—
205	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
206	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	62	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	62	2	no	—
215	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
216	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
217	C₄²	C₂²×D₃₁	1	1	yes	characteristic
218	C₄²	C₂²×D₃₁	1	1	yes	characteristic
219	C₂³	—	2	1	no	—
220	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	62	2	no	—
223	Q₈	—	62	2	no	—
224	Q₈	—	124	2	no	—
225	Q₈	#92495	1	2	yes	—
226	D₄	—	2	4	no	—
227	D₄	—	2	1	no	—
228	D₄	#92495	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	62	2	no	—
240	C₂×C₄	—	62	2	no	—
241	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
247	C₈	—	62	2	no	—
248	C₈	—	62	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	124	2	no	—
264	C₄	—	124	2	no	—
265	C₄	#11758528	1	2	yes	—
266	C₄	C₂×#92495	1	1	yes	characteristic
267	C₄	C₂×#92495	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	#14521560	1	1	yes	characteristic
271	C₂	C₂×#11758528	1	1	yes	characteristic
272	C₂	#14521847	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×#41×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#41×C₃₁	C₂²
hypercenter	C₄×#41	D₃₁
hypocenter	C₃₁	#3116
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14521606	C₂	semidirect
2. #14521390	C₂	semidirect
3. #14521344	C₂	semidirect
4. #14521339	C₂	semidirect
5. #14521259	C₂	semidirect
6. #14521254	C₂	semidirect
8. #14521154	C₂	semidirect
9. #14521149	C₂	semidirect
10. #14521106	C₂	semidirect
179. C₃₁	#3116	semidirect

Maximal subgroups

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

subgroup	index
#14521606	2
#14521390	2
#14521344	2
#14521339	2
#14521259	2
#14521254	2
#14521211	2
#14521154	2
#14521149	2
#14521106	2
C₄×#41×C₃₁	2
#3116	31