Subgroups — C₂×#72216719 #72337613

Order 4096.

At least 10502 conjugacy classes of subgroups; only the 339 normal subgroups are shown.

The subgroup lattice has 166 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#72216329	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#72216427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72210578	C₂	1	1	yes	maximal, characteristic
4	C₂×#72215653	C₂	1	2	yes	maximal, complemented
5	C₂×#110086	C₄	1	2	yes	complemented
6	C₂×#648360	C₂²	1	1	yes	characteristic
7	C₂×#233541	C₄	1	2	yes	complemented
8	C₂×#72210489	C₂	1	2	yes	maximal, complemented
9	C₂×#133822	C₂²	1	2	yes	—
10	C₂×#115690	C₂²	1	2	yes	—
11	C₂×#115690	C₂²	1	1	yes	characteristic
12	C₂×#6499867	C₂²	1	1	yes	characteristic
13	#72216719	C₂	1	8	yes	maximal, direct product factor
14	#72216427	C₂²	1	4	yes	complemented
15	#72216329	C₂²	1	4	yes	complemented
16	#72215653	C₂²	1	8	yes	complemented
17	#72210578	C₄	1	4	yes	complemented
18	#72210578	C₂²	1	4	yes	—
19	#72210489	C₂²	1	8	yes	complemented
20	C₂×#62817	D₄	1	1	yes	characteristic
21	C₂²×#2846	D₄	1	2	yes	—
22	C₂²×#2846	C₂³	1	1	yes	characteristic
23	C₂²×#2403	C₂×C₄	1	1	yes	characteristic, complemented
24	#6499867	C₂×C₄	1	2	yes	—
25	#6499867	D₄	1	2	yes	—
26	#6499867	D₄	1	2	yes	—
27	#6499867	C₂³	1	2	yes	—
28	C₂×#21746	C₂×C₄	1	2	yes	—
29	C₂×#21746	D₄	1	1	yes	characteristic
30	#648360	C₂×C₄	1	2	yes	complemented
31	#648360	C₂³	1	2	yes	—
32	C₂×C₂²⋊C₄²	C₂×C₄	1	1	yes	characteristic, complemented
33	#233541	C₂×C₄	1	8	yes	complemented
34	C₂×#8547	D₄	1	2	yes	complemented
35	C₂×#8541	D₄	1	2	yes	complemented
36	C₂×#8101	C₂×C₄	1	2	yes	complemented
37	#133822	D₄	1	4	yes	complemented
38	#133822	C₂³	1	4	yes	—
39	#115690	C₂×C₄	1	2	yes	complemented
40	#115690	C₂³	1	4	yes	—
41	#115690	C₂³	1	2	yes	—
42	#110086	C₂×C₄	1	8	yes	complemented
43	C₂⁸	C₂×D₄	1	1	yes	characteristic
44	#62817	C₂²⋊C₄	1	2	yes	—
45	#62817	C₂×D₄	1	2	yes	—
46	C₂³×#120	C₂²⋊C₄	1	2	yes	—
47	C₂³×#120	C₂×D₄	1	1	yes	characteristic
48	C₂³×#120	C₂×D₄	1	2	yes	—
49	C₂×#2846	C₂²⋊C₄	1	4	yes	—
50	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
51	C₂×#2846	C₂×D₄	1	1	yes	characteristic
52	C₂×#2846	C₂×D₄	1	2	yes	—
53	C₂×#2846	C₂×D₄	1	4	yes	—
54	C₂×#2846	C₂×D₄	1	1	yes	characteristic
55	C₂×#2846	#41	1	2	yes	—
56	C₂×#2846	C₂⁴	1	1	yes	characteristic
57	C₂×#2403	C₂²×C₄	1	2	yes	complemented
58	C₂²×#379	C₂²⋊C₄	1	2	yes	—
59	C₂²×#379	C₂²×C₄	1	1	yes	characteristic
60	C₂²×#379	#41	1	2	yes	—
61	#21746	C₂²⋊C₄	1	4	yes	—
62	#21746	C₂²⋊C₄	1	2	yes	—
63	#21746	C₂²×C₄	1	4	yes	—
64	#21746	C₂×D₄	1	2	yes	—
65	C₂²⋊C₄²	C₂²×C₄	1	2	yes	complemented
66	#8547	C₂×D₄	1	4	yes	complemented
67	#8541	C₂×D₄	1	4	yes	complemented
68	#8101	C₂²×C₄	1	4	yes	complemented
69	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂⁷	#121	1	2	yes	—
72	C₂⁷	#121	1	2	yes	—
73	C₂⁷	C₂²×D₄	1	1	yes	characteristic
74	C₂⁴×D₄	#99	1	2	yes	—
75	C₂⁴×D₄	#120	1	1	yes	characteristic
76	C₂²×#120	#99	1	4	yes	—
77	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
79	C₂²×#120	C₄×D₄	1	2	yes	—
80	C₂²×#120	#120	1	1	yes	characteristic
81	C₂²×#120	#120	1	2	yes	—
82	C₂²×#120	#121	1	2	yes	—
83	C₂²×#120	#123	1	1	yes	characteristic
84	C₂²×#120	C₂²×D₄	1	2	yes	—
85	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₂²⋊C₄	#99	1	2	yes	—
87	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
88	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
89	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
90	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#379	C₄×D₄	1	2	yes	—
92	C₂×#379	C₄×D₄	1	2	yes	—
93	C₂×#379	#121	1	2	yes	—
94	C₂×#379	#123	1	2	yes	—
95	C₂×#379	C₂³×C₄	1	1	yes	characteristic
96	C₂×#379	C₂×#41	1	2	yes	—
97	C₂⁶	#353	1	2	yes	complemented
98	C₂⁶	#386	1	1	yes	characteristic
99	C₂⁶	#386	1	2	yes	—
100	C₂⁶	#386	1	1	yes	characteristic
