Subgroups — C₄×#333×C₁₃₁ #45694113

Order 33536.

540 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×#333×C₁₃₁	C₂	1	2	yes	maximal
4	#1296×C₁₃₁	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₁	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₁₃₁	—	2	2	no	—
9	#391×C₁₃₁	C₄	1	2	yes	—
10	#389×C₁₃₁	—	2	2	no	—
11	#384×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₃₁	—	2	2	no	—
13	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
14	#336×C₁₃₁	—	2	2	no	—
15	#333×C₁₃₁	C₄	1	8	yes	direct product factor
16	C₂²×Q₈×C₁₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₁₃₁	—	2	4	no	—
18	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
19	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	4	yes	—
20	C₄×Q₈×C₁₃₁	—	4	4	no	—
21	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃₁	—	2	2	no	—
26	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₁₃₁	—	2	4	no	—
32	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
33	C₄×C₈×C₁₃₁	—	4	4	no	—
34	#95×C₁₃₁	—	2	4	no	—
35	C₂×Q₈×C₁₃₁	—	2	2	no	—
36	C₂×Q₈×C₁₃₁	—	4	2	no	—
37	C₂×Q₈×C₁₃₁	—	4	2	no	—
38	C₂×Q₈×C₁₃₁	C₄²	1	4	yes	—
39	C₂²×C₄×C₁₃₁	—	2	2	no	—
40	C₂²×C₄×C₁₃₁	—	2	2	no	—
41	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₃₁	—	2	8	no	—
47	C₂×C₈×C₁₃₁	—	4	4	no	—
48	C₂×C₈×C₁₃₁	—	4	4	no	—
49	C₄⋊C₄×C₁₃₁	—	2	8	no	—
50	C₄⋊C₄×C₁₃₁	—	4	2	no	—
51	C₄⋊C₄×C₁₃₁	—	4	4	no	—
52	C₄⋊C₄×C₁₃₁	—	4	4	no	—
53	C₄²×C₁₃₁	—	2	2	no	—
54	C₄²×C₁₃₁	—	2	4	no	—
55	C₄²×C₁₃₁	—	4	1	no	—
56	C₄²×C₁₃₁	—	4	4	no	—
57	C₄²×C₁₃₁	SD₁₆	1	4	yes	—
58	C₄²×C₁₃₁	Q₁₆	1	4	yes	—
59	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
60	C₄²×C₁₃₁	#41	1	2	yes	—
61	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₁₃₁	—	4	4	no	—
63	Q₈×C₁₃₁	—	4	4	no	—
64	C₂×C₄×C₁₃₁	—	2	1	no	—
65	C₂×C₄×C₁₃₁	—	2	2	no	—
66	C₂×C₄×C₁₃₁	—	2	1	no	—
67	C₂×C₄×C₁₃₁	—	2	2	no	—
68	C₂×C₄×C₁₃₁	—	2	4	no	—
69	C₂×C₄×C₁₃₁	—	2	1	no	—
70	C₂×C₄×C₁₃₁	—	2	2	no	—
71	C₂×C₄×C₁₃₁	—	4	2	no	—
72	C₂×C₄×C₁₃₁	—	4	2	no	—
73	C₂×C₄×C₁₃₁	—	4	2	no	—
74	C₂×C₄×C₁₃₁	—	4	4	no	—
75	C₂×C₄×C₁₃₁	#100	1	2	yes	—
76	C₂×C₄×C₁₃₁	#103	1	4	yes	—
77	C₂×C₄×C₁₃₁	#103	1	4	yes	—
78	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₁₃₁	#135	1	4	yes	—
84	C₈×C₁₃₁	—	4	8	no	—
85	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₁₃₁	C₂×#100	1	1	yes	characteristic
87	C₂²×C₁₃₁	#413	1	1	yes	characteristic
88	C₂²×C₁₃₁	C₂×#103	1	2	yes	—
89	C₂²×C₁₃₁	#416	1	2	yes	—
90	C₄×C₁₃₁	—	2	4	no	—
91	C₄×C₁₃₁	—	2	1	no	—
92	C₄×C₁₃₁	—	2	1	no	—
93	C₄×C₁₃₁	—	4	1	no	—
94	C₄×C₁₃₁	—	4	4	no	—
95	C₄×C₁₃₁	—	8	2	no	—
96	C₄×C₁₃₁	#333	1	4	yes	direct product factor
97	C₄×C₁₃₁	C₄×SD₁₆	1	4	yes	—
98	C₄×C₁₃₁	C₄×Q₁₆	1	4	yes	—
99	C₂×C₁₃₁	C₂×#333	1	1	yes	characteristic
100	C₂×C₁₃₁	#1542	1	2	yes	—
101	C₂×C₁₃₁	#1543	1	1	yes	characteristic
102	C₂×C₁₃₁	C₄×#100	1	1	yes	characteristic
103	C₂×C₁₃₁	C₄×#103	1	2	yes	—
104	C₄×#333	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₁₃₁	C₄×#333	1	1	yes	characteristic, direct product factor, Sylow
