Subgroups — C₂×#11762833 #14537339

Order 2176.

576 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#63223	C₂	1	2	yes	maximal
2	#11762833	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#3622	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₇	C₄	1	2	yes	—
6	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₄	1	4	yes	—
9	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	4	yes	—
12	C₂×#63223	—	2	8	no	—
13	C₂×#63223	C₂²	1	8	yes	—
14	C₂²×#3622	C₂²	1	4	yes	—
15	C₄⋊#3622	C₂²	1	16	yes	complemented
16	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	—	2	8	no	—
18	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₇	D₄	1	4	yes	—
24	C₂²×C₄×C₁₇	Q₈	1	4	yes	—
25	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
27	C₄²×C₁₇	C₂×C₄	1	4	yes	—
28	C₄²×C₁₇	C₂×C₄	1	4	yes	—
29	C₄²×C₁₇	C₂×C₄	1	4	yes	—
30	C₄²×C₁₇	C₂³	1	4	yes	—
31	#63223	—	2	16	no	—
32	C₂×#3622	—	2	8	no	—
33	C₂×#3622	C₂³	1	8	yes	—
34	C₂³×C₁₇	—	2	4	no	—
35	C₂³×C₁₇	C₄⋊C₄	1	4	yes	—
36	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
37	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	4	no	—
39	C₂×C₄×C₁₇	—	2	16	no	—
40	C₂×C₄×C₁₇	C₄⋊C₄	1	8	yes	—
41	C₂×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₇	M₁₆	1	8	yes	complemented
43	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₁₇	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
52	#3622	—	2	8	no	—
53	C₂×#423	—	17	1	no	maximal, Sylow
54	C₂²×C₁₇	—	2	4	no	—
55	C₂²×C₁₇	—	2	8	no	—
56	C₂²×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₁₇	C₂×M₁₆	1	4	yes	—
59	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₁₇	—	2	4	no	—
61	C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
62	C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
63	C₄×C₁₇	C₂×M₁₆	1	8	yes	complemented
64	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×M₁₆	—	17	2	no	—
67	C₂²×C₄²	D₁₇	1	1	yes	characteristic
68	#423	—	17	8	no	—
69	C₂×C₄⋊C₈	—	17	4	no	—
70	C₂×C₁₇	—	2	4	no	—
71	C₂×C₁₇	#423	1	4	yes	complemented
72	C₂×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₁₇	C₂²×M₁₆	1	2	yes	—
74	C₂³×C₄	Dic₁₇	1	2	yes	—
75	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
76	C₂×M₁₆	—	17	8	no	—
77	C₂×M₁₆	—	34	8	no	—
78	C₂²×C₈	—	17	4	no	—
79	C₂×C₄²	Dic₁₇	1	1	yes	characteristic
80	C₂×C₄²	Dic₁₇	1	4	yes	—
81	C₂×C₄²	Dic₁₇	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₁₇	1	4	yes	—
83	C₂×C₄²	C₂×D₁₇	1	2	yes	—
84	C₄⋊C₈	—	17	16	no	—
85	C₁₇	C₂×#423	1	1	yes	characteristic, complemented, Sylow
86	C₂⁴	C₂×Dic₁₇	1	1	yes	characteristic
87	C₂²×C₄	—	2	8	no	—
88	C₂²×C₄	Dic₃₄	1	4	yes	—
89	C₂²×C₄	D₆₈	1	4	yes	—
90	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₇	1	8	yes	—
92	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
97	M₁₆	—	34	16	no	—
98	C₂×C₈	—	17	8	no	—
99	C₂×C₈	—	34	8	no	—
100	C₄²	C₂×Dic₁₇	1	4	yes	—
101	C₄²	C₂×Dic₁₇	1	4	yes	—
102	C₄²	C₂×Dic₁₇	1	4	yes	—
103	C₄²	C₂²×D₁₇	1	4	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	C₄⋊Dic₁₇	1	4	yes	—
106	C₂³	C₂²×Dic₁₇	1	2	yes	—
107	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	16	no	—
110	C₂×C₄	#63223	1	8	yes	complemented
111	C₂×C₄	C₄⋊Dic₁₇	1	8	yes	—
112	C₂×C₄	C₄⋊Dic₁₇	1	4	yes	—
113	C₂×C₄	C₂×Dic₃₄	1	4	yes	—
114	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
115	C₂×C₄	C₂×D₆₈	1	2	yes	—
116	C₂×C₄	C₂×D₆₈	1	4	yes	—
117	C₂×C₄	C₂²×Dic₁₇	1	4	yes	—
118	C₂×C₄	C₂²×Dic₁₇	1	4	yes	—
119	C₂×C₄	C₂²×Dic₁₇	1	2	yes	—
120	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
121	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
122	C₈	—	34	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	8	no	—
125	C₂²	C₂×#63223	1	4	yes	—
126	C₂²	C₂×C₄⋊Dic₁₇	1	4	yes	—
127	C₂²	C₂×C₄⋊Dic₁₇	1	2	yes	—
128	C₂²	C₂³×Dic₁₇	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	C₂×#63223	1	8	yes	complemented
131	C₄	C₂×C₄⋊Dic₁₇	1	4	yes	—
132	C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
133	C₄	C₂²×Dic₃₄	1	1	yes	characteristic
134	C₄	C₂²×D₆₈	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#11762833	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
138	C₂	C₂²×#63223	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂²×C₄²×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₄
prime radical	C₂⁴×C₁₇	C₂×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₇
hypocenter	C₁₇	C₂×#423
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #11762833	C₂	direct (N × H)
3. C₂×C₄⋊#3622	C₂	semidirect
15. C₄⋊#3622	C₂²	semidirect
42. C₂×C₄×C₁₇	M₁₆	semidirect
63. C₄×C₁₇	C₂×M₁₆	semidirect
71. C₂×C₁₇	#423	semidirect
85. C₁₇	C₂×#423	semidirect
110. C₂×C₄	#63223	semidirect
130. C₄	C₂×#63223	semidirect
136. C₂	#11762833	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×#63223	2
#11762833	2
C₂×C₄⋊#3622	2
C₂²×C₄²×C₁₇	2
C₂×#423	17

Subgroups — C2×#11762833 #14537339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762833 #14537339