Subgroups — C₂×#1537×C₁₃₁ #45703545

Order 33536.

954 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₁₃₁	C₂	1	2	yes	maximal
3	#1537×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#330×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#533×C₁₃₁	—	2	4	no	—
6	#533×C₁₃₁	C₄	1	4	yes	—
7	#533×C₁₃₁	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₁₃₁	C₄	1	2	yes	—
9	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
11	C₂×C₄×Q₈×C₁₃₁	—	2	2	no	—
12	C₂×#117×C₁₃₁	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
14	#432×C₁₃₁	—	2	4	no	—
15	#432×C₁₃₁	C₂²	1	8	yes	—
16	C₂×C₄⋊C₈×C₁₃₁	—	2	2	no	—
17	C₂×C₂²⋊C₈×C₁₃₁	—	2	2	no	—
18	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
19	#330×C₁₃₁	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₁₃₁	—	2	2	no	—
21	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₁	D₄	1	2	yes	—
23	C₂²×C₈×C₁₃₁	—	2	4	no	—
24	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
25	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	8	yes	—
26	#125×C₁₃₁	—	2	4	no	—
27	#125×C₁₃₁	C₂×C₄	1	4	yes	—
28	#125×C₁₃₁	C₂³	1	4	yes	—
29	C₂²⋊Q₈×C₁₃₁	—	2	8	no	—
30	C₂²⋊Q₈×C₁₃₁	—	2	8	no	—
31	C₄×Q₈×C₁₃₁	—	2	8	no	—
32	C₄×Q₈×C₁₃₁	—	2	8	no	—
33	#117×C₁₃₁	—	2	4	no	—
34	#117×C₁₃₁	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
39	C₂×C₄²×C₁₃₁	—	2	2	no	—
40	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃₁	D₄	1	4	yes	—
44	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
46	C₄⋊C₈×C₁₃₁	—	2	8	no	—
47	C₂²⋊C₈×C₁₃₁	—	2	8	no	—
48	#97×C₁₃₁	—	2	4	no	—
49	#97×C₁₃₁	C₂³	1	8	yes	—
50	C₂⁴×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₁	—	2	8	no	—
52	C₂×Q₈×C₁₃₁	—	4	8	no	—
53	C₂²×C₄×C₁₃₁	—	2	4	no	—
54	C₂²×C₄×C₁₃₁	—	2	4	no	—
55	C₂²×C₄×C₁₃₁	—	4	4	no	—
56	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₁₃₁	—	2	8	no	—
65	C₂×C₈×C₁₃₁	—	4	8	no	—
66	C₄⋊C₄×C₁₃₁	—	2	8	no	—
67	C₄⋊C₄×C₁₃₁	—	2	8	no	—
68	C₄⋊C₄×C₁₃₁	—	2	4	no	—
69	C₄⋊C₄×C₁₃₁	—	2	8	no	—
70	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
71	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
72	C₄²×C₁₃₁	—	2	8	no	—
73	C₄²×C₁₃₁	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
79	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
80	C₂³×C₁₃₁	—	2	2	no	—
81	C₂³×C₁₃₁	—	2	4	no	—
82	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₁₃₁	—	4	16	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	2	4	no	—
87	C₂×C₄×C₁₃₁	—	2	4	no	—
88	C₂×C₄×C₁₃₁	—	2	4	no	—
89	C₂×C₄×C₁₃₁	—	2	4	no	—
90	C₂×C₄×C₁₃₁	—	2	8	no	—
91	C₂×C₄×C₁₃₁	—	4	4	no	—
92	C₂×C₄×C₁₃₁	—	4	4	no	—
93	C₂×C₄×C₁₃₁	—	4	8	no	—
94	C₂×C₄×C₁₃₁	#101	1	4	yes	—
95	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
101	C₈×C₁₃₁	—	4	8	no	—
102	C₂²×C₁₃₁	—	2	2	no	—
103	C₂²×C₁₃₁	—	2	4	no	—
104	C₂²×C₁₃₁	—	4	4	no	—
105	C₂²×C₁₃₁	C₂×#101	1	2	yes	—
106	C₂²×C₁₃₁	#421	1	4	yes	—
107	C₂²×C₁₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₁₃₁	—	2	1	no	—
109	C₄×C₁₃₁	—	2	1	no	—
110	C₄×C₁₃₁	—	2	2	no	—
111	C₄×C₁₃₁	—	2	2	no	—
112	C₄×C₁₃₁	—	2	4	no	—
113	C₄×C₁₃₁	—	4	8	no	—
114	C₄×C₁₃₁	C₂×#101	1	4	yes	—
115	C₂×C₁₃₁	—	4	2	no	—
116	C₂×C₁₃₁	#1537	1	4	yes	direct product factor
117	C₂×C₁₃₁	C₂²×#101	1	1	yes	characteristic
118	C₂×C₁₃₁	C₂×#421	1	2	yes	—
119	C₂×#1537	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₁₃₁	C₂×#1537	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#533	C₂×C₁₃₁	1	1	yes	characteristic
122	C₂×#432	C₂×C₁₃₁	1	2	yes	—
123	#1537	C₂×C₁₃₁	1	8	yes	direct product factor
