Subgroups — C₂×#14528936 #17034703

Order 4352.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#3622	C₂	1	6	yes	maximal
2	#14528936	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄³×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#3622	C₂²	1	4	yes	—
5	C₂×C₄⋊#3622	—	2	24	no	—
6	C₂×C₄⋊#3622	C₂²	1	24	yes	—
7	C₂²×C₄²×C₁₇	C₄	1	3	yes	—
8	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₁₇	C₂²	1	3	yes	—
10	C₄³×C₁₇	C₄	1	4	yes	—
11	C₄³×C₁₇	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₇	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₁₇	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	—	2	12	no	—
16	C₂×C₄²×C₁₇	C₈	1	8	yes	complemented
17	C₂×C₄²×C₁₇	C₂×C₄	1	12	yes	—
18	C₂×C₄²×C₁₇	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₇	D₄	1	12	yes	—
21	C₂×C₄²×C₁₇	Q₈	1	12	yes	—
22	C₂×C₄²×C₁₇	C₂³	1	6	yes	—
23	C₂²×#3622	—	2	24	no	—
24	C₂²×#3622	C₂³	1	8	yes	—
25	C₄⋊#3622	—	2	48	no	—
26	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	—	2	12	no	—
28	C₂²×C₄×C₁₇	C₄⋊C₄	1	12	yes	—
29	C₂²×C₄×C₁₇	C₄⋊C₄	1	12	yes	—
30	C₂²×C₄×C₁₇	C₂×C₈	1	12	yes	—
31	C₂²×C₄×C₁₇	M₁₆	1	12	yes	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₁₇	C₂×Q₈	1	6	yes	—
37	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
38	C₄²×C₁₇	—	2	24	no	—
39	C₄²×C₁₇	C₄⋊C₄	1	24	yes	—
40	C₄²×C₁₇	C₂×C₈	1	16	yes	complemented
41	C₄²×C₁₇	C₂×D₄	1	12	yes	—
42	C₄²×C₁₇	C₂×Q₈	1	12	yes	—
43	C₂×#3622	—	2	24	no	—
44	C₂×#3622	—	4	16	no	—
45	C₂×#1842	—	17	1	no	maximal, Sylow
46	C₂³×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
47	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
48	C₂³×C₁₇	C₂×M₁₆	1	6	yes	—
49	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	—	2	12	no	—
51	C₂×C₄×C₁₇	—	2	24	no	—
52	C₂×C₄×C₁₇	C₄⋊C₈	1	48	yes	complemented
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	12	yes	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
55	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	12	yes	—
56	C₂×C₄×C₁₇	#127	1	4	yes	—
57	C₂×C₄×C₁₇	C₄⋊Q₈	1	12	yes	—
58	C₂×C₄×C₁₇	C₂²×C₈	1	12	yes	—
59	C₂×C₄×C₁₇	C₂×M₁₆	1	12	yes	—
60	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
61	C₂×C₄×C₁₇	C₂²×Q₈	1	3	yes	—
62	#3622	—	4	8	no	—
63	C₂²×C₄⋊C₈	—	17	6	no	—
64	C₂×C₄³	D₁₇	1	1	yes	characteristic
65	#1842	—	17	8	no	—
66	C₂²×C₁₇	#384	1	4	yes	—
67	C₂²×C₁₇	C₂×C₄⋊C₈	1	12	yes	—
68	C₂²×C₁₇	#423	1	12	yes	—
69	C₂²×C₁₇	C₂²×C₄⋊C₄	1	3	yes	—
70	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
71	C₂²×C₁₇	C₂²×M₁₆	1	3	yes	—
72	C₄×C₁₇	—	2	12	no	—
73	C₄×C₁₇	#384	1	4	yes	—
74	C₄×C₁₇	C₂×C₄⋊C₈	1	24	yes	complemented
75	C₄×C₁₇	C₂×#127	1	1	yes	characteristic
76	C₄×C₁₇	C₂×C₄⋊Q₈	1	3	yes	—
77	C₂³×C₈	—	17	4	no	—
78	C₂²×C₄²	Dic₁₇	1	3	yes	—
79	C₂²×C₄²	Dic₁₇	1	1	yes	characteristic
80	C₂²×C₄²	C₂×D₁₇	1	3	yes	—
81	C₂×C₄⋊C₈	—	17	24	no	—
82	C₂×C₄⋊C₈	—	34	24	no	—
83	C₄³	Dic₁₇	1	4	yes	—
84	C₄³	C₂×D₁₇	1	4	yes	—
85	C₂×C₁₇	#1842	1	8	yes	complemented
86	C₂×C₁₇	C₂×#384	1	1	yes	characteristic