101	C₂⁶	C₂×#99	1	2	yes	—
102	C₂⁶	C₂×#99	1	2	yes	—
103	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
104	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
105	C₂⁶	C₂×#121	1	2	yes	—
106	C₂³×D₄	#353	1	4	yes	complemented
107	C₂³×D₄	#386	1	1	yes	characteristic
108	C₂³×D₄	C₂×#99	1	4	yes	—
109	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
111	C₂³×D₄	C₂×#120	1	1	yes	characteristic
112	C₂⁴×C₄	#353	1	2	yes	complemented
113	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
115	C₂²×C₂²⋊C₄	#386	1	2	yes	—
116	C₂²×C₂²⋊C₄	C₂×#99	1	4	yes	—
117	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	2	yes	—
118	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
119	C₂⁵	#1889	1	1	yes	characteristic
120	C₂⁵	#1889	1	1	yes	characteristic
121	C₂⁵	#1889	1	1	yes	characteristic
122	C₂⁵	#1898	1	1	yes	characteristic
123	C₂⁵	#1898	1	1	yes	characteristic
124	C₂⁵	#1898	1	2	yes	—
125	C₂⁵	#1913	1	1	yes	characteristic
126	C₂⁵	#1913	1	1	yes	characteristic
127	C₂⁵	C₂×#353	1	2	yes	—
128	C₂⁵	C₂×#353	1	2	yes	—
129	C₂⁵	C₂×#353	1	4	yes	complemented
130	C₂⁵	#2127	1	2	yes	—
131	C₂⁵	#2127	1	2	yes	—
132	C₂⁵	C₂×#386	1	1	yes	characteristic
133	C₂⁵	C₂²×#99	1	2	yes	—
134	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
135	C₂³×C₄	#1889	1	1	yes	characteristic
136	C₂³×C₄	C₂×#353	1	4	yes	complemented
137	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
138	C₂⁴	#9495	1	1	yes	characteristic
139	C₂⁴	#9495	1	1	yes	characteristic
140	C₂⁴	#9495	1	1	yes	characteristic
141	C₂⁴	#9495	1	1	yes	characteristic
142	C₂⁴	#13088	1	2	yes	—
143	C₂⁴	#13088	1	2	yes	—
144	C₂⁴	#13088	1	2	yes	—
145	C₂⁴	#13088	1	2	yes	—
146	C₂⁴	C₂×#1889	1	1	yes	characteristic
147	C₂⁴	C₂×#1898	1	1	yes	characteristic
148	C₂⁴	C₂×#1913	1	1	yes	characteristic
149	C₂⁴	C₂²×#353	1	2	yes	—
150	C₂⁴	C₂×#2127	1	2	yes	—
151	C₂³	#139718	1	1	yes	characteristic
152	C₂³	#139718	1	2	yes	—
153	C₂³	#139718	1	1	yes	characteristic
154	C₂³	#139743	1	2	yes	—
155	C₂³	#139743	1	2	yes	—
156	C₂³	C₂×#9495	1	1	yes	characteristic
157	C₂³	C₂×#13088	1	2	yes	—
158	C₂²	#72216717	1	1	yes	characteristic
159	C₂²	#72216717	1	2	yes	—
160	C₂²	#72216717	1	1	yes	characteristic
161	C₂²	C₂×#139743	1	2	yes	—
162	C₂²	C₂×#139718	1	1	yes	characteristic
163	C₂	#72216719	1	2	yes	direct product factor
164	C₂	C₂×#72216717	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72216717
commutator	C₂×#379	C₂³×C₄
Frattini, cosocle	C₂×#2846	C₂⁴
prime radical	C₂×#72210578	C₂
LCS	G ▷ C₂×#379 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ C₂²×#2846 ◁ G
LECS	G ▷ C₂×#2846 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#72216329	C₂	semidirect
2. C₂×#72216427	C₂	semidirect
4. C₂×#72215653	C₂	semidirect
5. C₂×#110086	C₄	semidirect
7. C₂×#233541	C₄	semidirect
8. C₂×#72210489	C₂	semidirect
13. #72216719	C₂	direct (N × H)
14. #72216427	C₂²	semidirect
15. #72216329	C₂²	semidirect
16. #72215653	C₂²	semidirect
17. #72210578	C₄	semidirect
19. #72210489	C₂²	semidirect
23. C₂²×#2403	C₂×C₄	semidirect
30. #648360	C₂×C₄	semidirect
32. C₂×C₂²⋊C₄²	C₂×C₄	semidirect
33. #233541	C₂×C₄	semidirect
34. C₂×#8547	D₄	semidirect
35. C₂×#8541	D₄	semidirect
36. C₂×#8101	C₂×C₄	semidirect
37. #133822	D₄	semidirect
39. #115690	C₂×C₄	semidirect
42. #110086	C₂×C₄	semidirect
57. C₂×#2403	C₂²×C₄	semidirect
65. C₂²⋊C₄²	C₂²×C₄	semidirect
66. #8547	C₂×D₄	semidirect
67. #8541	C₂×D₄	semidirect
68. #8101	C₂²×C₄	semidirect
97. C₂⁶	#353	semidirect
106. C₂³×D₄	#353	semidirect
112. C₂⁴×C₄	#353	semidirect
129. C₂⁵	C₂×#353	semidirect
136. C₂³×C₄	C₂×#353	semidirect
163. C₂	#72216719	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×#72216329	2
C₂×#72216427	2
C₂×#72210578	2
C₂×#72215653	2
C₂×#72210489	2
#72216719	2

Subgroups — C2×#72216719 #72337613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72216719 #72337613