Subgroups — C₂×#1836×C₁₃₁ #45706645

Order 33536.

1054 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1836×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×#340×C₁₃₁	C₂	1	4	yes	maximal
6	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₃₁	—	2	2	no	—
8	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×#97×C₁₃₁	C₄	1	8	yes	complemented
12	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
13	#384×C₁₃₁	—	2	2	no	—
14	#384×C₁₃₁	C₂²	1	4	yes	—
15	#382×C₁₃₁	—	2	2	no	—
16	#382×C₁₃₁	C₂²	1	4	yes	—
17	C₂×#95×C₁₃₁	—	2	2	no	—
18	#340×C₁₃₁	C₂²	1	16	yes	—
19	C₂³×C₄×C₁₃₁	—	2	2	no	—
20	C₂³×C₄×C₁₃₁	—	2	2	no	—
21	C₂³×C₄×C₁₃₁	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₃₁	—	2	4	no	—
24	C₂²×C₈×C₁₃₁	—	2	4	no	—
25	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	4	8	no	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃₁	Q₈	1	4	yes	—
38	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
39	#97×C₁₃₁	C₂×C₄	1	16	yes	complemented
40	#95×C₁₃₁	—	2	8	no	—
41	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	—	2	4	no	—
43	C₂²×C₄×C₁₃₁	—	2	4	no	—
44	C₂²×C₄×C₁₃₁	—	2	4	no	—
45	C₂²×C₄×C₁₃₁	—	2	4	no	—
46	C₂²×C₄×C₁₃₁	—	4	4	no	—
47	C₂²×C₄×C₁₃₁	—	4	4	no	—
48	C₂²×C₄×C₁₃₁	—	4	8	no	—
49	C₂²×C₄×C₁₃₁	—	4	4	no	—
50	C₂²×C₄×C₁₃₁	—	4	4	no	—
51	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	8	yes	—
52	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₁₃₁	#41	1	4	yes	—
57	C₂²×C₄×C₁₃₁	#41	1	4	yes	—
58	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	—	2	16	no	—
60	C₂×C₈×C₁₃₁	—	2	4	no	—
61	C₂×C₈×C₁₃₁	—	2	4	no	—
62	C₂×C₈×C₁₃₁	C₂²×C₄	1	8	yes	—
63	C₄⋊C₄×C₁₃₁	—	2	8	no	—
64	C₄⋊C₄×C₁₃₁	—	2	8	no	—
65	C₄⋊C₄×C₁₃₁	—	4	16	no	—
66	C₄⋊C₄×C₁₃₁	—	4	4	no	—
67	C₄⋊C₄×C₁₃₁	—	4	4	no	—
68	C₄²×C₁₃₁	C₄⋊C₄	1	8	yes	—
69	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
70	C₄²×C₁₃₁	C₂×Q₈	1	4	yes	—
71	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₁₃₁	#124	1	2	yes	—
73	C₂³×C₁₃₁	#124	1	2	yes	—
74	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
75	C₂³×C₁₃₁	#137	1	4	yes	—
76	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₁	—	2	4	no	—
78	C₂×C₄×C₁₃₁	—	2	4	no	—
79	C₂×C₄×C₁₃₁	—	2	16	no	—
80	C₂×C₄×C₁₃₁	—	4	4	no	—
81	C₂×C₄×C₁₃₁	—	4	4	no	—
82	C₂×C₄×C₁₃₁	—	4	8	no	—
83	C₂×C₄×C₁₃₁	—	4	4	no	—
84	C₂×C₄×C₁₃₁	—	4	4	no	—
85	C₂×C₄×C₁₃₁	—	8	8	no	—
86	C₂×C₄×C₁₃₁	—	8	8	no	—
87	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	8	yes	—
88	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₃₁	#117	1	8	yes	—
90	C₂×C₄×C₁₃₁	#124	1	2	yes	—
91	C₂×C₄×C₁₃₁	#124	1	2	yes	—
92	C₂×C₄×C₁₃₁	#125	1	8	yes	—
93	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
97	C₈×C₁₃₁	—	2	8	no	—
98	C₂²×C₁₃₁	#382	1	4	yes	—
99	C₂²×C₁₃₁	#428	1	8	yes	—
100	C₂²×C₁₃₁	#489	1	8	yes	—
101	C₂²×C₁₃₁	#490	1	4	yes	—
102	C₂²×C₁₃₁	#491	1	4	yes	—
103	C₂²×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
105	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
106	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
107	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
108	C₄×C₁₃₁	—	2	8	no	—
109	C₄×C₁₃₁	—	8	4	no	—
110	C₄×C₁₃₁	—	8	4	no	—
111	C₄×C₁₃₁	#382	1	4	yes	—
112	C₄×C₁₃₁	C₂×#117	1	2	yes	—
113	C₄×C₁₃₁	C₂×#125	1	2	yes	—
114	C₂×C₁₃₁	#1836	1	8	yes	direct product factor
115	C₂×C₁₃₁	C₂×#382	1	1	yes	characteristic
116	C₂×C₁₃₁	C₂×#428	1	2	yes	—
117	C₂×C₁₃₁	C₂×#489	1	2	yes	—
118	C₂×C₁₃₁	C₂×#490	1	1	yes	characteristic
119	C₂×C₁₃₁	C₂×#491	1	1	yes	characteristic
120	C₂×#1836	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₃₁	C₂×#1836	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×#97	C₂×C₁₃₁	1	1	yes	characteristic
123	C₂×#384	C₂×C₁₃₁	1	1	yes	characteristic
124	C₂×#382	C₂×C₁₃₁	1	1	yes	characteristic
125	#1836	C₂×C₁₃₁	1	8	yes	direct product factor
126	C₂×#340	C₂×C₁₃₁	1	4	yes	—
127	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
128	C₂²×C₄⋊C₄	—	2	2	no	—
