Subgroups — C₂×#1388 #12570

Order 256.

277 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×#108	C₂	1	1	yes	maximal, characteristic
4	#1388	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆	—	2	1	no	—
6	C₂²×M₁₆	—	2	1	no	—
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×M₃₂	—	2	1	no	—
9	C₂×M₃₂	—	2	1	no	—
10	C₂×M₃₂	—	2	1	no	—
11	C₂×M₃₂	C₄	1	2	yes	—
12	C₂×M₃₂	C₄	1	2	yes	—
13	C₂×M₃₂	C₂²	1	2	yes	—
14	C₂²×C₁₆	—	2	1	no	—
15	C₂×#108	—	2	1	no	—
16	C₂×#108	—	2	1	no	—
17	C₂×#108	—	2	1	no	—
18	C₂×#108	—	2	1	no	—
19	C₂×#108	—	2	1	no	—
20	C₂×#108	—	2	1	no	—
21	C₂×#108	C₄	1	2	yes	—
22	C₂×#108	C₄	1	2	yes	—
23	C₂×#108	C₄	1	2	yes	—
24	C₂×#108	C₄	1	2	yes	—
25	C₂×#108	C₂²	1	2	yes	—
26	C₂×#108	C₂²	1	2	yes	—
27	C₂³×C₄	D₄	1	1	yes	characteristic
28	C₂×M₁₆	—	2	2	no	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂×M₁₆	—	4	1	no	—
35	C₂×M₁₆	—	4	1	no	—
36	C₂×M₁₆	—	4	1	no	—
37	C₂×M₁₆	—	4	1	no	—
38	C₂×M₁₆	—	4	1	no	—
39	C₂×M₁₆	—	4	1	no	—
40	C₂²×C₈	—	4	1	no	—
41	C₂²×C₈	—	4	1	no	—
42	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₈	D₄	1	1	yes	characteristic
51	C₂²×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₈	Q₈	1	1	yes	characteristic
54	C₂²×C₈	Q₈	1	1	yes	characteristic
55	C₂²×C₈	C₂³	1	1	yes	characteristic
56	M₃₂	—	2	2	no	—
57	M₃₂	—	2	2	no	—
58	M₃₂	—	2	2	no	—
59	M₃₂	C₂×C₄	1	4	yes	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	#108	—	2	2	no	—
64	#108	—	2	2	no	—
65	#108	—	2	2	no	—
66	#108	—	2	2	no	—
67	#108	—	2	2	no	—
68	#108	—	2	2	no	—
69	#108	C₂×C₄	1	4	yes	—
70	#108	C₂×C₄	1	4	yes	—
71	C₂⁴	Q₁₆	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	D₈	1	1	yes	characteristic
79	C₂²×C₄	D₈	1	1	yes	characteristic
80	C₂²×C₄	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄	SD₁₆	1	1	yes	characteristic
84	C₂²×C₄	Q₁₆	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	M₁₆	—	2	4	no	—
88	M₁₆	—	4	2	no	—
89	M₁₆	—	4	2	no	—
90	M₁₆	—	4	2	no	—
91	M₁₆	—	4	2	no	—
92	M₁₆	—	4	2	no	—
93	M₁₆	—	4	2	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₄²	1	1	yes	characteristic
101	C₂×C₈	C₄²	1	1	yes	characteristic
102	C₂×C₈	C₄²	1	1	yes	characteristic
103	C₂×C₈	C₄²	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₂³	—	4	2	no	—
130	C₂³	#103	1	1	yes	characteristic
131	C₂³	#103	1	1	yes	characteristic
132	C₂³	#103	1	1	yes	characteristic
133	C₂³	#103	1	1	yes	characteristic
134	C₂³	#107	1	1	yes	characteristic
135	C₂³	#107	1	1	yes	characteristic
136	C₂³	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄	#95	1	1	yes	characteristic
138	C₂×C₄	#95	1	1	yes	characteristic
139	C₂×C₄	#95	1	1	yes	characteristic
140	C₂×C₄	#95	1	1	yes	characteristic
141	C₂×C₄	#102	1	1	yes	characteristic
142	C₂×C₄	#102	1	1	yes	characteristic
143	C₂×C₄	#102	1	1	yes	characteristic
144	C₂×C₄	#102	1	1	yes	characteristic
145	C₂×C₄	#102	1	1	yes	characteristic
146	C₂×C₄	#102	1	1	yes	characteristic
147	C₂×C₄	#102	1	1	yes	characteristic
148	C₂×C₄	#102	1	1	yes	characteristic
149	C₂×C₄	#103	1	1	yes	characteristic
150	C₂×C₄	#103	1	1	yes	characteristic
151	C₂×C₄	#103	1	1	yes	characteristic
152	C₂×C₄	#103	1	1	yes	characteristic
153	C₂×C₄	#106	1	1	yes	characteristic
154	C₂×C₄	#106	1	1	yes	characteristic
155	C₂×C₄	#106	1	1	yes	characteristic
156	C₂×C₄	#106	1	1	yes	characteristic
157	C₂×C₄	#107	1	1	yes	characteristic
158	C₂×C₄	#107	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
165	C₈	—	4	4	no	—
166	C₈	—	4	4	no	—
167	C₈	#95	1	1	yes	characteristic
168	C₈	#95	1	1	yes	characteristic
169	C₈	#95	1	1	yes	characteristic
170	C₈	#95	1	1	yes	characteristic
171	C₈	C₂×C₄²	1	1	yes	characteristic
172	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	#340	1	1	yes	characteristic
183	C₂²	#340	1	1	yes	characteristic
184	C₂²	#340	1	1	yes	characteristic
185	C₂²	#340	1	1	yes	characteristic
186	C₂²	C₂×#103	1	1	yes	characteristic
187	C₂²	C₂×#103	1	1	yes	characteristic
188	C₂²	C₂×#107	1	1	yes	characteristic
189	C₄	#340	1	1	yes	characteristic
190	C₄	#340	1	1	yes	characteristic
191	C₄	#340	1	1	yes	characteristic
192	C₄	#340	1	1	yes	characteristic
193	C₄	C₂×#95	1	1	yes	characteristic
194	C₄	C₂×#102	1	1	yes	characteristic
195	C₄	C₂×#102	1	1	yes	characteristic
196	C₄	C₂×#106	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#1388	1	2	yes	direct product factor
204	C₂	C₂×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#102
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	#340
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1388	C₂	direct (N × H)
203. C₂	#1388	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×M₃₂	2
C₂²×#108	2
#1388	2

Subgroups — C2×#1388 #12570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1388 #12570