Subgroups — C₂×#7139 #126554

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	2	yes	maximal
2	C₂²×#322	C₂	1	1	yes	maximal, characteristic
3	#7139	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂	C₄	1	4	yes	—
5	C₂²×C₄×C₈	C₄	1	2	yes	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₄×M₃₂	—	2	2	no	—
8	C₄×M₃₂	—	2	2	no	—
9	C₄×M₃₂	C₄	1	4	yes	—
10	C₄×M₃₂	C₂²	1	4	yes	—
11	C₂×#346	—	2	2	no	—
12	C₂×C₄×C₁₆	—	2	2	no	—
13	C₂×#322	—	2	2	no	—
14	C₂×#322	—	2	2	no	—
15	C₂×#322	C₄	1	2	yes	—
16	C₂×#322	C₂²	1	2	yes	—
17	C₂³×C₈	C₈	1	4	yes	—
18	C₂³×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×M₃₂	—	2	4	no	—
21	C₂×M₃₂	—	2	8	no	—
22	C₂×M₃₂	C₈	1	16	yes	complemented
23	C₂×M₃₂	C₂×C₄	1	8	yes	—
24	C₂²×C₁₆	—	2	4	no	—
25	C₂×C₄×C₈	—	2	2	no	—
26	C₂×C₄×C₈	—	2	2	no	—
27	C₂×C₄×C₈	—	2	2	no	—
28	C₂×C₄×C₈	—	2	2	no	—
29	C₂×C₄×C₈	C₈	1	4	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#346	—	2	4	no	—
41	#346	—	2	4	no	—
42	C₄×C₁₆	—	2	4	no	—
43	C₄×C₁₆	—	2	4	no	—
44	#322	—	2	4	no	—
45	#322	—	2	4	no	—
46	C₂³×C₄	C₄²	1	1	yes	characteristic
47	C₂³×C₄	C₂×C₈	1	2	yes	—
48	C₂²×C₈	—	2	4	no	—
49	C₂²×C₈	—	2	4	no	—
50	C₂²×C₈	—	4	8	no	—
51	C₂²×C₈	C₄²	1	2	yes	—
52	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₈	C₄⋊C₄	1	2	yes	—
55	C₂²×C₈	C₄⋊C₄	1	2	yes	—
56	C₂²×C₈	C₂×C₈	1	8	yes	—
57	C₂²×C₈	C₂×C₈	1	8	yes	—
58	C₂²×C₈	M₁₆	1	8	yes	—
59	C₂²×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₄²	C₄²	1	1	yes	characteristic
61	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂×C₈	1	2	yes	—
66	C₂×C₄²	M₁₆	1	2	yes	—
67	C₂×C₄²	M₁₆	1	2	yes	—
68	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
69	M₃₂	—	2	16	no	—
70	M₃₂	—	2	8	no	—
71	M₃₂	C₂×C₈	1	16	yes	complemented
72	C₂×C₁₆	—	2	16	no	—
73	C₂×C₁₆	—	2	8	no	—
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	2	4	no	—
76	C₄×C₈	—	2	4	no	—
77	C₄×C₈	—	2	4	no	—
78	C₄×C₈	C₂²⋊C₄	1	2	yes	—
79	C₄×C₈	C₂²⋊C₄	1	2	yes	—
80	C₄×C₈	C₂²⋊C₄	1	2	yes	—
81	C₄×C₈	C₄⋊C₄	1	2	yes	—
82	C₄×C₈	C₄⋊C₄	1	2	yes	—
83	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₈	C₂×D₄	1	2	yes	—
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₂⁴	C₄×C₈	1	1	yes	characteristic
89	C₂²×C₄	—	2	8	no	—
90	C₂²×C₄	#95	1	1	yes	characteristic
91	C₂²×C₄	#95	1	1	yes	characteristic
92	C₂²×C₄	C₄×C₈	1	2	yes	—
93	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
94	C₂²×C₄	C₄×C₈	1	4	yes	—
95	C₂²×C₄	#97	1	1	yes	characteristic
96	C₂²×C₄	#97	1	1	yes	characteristic
97	C₂²×C₄	#97	1	4	yes	—
98	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
99	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
100	C₂²×C₄	C₄⋊C₈	1	2	yes	—
101	C₂²×C₄	C₄⋊C₈	1	2	yes	—