129	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
131	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
132	C₂×#97	C₄×C₁₃₁	1	8	yes	complemented
133	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
134	#384	—	2	2	no	—
135	#384	C₂²×C₁₃₁	1	4	yes	—
136	#382	—	2	2	no	—
137	#382	C₂²×C₁₃₁	1	4	yes	—
138	C₂×#95	—	2	2	no	—
139	#340	C₂²×C₁₃₁	1	16	yes	—
140	C₂³×C₄	—	2	2	no	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	D₄×C₁₃₁	1	2	yes	—
143	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
147	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	8	no	—
149	C₂×C₄⋊C₄	—	2	8	no	—
150	C₂×C₄⋊C₄	—	2	8	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	4	8	no	—
154	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
155	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
156	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
157	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
158	C₂×C₄²	Q₈×C₁₃₁	1	4	yes	—
159	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
160	#97	C₂×C₄×C₁₃₁	1	16	yes	complemented
161	#95	—	2	8	no	—
162	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	—	4	8	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	8	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
175	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
176	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
177	C₂²×C₄	#41×C₁₃₁	1	4	yes	—
178	C₂²×C₄	#41×C₁₃₁	1	4	yes	—
179	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
180	C₂×C₈	—	2	16	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×C₄×C₁₃₁	1	8	yes	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	4	16	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	C₄⋊C₄×C₁₃₁	1	8	yes	—
190	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
191	C₄²	C₂×Q₈×C₁₃₁	1	4	yes	—
192	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
193	C₂³	#124×C₁₃₁	1	2	yes	—
194	C₂³	#124×C₁₃₁	1	2	yes	—
195	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
196	C₂³	#137×C₁₃₁	1	4	yes	—
197	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	8	8	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	8	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
210	C₂×C₄	#117×C₁₃₁	1	8	yes	—
211	C₂×C₄	#124×C₁₃₁	1	2	yes	—
212	C₂×C₄	#124×C₁₃₁	1	2	yes	—
213	C₂×C₄	#125×C₁₃₁	1	8	yes	—
214	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
217	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
218	C₈	—	2	8	no	—
219	C₂²	#382×C₁₃₁	1	4	yes	—
220	C₂²	#428×C₁₃₁	1	8	yes	—
221	C₂²	#489×C₁₃₁	1	8	yes	—
222	C₂²	#490×C₁₃₁	1	4	yes	—
223	C₂²	#491×C₁₃₁	1	4	yes	—
224	C₂²	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
225	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
226	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
227	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
228	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
229	C₄	—	2	8	no	—
230	C₄	—	8	4	no	—
231	C₄	—	8	4	no	—
232	C₄	#382×C₁₃₁	1	4	yes	—
233	C₄	C₂×#117×C₁₃₁	1	2	yes	—
234	C₄	C₂×#125×C₁₃₁	1	2	yes	—
235	C₂	#1836×C₁₃₁	1	8	yes	direct product factor
236	C₂	C₂×#382×C₁₃₁	1	1	yes	characteristic
237	C₂	C₂×#428×C₁₃₁	1	2	yes	—
238	C₂	C₂×#489×C₁₃₁	1	2	yes	—
239	C₂	C₂×#490×C₁₃₁	1	1	yes	characteristic
240	C₂	C₂×#491×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1836×C₁₃₁	C₂	direct (N × H)
11. C₂×#97×C₁₃₁	C₄	semidirect
39. #97×C₁₃₁	C₂×C₄	semidirect
114. C₂×C₁₃₁	#1836	direct (N × H)
120. C₂×#1836	C₁₃₁	direct (N × H)
121. C₁₃₁	C₂×#1836	direct (N × H)
125. #1836	C₂×C₁₃₁	direct (N × H)
132. C₂×#97	C₄×C₁₃₁	semidirect
160. #97	C₂×C₄×C₁₃₁	semidirect
235. C₂	#1836×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
C₂²×#97×C₁₃₁	2
C₂×#384×C₁₃₁	2
C₂×#382×C₁₃₁	2
#1836×C₁₃₁	2
C₂×#340×C₁₃₁	2
C₂×#1836	131

Subgroups — C2×#1836×C131 #45706645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1836×C₁₃₁ #45706645