Subgroups — C₂×C₄×#347 #492829

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 185 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#347	C₂	1	4	yes	maximal
2	C₂×C₄×M₃₂	C₂	1	2	yes	maximal
3	C₂×C₄×#97	C₂	1	1	yes	maximal, characteristic
4	C₄×#347	C₂	1	8	yes	maximal, direct product factor
5	C₂²×M₃₂	C₂²	1	4	yes	—
6	C₂²×#97	C₄	1	2	yes	—
7	C₂²×#97	C₂²	1	2	yes	—
8	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄³	C₄	1	1	yes	characteristic
11	C₂×#347	C₄	1	32	yes	direct product factor
12	C₂×#347	C₂²	1	16	yes	—
13	C₄×M₃₂	C₄	1	8	yes	complemented
14	C₄×M₃₂	C₂²	1	8	yes	—
15	C₂×#346	C₄	1	4	yes	complemented
16	C₂×C₄×C₁₆	C₄	1	4	yes	complemented
17	C₄×#97	C₄	1	4	yes	—
18	C₄×#97	C₂²	1	4	yes	—
19	C₂³×C₈	C₂×C₄	1	2	yes	—
20	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₈	C₂³	1	1	yes	characteristic
22	C₂²×C₄²	—	2	2	no	—
23	C₂²×C₄²	C₂×C₄	1	2	yes	—
24	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×M₃₂	C₂×C₄	1	32	yes	—
26	C₂×M₃₂	C₂×C₄	1	8	yes	—
27	C₂×M₃₂	C₂³	1	8	yes	—
28	C₂²×C₁₆	C₂×C₄	1	8	yes	—
29	C₂×#97	—	2	8	no	—
30	C₂×#97	C₂×C₄	1	8	yes	—
31	C₂×#97	C₂×C₄	1	4	yes	—
32	C₂×#97	C₂×C₄	1	16	yes	—
33	C₂×#97	C₂³	1	4	yes	—
34	C₂×C₄×C₈	—	2	2	no	—
35	C₂×C₄×C₈	—	2	2	no	—
36	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
40	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₈	C₂³	1	2	yes	—
42	C₄³	—	2	2	no	—
43	C₄³	C₂×C₄	1	4	yes	—
44	#347	C₂×C₄	1	64	yes	direct product factor
45	#346	C₂×C₄	1	16	yes	complemented
46	C₄×C₁₆	C₂×C₄	1	16	yes	complemented
47	C₂³×C₄	—	2	2	no	—
48	C₂³×C₄	M₁₆	1	2	yes	—
49	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₂²×C₈	—	2	4	no	—
53	C₂²×C₈	C₄²	1	8	yes	—
54	C₂²×C₈	C₄²	1	2	yes	—
55	C₂²×C₈	C₄²	1	2	yes	—
56	C₂²×C₈	M₁₆	1	8	yes	—
57	C₂²×C₈	C₂²×C₄	1	4	yes	—
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₄	1	2	yes	—
60	C₂²×C₈	C₂²×C₄	1	2	yes	—
61	C₂²×C₈	C₂²×C₄	1	8	yes	—
62	C₂²×C₈	C₂²×C₄	1	2	yes	—
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₂×C₄²	—	2	2	no	—
65	C₂×C₄²	—	2	4	no	—
66	C₂×C₄²	—	2	8	no	—
67	C₂×C₄²	—	4	8	no	—
68	C₂×C₄²	C₄²	1	8	yes	—
69	C₂×C₄²	C₄²	1	1	yes	characteristic
70	C₂×C₄²	C₄²	1	2	yes	—
71	C₂×C₄²	C₄²	1	1	yes	characteristic
72	C₂×C₄²	M₁₆	1	2	yes	—
73	C₂×C₄²	M₁₆	1	2	yes	—
74	C₂×C₄²	M₁₆	1	2	yes	—
75	C₂×C₄²	C₂²×C₄	1	4	yes	—
76	C₂×C₄²	C₂²×C₄	1	2	yes	—
77	M₃₂	C₄²	1	32	yes	complemented
78	M₃₂	C₂²×C₄	1	32	yes	—
79	C₂×C₁₆	C₄²	1	32	yes	complemented
80	C₂×C₁₆	C₂²×C₄	1	16	yes	—
81	#97	—	2	16	no	—
82	#97	C₄²	1	16	yes	—
83	#97	C₂²×C₄	1	16	yes	—
84	C₄×C₈	—	2	8	no	—
85	C₄×C₈	C₄²	1	4	yes	—
86	C₄×C₈	C₄²	1	4	yes	—
87	C₄×C₈	C₂²×C₄	1	4	yes	—
88	C₄×C₈	C₂²×C₄	1	4	yes	—
89	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
90	C₂²×C₄	—	2	8	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	—	4	8	no	—
94	C₂²×C₄	#97	1	2	yes	—
95	C₂²×C₄	#97	1	4	yes	—
96	C₂²×C₄	#97	1	2	yes	—
97	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₄²	1	4	yes	—
100	C₂²×C₄	C₂×M₁₆	1	4	yes	—
