Subgroups — #2544×C₁₃₁ #30662727

Order 16768.

704 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#394×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#391×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#390×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#386×C₁₃₁	C₂	1	4	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₃₁	C₂²	1	4	yes	—
13	C₂²×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃₁	C₂²	1	2	yes	—
15	#127×C₁₃₁	—	2	4	no	—
16	#124×C₁₃₁	—	2	4	no	—
17	#124×C₁₃₁	—	2	4	no	—
18	C₄×D₄×C₁₃₁	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
26	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
29	#95×C₁₃₁	C₂²	1	4	yes	—
30	#95×C₁₃₁	C₂²	1	2	yes	—
31	C₂⁴×C₁₃₁	C₂³	1	2	yes	—
32	C₂⁴×C₁₃₁	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₃₁	—	2	2	no	—
34	C₂×Q₈×C₁₃₁	—	2	2	no	—
35	C₂×Q₈×C₁₃₁	—	2	2	no	—
36	C₂×D₄×C₁₃₁	—	2	8	no	—
37	C₂×D₄×C₁₃₁	—	2	2	no	—
38	C₂×D₄×C₁₃₁	—	2	2	no	—
39	C₂×D₄×C₁₃₁	—	2	2	no	—
40	C₂×D₄×C₁₃₁	—	2	2	no	—
41	C₂×D₄×C₁₃₁	—	2	8	no	—
42	C₂×D₄×C₁₃₁	—	2	2	no	—
43	C₂×D₄×C₁₃₁	—	4	4	no	—
44	C₂×D₄×C₁₃₁	D₄	1	4	yes	—
45	C₂²×C₄×C₁₃₁	—	2	4	no	—
46	C₂²×C₄×C₁₃₁	—	4	4	no	—
47	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₃₁	—	2	2	no	—
55	C₄⋊C₄×C₁₃₁	—	2	4	no	—
56	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
57	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
58	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
59	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
60	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
61	C₄²×C₁₃₁	—	2	2	no	—
62	C₄²×C₁₃₁	—	2	2	no	—
63	C₄²×C₁₃₁	D₄	1	4	yes	complemented
64	C₂³×C₁₃₁	—	2	2	no	—
65	C₂³×C₁₃₁	—	2	2	no	—
66	C₂³×C₁₃₁	—	2	4	no	—
67	C₂³×C₁₃₁	—	2	4	no	—
68	C₂³×C₁₃₁	—	2	2	no	—
69	C₂³×C₁₃₁	—	2	2	no	—
70	C₂³×C₁₃₁	—	2	2	no	—
71	C₂³×C₁₃₁	—	2	4	no	—
72	C₂³×C₁₃₁	—	2	4	no	—
73	C₂³×C₁₃₁	C₂×D₄	1	4	yes	—
74	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
75	Q₈×C₁₃₁	—	4	4	no	—
76	D₄×C₁₃₁	—	2	8	no	—
77	D₄×C₁₃₁	—	4	16	no	—
78	D₄×C₁₃₁	—	4	4	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	2	no	—
81	C₂×C₄×C₁₃₁	—	2	2	no	—
82	C₂×C₄×C₁₃₁	—	2	2	no	—
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	2	2	no	—
87	C₂×C₄×C₁₃₁	—	2	2	no	—
88	C₂×C₄×C₁₃₁	—	2	2	no	—
89	C₂×C₄×C₁₃₁	—	2	2	no	—
90	C₂×C₄×C₁₃₁	—	2	4	no	—
91	C₂×C₄×C₁₃₁	—	4	8	no	—
92	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	#41	1	4	yes	—
97	C₂×C₄×C₁₃₁	#41	1	2	yes	—
98	C₂²×C₁₃₁	—	2	4	no	—
99	C₂²×C₁₃₁	—	2	8	no	—
100	C₂²×C₁₃₁	—	4	2	no	—
101	C₂²×C₁₃₁	—	4	2	no	—
102	C₂²×C₁₃₁	—	4	4	no	—
103	C₂²×C₁₃₁	—	4	2	no	—
104	C₂²×C₁₃₁	—	4	2	no	—
105	C₂²×C₁₃₁	—	4	2	no	—
106	C₂²×C₁₃₁	—	4	4	no	—
107	C₂²×C₁₃₁	—	4	4	no	—
108	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
112	C₂²×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
113	C₄×C₁₃₁	—	2	2	no	—
114	C₄×C₁₃₁	—	2	2	no	—
115	C₄×C₁₃₁	—	2	4	no	—
116	C₄×C₁₃₁	—	4	2	no	—
117	C₄×C₁₃₁	—	4	2	no	—
118	C₄×C₁₃₁	—	4	2	no	—
119	C₄×C₁₃₁	#121	1	4	yes	—
120	C₂×C₁₃₁	—	4	4	no	—
121	C₂×C₁₃₁	—	8	2	no	—
122	C₂×C₁₃₁	C₂×#121	1	1	yes	characteristic
123	C₂×C₁₃₁	#532	1	1	yes	characteristic
124	C₂×C₁₃₁	#535	1	1	yes	characteristic
125	C₂×C₁₃₁	#546	1	2	yes	—
126	C₂×C₁₃₁	#559	1	2	yes	—
127	C₁₃₁	#2544	1	1	yes	characteristic, direct product factor, Sylow
128	#2544	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×#127	C₂×C₁₃₁	1	1	yes	characteristic
130	C₂×#124	C₂×C₁₃₁	1	1	yes	characteristic, complemented