102	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₄	C₂²×C₈	1	2	yes	—
104	C₂×C₈	—	2	8	no	—
105	C₂×C₈	—	4	8	no	—
106	C₂×C₈	—	4	8	no	—
107	C₂×C₈	—	4	8	no	—
108	C₂×C₈	#95	1	2	yes	—
109	C₂×C₈	#95	1	2	yes	—
110	C₂×C₈	C₄×C₈	1	4	yes	—
111	C₂×C₈	#97	1	4	yes	—
112	C₂×C₈	C₂²⋊C₈	1	8	yes	—
113	C₂×C₈	C₂²⋊C₈	1	8	yes	—
114	C₂×C₈	C₄⋊C₈	1	8	yes	—
115	C₂×C₈	C₄⋊C₈	1	8	yes	—
116	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₈	C₂²×C₈	1	4	yes	—
119	C₂×C₈	C₂×M₁₆	1	4	yes	—
120	C₄²	#95	1	1	yes	characteristic
121	C₄²	#95	1	1	yes	characteristic
122	C₄²	#97	1	2	yes	—
123	C₄²	C₂²⋊C₈	1	2	yes	—
124	C₄²	C₂²⋊C₈	1	2	yes	—
125	C₄²	C₄⋊C₈	1	2	yes	—
126	C₄²	C₄⋊C₈	1	2	yes	—
127	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₄²	C₂×M₁₆	1	2	yes	—
130	C₁₆	—	2	16	no	—
131	C₂³	—	2	4	no	—
132	C₂³	C₈²	1	2	yes	—
133	C₂³	#322	1	2	yes	—
134	C₂³	#336	1	1	yes	characteristic
135	C₂³	#336	1	1	yes	characteristic
136	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	C₈²	1	2	yes	—
143	C₂×C₄	#322	1	4	yes	—
144	C₂×C₄	#322	1	4	yes	—
145	C₂×C₄	#322	1	2	yes	—
146	C₂×C₄	#336	1	2	yes	—
147	C₂×C₄	#336	1	2	yes	—
148	C₂×C₄	#336	1	4	yes	—
149	C₂×C₄	#336	1	4	yes	—
150	C₂×C₄	#336	1	1	yes	characteristic
151	C₂×C₄	#336	1	1	yes	characteristic
152	C₂×C₄	#349	1	8	yes	—
153	C₂×C₄	C₂×#95	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
155	C₂×C₄	C₂×#97	1	1	yes	characteristic
156	C₂×C₄	C₂×#97	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊C₈	1	2	yes	—
158	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
159	C₈	—	4	8	no	—
160	C₈	#336	1	4	yes	—
161	C₈	#336	1	4	yes	—
162	C₈	C₂×C₂²⋊C₈	1	4	yes	—
163	C₈	C₂×C₄⋊C₈	1	4	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	2	4	no	—
168	C₂²	#1273	1	2	yes	—
169	C₂²	#1273	1	2	yes	—
170	C₂²	#1377	1	4	yes	—
171	C₂²	C₂×C₈²	1	1	yes	characteristic
172	C₂²	C₂×#322	1	1	yes	characteristic
173	C₂²	C₂×#336	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	#1273	1	2	yes	—
177	C₄	#1273	1	2	yes	—
178	C₄	#1377	1	4	yes	—
179	C₄	C₂×#322	1	2	yes	—
180	C₄	C₂×#336	1	2	yes	—
181	C₄	C₂×#349	1	4	yes	—
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	2	2	no	—
185	C₂	#7139	1	4	yes	direct product factor
186	C₂	C₂×#1273	1	1	yes	characteristic
187	C₂	C₂×#1377	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊C₄
commutator	C₂²	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#336
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #7139	C₂	direct (N × H)
22. C₂×M₃₂	C₈	semidirect
71. M₃₂	C₂×C₈	semidirect
185. C₂	#7139	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×C₄×M₃₂	2
C₂²×#322	2
#7139	2

Subgroups — C2×#7139 #126554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7139 #126554