101	C₂²×C₄	C₂×M₁₆	1	4	yes	—
102	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
103	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
104	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	2	8	no	—
110	C₂×C₈	#97	1	8	yes	—
111	C₂×C₈	#97	1	8	yes	—
112	C₂×C₈	C₂×C₄²	1	8	yes	—
113	C₂×C₈	C₂×C₄²	1	2	yes	—
114	C₂×C₈	C₂×C₄²	1	4	yes	—
115	C₂×C₈	C₂×C₄²	1	8	yes	—
116	C₂×C₈	C₂×C₄²	1	8	yes	—
117	C₂×C₈	C₂×C₄²	1	4	yes	—
118	C₂×C₈	C₂×M₁₆	1	8	yes	—
119	C₂×C₈	C₂³×C₄	1	2	yes	—
120	C₂×C₈	C₂³×C₄	1	4	yes	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	8	no	—
123	C₄²	—	4	16	no	—
124	C₄²	#97	1	2	yes	—
125	C₄²	#97	1	4	yes	—
126	C₄²	#97	1	2	yes	—
127	C₄²	C₂×C₄²	1	8	yes	—
128	C₄²	C₂×C₄²	1	2	yes	—
129	C₄²	C₂×C₄²	1	2	yes	—
130	C₄²	C₂×M₁₆	1	4	yes	—
131	C₁₆	C₂×C₄²	1	32	yes	complemented
132	C₂³	—	2	4	no	—
133	C₂³	C₂×#97	1	1	yes	characteristic
134	C₂³	C₂×#97	1	1	yes	characteristic
135	C₂³	C₄×M₁₆	1	4	yes	—
136	C₂³	C₂²×M₁₆	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	4	8	no	—
142	C₂×C₄	—	4	16	no	—
143	C₂×C₄	#347	1	8	yes	direct product factor
144	C₂×C₄	C₄³	1	2	yes	—
145	C₂×C₄	C₄³	1	2	yes	—
146	C₂×C₄	C₂×#97	1	4	yes	—
147	C₂×C₄	C₂×#97	1	2	yes	—
148	C₂×C₄	C₂×#97	1	2	yes	—
149	C₂×C₄	C₂×#97	1	4	yes	—
150	C₂×C₄	C₂×#97	1	1	yes	characteristic
151	C₂×C₄	C₂×#97	1	1	yes	characteristic
152	C₂×C₄	C₄×M₁₆	1	4	yes	—
153	C₂×C₄	C₄×M₁₆	1	4	yes	—
154	C₂×C₄	C₄×M₁₆	1	4	yes	—
155	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₄²	1	2	yes	—
157	C₂×C₄	C₂²×M₁₆	1	2	yes	—
158	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
159	C₈	—	2	8	no	—
160	C₈	C₄³	1	4	yes	—
161	C₈	C₂×#97	1	8	yes	—
162	C₈	C₂²×C₄²	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	8	no	—
166	C₂²	C₄×#97	1	2	yes	—
167	C₂²	C₄×#97	1	2	yes	—
168	C₂²	C₂×#347	1	4	yes	—
169	C₂²	C₂²×#97	1	1	yes	characteristic
170	C₂²	C₂×C₄×M₁₆	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	4	8	no	—
173	C₄	C₄×#97	1	2	yes	—
174	C₄	C₄×#97	1	2	yes	—
175	C₄	C₂×#347	1	8	yes	direct product factor
176	C₄	C₂×C₄³	1	1	yes	characteristic
177	C₄	C₂²×#97	1	1	yes	characteristic
178	C₄	C₂×C₄×M₁₆	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	2	no	—
181	C₂	C₄×#347	1	4	yes	direct product factor
182	C₂	C₂×C₄×#97	1	1	yes	characteristic
183	C₂	C₂²×#347	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ 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. C₄×#347	C₂	direct (N × H)
11. C₂×#347	C₄	direct (N × H)
13. C₄×M₃₂	C₄	semidirect
15. C₂×#346	C₄	semidirect
16. C₂×C₄×C₁₆	C₄	semidirect
44. #347	C₂×C₄	direct (N × H)
45. #346	C₂×C₄	semidirect
46. C₄×C₁₆	C₂×C₄	semidirect
77. M₃₂	C₄²	semidirect
79. C₂×C₁₆	C₄²	semidirect
131. C₁₆	C₂×C₄²	semidirect
143. C₂×C₄	#347	direct (N × H)
175. C₄	C₂×#347	direct (N × H)
181. C₂	C₄×#347	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×#347	2
C₂×C₄×M₃₂	2
C₂×C₄×#97	2
C₄×#347	2

Subgroups — C2×C4×#347 #492829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#347 #492829