131	C₂×#124	C₂×C₁₃₁	1	1	yes	characteristic, complemented
132	C₂×C₄×D₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
133	#394	C₂×C₁₃₁	1	4	yes	complemented
134	#391	C₂×C₁₃₁	1	1	yes	characteristic, complemented
135	#390	C₂×C₁₃₁	1	1	yes	characteristic, complemented
136	#386	C₂×C₁₃₁	1	4	yes	complemented
137	C₄×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
138	C₂²×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	C₂²×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
140	C₂²×D₄	C₂²×C₁₃₁	1	4	yes	—
141	C₂²×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₁₃₁	1	2	yes	—
143	#127	—	2	4	no	—
144	#124	—	2	4	no	—
145	#124	—	2	4	no	—
146	C₄×D₄	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
154	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
156	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
157	#95	C₂²×C₁₃₁	1	2	yes	—
158	#95	C₂²×C₁₃₁	1	4	yes	—
159	C₂⁴	C₂³×C₁₃₁	1	2	yes	—
160	C₂⁴	C₂³×C₁₃₁	1	2	yes	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×D₄	—	2	8	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	8	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	D₄×C₁₃₁	1	4	yes	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
176	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
177	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
178	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
181	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	4	4	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	D₄×C₁₃₁	1	4	yes	complemented
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	4	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₂×D₄×C₁₃₁	1	4	yes	—
202	C₂³	C₂⁴×C₁₃₁	1	1	yes	characteristic
203	Q₈	—	4	4	no	—
204	D₄	—	2	8	no	—
205	D₄	—	4	16	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
221	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
224	C₂×C₄	#41×C₁₃₁	1	4	yes	—
225	C₂×C₄	#41×C₁₃₁	1	2	yes	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	8	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
237	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
238	C₂²	C₂×#41×C₁₃₁	1	1	yes	characteristic
239	C₂²	C₂×#41×C₁₃₁	1	2	yes	—
240	C₂²	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	4	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₄	#121×C₁₃₁	1	4	yes	—
248	C₂	—	4	4	no	—
249	C₂	—	8	2	no	—
250	C₂	C₂×#121×C₁₃₁	1	1	yes	characteristic
251	C₂	#532×C₁₃₁	1	1	yes	characteristic
252	C₂	#535×C₁₃₁	1	1	yes	characteristic
253	C₂	#546×C₁₃₁	1	2	yes	—
254	C₂	#559×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#124×C₁₃₁	C₂	semidirect
3. C₂×#124×C₁₃₁	C₂	semidirect
4. C₂×C₄×D₄×C₁₃₁	C₂	semidirect
5. #394×C₁₃₁	C₂	semidirect
6. #391×C₁₃₁	C₂	semidirect
7. #390×C₁₃₁	C₂	semidirect
8. #386×C₁₃₁	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₃₁	C₂	semidirect
63. C₄²×C₁₃₁	D₄	semidirect
127. C₁₃₁	#2544	direct (N × H)
128. #2544	C₁₃₁	direct (N × H)
130. C₂×#124	C₂×C₁₃₁	semidirect
131. C₂×#124	C₂×C₁₃₁	semidirect
132. C₂×C₄×D₄	C₂×C₁₃₁	semidirect
133. #394	C₂×C₁₃₁	semidirect
134. #391	C₂×C₁₃₁	semidirect
135. #390	C₂×C₁₃₁	semidirect
136. #386	C₂×C₁₃₁	semidirect
137. C₄×C₄⋊C₄	C₂×C₁₃₁	semidirect
191. C₄²	D₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₁₃₁	2
C₂×#124×C₁₃₁	2
C₂×C₄×D₄×C₁₃₁	2
#394×C₁₃₁	2
#391×C₁₃₁	2
#390×C₁₃₁	2
#386×C₁₃₁	2
C₄×C₄⋊C₄×C₁₃₁	2
#2544	131

Subgroups — #2544×C131 #30662727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2544×C₁₃₁ #30662727