87	C₂×C₁₇	C₂²×C₄⋊C₈	1	3	yes	—
88	C₂×C₁₇	C₂×#423	1	3	yes	—
89	C₂³×C₄	C₂×Dic₁₇	1	3	yes	—
90	C₂³×C₄	C₂×Dic₁₇	1	3	yes	—
91	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
92	C₂²×C₈	—	17	8	no	—
93	C₂²×C₈	—	34	24	no	—
94	C₂×C₄²	—	2	12	no	—
95	C₂×C₄²	#3622	1	8	yes	complemented
96	C₂×C₄²	Dic₃₄	1	12	yes	—
97	C₂×C₄²	D₆₈	1	12	yes	—
98	C₂×C₄²	C₂×Dic₁₇	1	12	yes	—
99	C₂×C₄²	C₂×Dic₁₇	1	4	yes	—
100	C₂×C₄²	C₂×Dic₁₇	1	6	yes	—
101	C₂×C₄²	C₂²×D₁₇	1	6	yes	—
102	C₄⋊C₈	—	34	48	no	—
103	C₁₇	C₂×#1842	1	1	yes	characteristic, complemented, Sylow
104	C₂⁴	C₂²×Dic₁₇	1	1	yes	characteristic
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	C₂×#3622	1	12	yes	—
107	C₂²×C₄	#63223	1	12	yes	—
108	C₂²×C₄	C₄⋊Dic₁₇	1	12	yes	—
109	C₂²×C₄	C₄⋊Dic₁₇	1	12	yes	—
110	C₂²×C₄	C₂×Dic₃₄	1	6	yes	—
111	C₂²×C₄	C₂×D₆₈	1	6	yes	—
112	C₂²×C₄	C₂²×Dic₁₇	1	6	yes	—
113	C₂²×C₄	C₂²×Dic₁₇	1	6	yes	—
114	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
115	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
116	C₂×C₈	—	34	24	no	—
117	C₂×C₈	—	68	16	no	—
118	C₄²	—	2	24	no	—
119	C₄²	C₂×#3622	1	16	yes	complemented
120	C₄²	C₄⋊Dic₁₇	1	24	yes	—
121	C₄²	C₂×Dic₃₄	1	12	yes	—
122	C₄²	C₂×D₆₈	1	12	yes	—
123	C₂³	C₂²×#3622	1	2	yes	—
124	C₂³	C₂×#63223	1	6	yes	—
125	C₂³	C₂×C₄⋊Dic₁₇	1	6	yes	—
126	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	24	no	—
129	C₂×C₄	C₄⋊#3622	1	48	yes	complemented
130	C₂×C₄	#10587555	1	12	yes	—
131	C₂×C₄	#10587560	1	4	yes	—
132	C₂×C₄	C₂²×#3622	1	12	yes	—
133	C₂×C₄	C₂×#63223	1	12	yes	—
134	C₂×C₄	C₂×C₄⋊Dic₁₇	1	12	yes	—
135	C₂×C₄	C₂×C₄⋊Dic₁₇	1	12	yes	—
136	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
137	C₂×C₄	C₂²×Dic₃₄	1	3	yes	—
138	C₂×C₄	C₂²×D₆₈	1	3	yes	—
139	C₈	—	68	8	no	—
140	C₂²	#11762679	1	4	yes	—
141	C₂²	C₂×C₄⋊#3622	1	12	yes	—
142	C₂²	#11762833	1	12	yes	—
143	C₂²	C₂²×C₄⋊Dic₁₇	1	3	yes	—
144	C₂²	C₂³×#3622	1	1	yes	characteristic
145	C₂²	C₂²×#63223	1	3	yes	—
146	C₄	—	2	12	no	—
147	C₄	#11762679	1	4	yes	—
148	C₄	C₂×C₄⋊#3622	1	24	yes	complemented
149	C₄	C₂×#10587560	1	1	yes	characteristic
150	C₄	C₂×#10587555	1	3	yes	—
151	C₂	#14528936	1	8	yes	direct product factor
152	C₂	C₂×#11762679	1	1	yes	characteristic
153	C₂	C₂²×C₄⋊#3622	1	3	yes	—
154	C₂	C₂×#11762833	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×C₄³×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄³	D₁₇
hypocenter	C₁₇	C₂×#1842
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #14528936	C₂	direct (N × H)
16. C₂×C₄²×C₁₇	C₈	semidirect
40. C₄²×C₁₇	C₂×C₈	semidirect
52. C₂×C₄×C₁₇	C₄⋊C₈	semidirect
74. C₄×C₁₇	C₂×C₄⋊C₈	semidirect
85. C₂×C₁₇	#1842	semidirect
95. C₂×C₄²	#3622	semidirect
103. C₁₇	C₂×#1842	semidirect
119. C₄²	C₂×#3622	semidirect
129. C₂×C₄	C₄⋊#3622	semidirect
148. C₄	C₂×C₄⋊#3622	semidirect
151. C₂	#14528936	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×C₄⋊#3622	2
#14528936	2
C₂×C₄³×C₁₇	2
C₂×#1842	17

Subgroups — C2×#14528936 #17034703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528936 #17034703