106	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	1	yes	characteristic
107	C₄×C₄⋊C₈	C₂×C₁₃₁	1	2	yes	—
108	C₂×#333	C₂×C₁₃₁	1	2	yes	—
109	#1296	C₂×C₁₃₁	1	2	yes	—
110	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
111	C₂×C₄×Q₈	C₄×C₁₃₁	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	4	yes	—
113	C₂×C₄×C₈	—	2	2	no	—
114	#391	C₄×C₁₃₁	1	2	yes	—
115	#389	—	2	2	no	—
116	#384	C₂²×C₁₃₁	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	2	2	no	—
118	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	#333	C₄×C₁₃₁	1	8	yes	direct product factor
121	C₂²×Q₈	C₂×C₄×C₁₃₁	1	2	yes	—
122	C₂²×C₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	4	yes	—
125	C₄×Q₈	—	4	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
133	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
134	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
135	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
138	C₄×C₈	—	4	4	no	—
139	#95	—	2	4	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	C₄²×C₁₃₁	1	4	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
148	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	4	no	—
162	C₄²	SD₁₆×C₁₃₁	1	4	yes	—
163	C₄²	Q₁₆×C₁₃₁	1	4	yes	—
164	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
165	C₄²	#41×C₁₃₁	1	2	yes	—
166	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
167	Q₈	—	4	4	no	—
168	Q₈	—	4	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	#100×C₁₃₁	1	2	yes	—
181	C₂×C₄	#103×C₁₃₁	1	4	yes	—
182	C₂×C₄	#103×C₁₃₁	1	4	yes	—
183	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
184	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
185	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
186	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	2	yes	—
187	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
188	C₂×C₄	#135×C₁₃₁	1	4	yes	—
189	C₈	—	4	8	no	—
190	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
191	C₂²	C₂×#100×C₁₃₁	1	1	yes	characteristic
192	C₂²	#413×C₁₃₁	1	1	yes	characteristic
193	C₂²	C₂×#103×C₁₃₁	1	2	yes	—
194	C₂²	#416×C₁₃₁	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	#333×C₁₃₁	1	4	yes	direct product factor
202	C₄	C₄×SD₁₆×C₁₃₁	1	4	yes	—
203	C₄	C₄×Q₁₆×C₁₃₁	1	4	yes	—
204	C₂	C₂×#333×C₁₃₁	1	1	yes	characteristic
205	C₂	#1542×C₁₃₁	1	2	yes	—
206	C₂	#1543×C₁₃₁	1	1	yes	characteristic
207	C₂	C₄×#100×C₁₃₁	1	1	yes	characteristic
208	C₂	C₄×#103×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle, prime radical	C₂³×C₁₃₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. #333×C₁₃₁	C₄	direct (N × H)
96. C₄×C₁₃₁	#333	direct (N × H)
104. C₄×#333	C₁₃₁	direct (N × H)
105. C₁₃₁	C₄×#333	direct (N × H)
120. #333	C₄×C₁₃₁	direct (N × H)
201. C₄	#333×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₃₁	2
C₄×C₄⋊C₈×C₁₃₁	2
C₂×#333×C₁₃₁	2
#1296×C₁₃₁	2
C₄×#333	131

Subgroups — C4×#333×C131 #45694113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#333×C₁₃₁ #45694113