124	C₂×#330	C₂×C₁₃₁	1	4	yes	complemented
125	#533	—	2	4	no	—
126	#533	C₄×C₁₃₁	1	4	yes	—
127	#533	C₂²×C₁₃₁	1	4	yes	—
128	C₂×C₄⋊Q₈	C₄×C₁₃₁	1	2	yes	—
129	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
130	C₂×C₂²⋊Q₈	—	2	2	no	—
131	C₂×C₄×Q₈	—	2	2	no	—
132	C₂×#117	C₄×C₁₃₁	1	2	yes	—
133	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
134	#432	—	2	4	no	—
135	#432	C₂²×C₁₃₁	1	8	yes	—
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₂²⋊C₈	—	2	2	no	—
138	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
139	#330	C₂²×C₁₃₁	1	16	yes	complemented
140	C₂²×Q₈	—	2	2	no	—
141	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
142	C₂³×C₄	D₄×C₁₃₁	1	2	yes	—
143	C₂²×C₈	—	2	4	no	—
144	C₄⋊Q₈	—	2	4	no	—
145	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	8	yes	—
146	#125	—	2	4	no	—
147	#125	C₂×C₄×C₁₃₁	1	4	yes	—
148	#125	C₂³×C₁₃₁	1	4	yes	—
149	C₂²⋊Q₈	—	2	8	no	—
150	C₂²⋊Q₈	—	2	8	no	—
151	C₄×Q₈	—	2	8	no	—
152	C₄×Q₈	—	2	8	no	—
153	#117	—	2	4	no	—
154	#117	C₂×C₄×C₁₃₁	1	8	yes	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
158	C₂×C₂²⋊C₄	—	2	2	no	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
162	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
163	C₂×C₄²	D₄×C₁₃₁	1	4	yes	—
164	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
166	C₄⋊C₈	—	2	8	no	—
167	C₂²⋊C₈	—	2	8	no	—
168	#97	—	2	4	no	—
169	#97	C₂³×C₁₃₁	1	8	yes	—
170	C₂⁴	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	—	4	8	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	4	yes	—
179	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	4	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	8	no	—
190	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
191	C₂²⋊C₄	—	2	8	no	—
192	C₄²	—	2	8	no	—
193	C₄²	C₂²⋊C₄×C₁₃₁	1	4	yes	—
194	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
195	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
196	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
197	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
198	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
199	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
203	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
204	Q₈	—	4	16	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	#101×C₁₃₁	1	4	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	4	yes	—
217	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
219	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
221	C₈	—	4	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×#101×C₁₃₁	1	2	yes	—
226	C₂²	#421×C₁₃₁	1	4	yes	—
227	C₂²	C₂²×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	8	no	—
234	C₄	C₂×#101×C₁₃₁	1	4	yes	—
235	C₂	—	4	2	no	—
236	C₂	#1537×C₁₃₁	1	4	yes	direct product factor
237	C₂	C₂²×#101×C₁₃₁	1	1	yes	characteristic
238	C₂	C₂×#421×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
prime radical	C₂⁴×C₁₃₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1537×C₁₃₁	C₂	direct (N × H)
4. C₂×#330×C₁₃₁	C₂	semidirect
19. #330×C₁₃₁	C₂²	semidirect
116. C₂×C₁₃₁	#1537	direct (N × H)
119. C₂×#1537	C₁₃₁	direct (N × H)
120. C₁₃₁	C₂×#1537	direct (N × H)
123. #1537	C₂×C₁₃₁	direct (N × H)
124. C₂×#330	C₂×C₁₃₁	semidirect
139. #330	C₂²×C₁₃₁	semidirect
236. C₂	#1537×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
C₂×#533×C₁₃₁	2
C₂×#432×C₁₃₁	2
#1537×C₁₃₁	2
C₂×#330×C₁₃₁	2
C₂×#1537	131

Subgroups — C2×#1537×C131 #45703545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1537×C₁₃₁ #45703545