Subgroups — C₂×C₄²×Q₈×C₁₇ #16891736

Order 4352.

3966 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₇	C₂	1	3	yes	maximal
2	C₄²×Q₈×C₁₇	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊C₄×C₁₇	C₂	1	9	yes	maximal
4	C₂×C₄³×C₁₇	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₁₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₇	C₄	1	48	yes	direct product factor
7	C₂×C₄×Q₈×C₁₇	C₂²	1	24	yes	—
8	C₂²×C₄⋊C₄×C₁₇	C₂²	1	9	yes	—
9	C₂²×C₄²×C₁₇	C₂²	1	9	yes	—
10	C₂²×C₄²×C₁₇	C₂²	1	9	yes	—
11	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₇	C₂²	1	72	yes	—
13	C₂×C₁₇×#95	C₂²	1	6	yes	—
14	C₄³×C₁₇	C₂²	1	24	yes	—
15	C₂²×Q₈×C₁₇	C₂×C₄	1	24	yes	—
16	C₂²×Q₈×C₁₇	C₂³	1	4	yes	—
17	C₂³×C₄×C₁₇	C₂³	1	9	yes	—
18	C₂³×C₄×C₁₇	C₂³	1	3	yes	—
19	C₂³×C₄×C₁₇	C₂³	1	3	yes	—
20	C₄×Q₈×C₁₇	C₂×C₄	1	192	yes	direct product factor
21	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	72	yes	—
22	C₂×C₄⋊C₄×C₁₇	C₂³	1	36	yes	—
23	C₂×C₄²×C₁₇	—	2	36	no	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	72	yes	—
25	C₂×C₄²×C₁₇	Q₈	1	8	yes	direct product factor
26	C₂×C₄²×C₁₇	C₂³	1	36	yes	—
27	C₂×C₄²×C₁₇	C₂³	1	36	yes	—
28	C₂×C₄²×C₁₇	C₂³	1	4	yes	—
29	C₁₇×#95	C₂³	1	24	yes	—
30	C₂⁴×C₁₇	C₂⁴	1	1	yes	characteristic
31	C₂×Q₈×C₁₇	C₄²	1	64	yes	direct product factor
32	C₂×Q₈×C₁₇	C₂²×C₄	1	48	yes	—
33	C₂²×C₄×C₁₇	—	2	36	no	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	36	yes	—
35	C₂²×C₄×C₁₇	C₂²×C₄	1	36	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	12	yes	—
37	C₂²×C₄×C₁₇	C₂×Q₈	1	12	yes	—
38	C₂²×C₄×C₁₇	#41	1	12	yes	—
39	C₂²×C₄×C₁₇	C₂⁴	1	18	yes	—
40	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
41	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
42	C₄⋊C₄×C₁₇	C₂²×C₄	1	144	yes	—
43	C₄²×C₁₇	—	2	72	no	—
44	C₄²×C₁₇	C₂²×C₄	1	144	yes	—
45	C₄²×C₁₇	C₂×Q₈	1	16	yes	direct product factor
46	C₂×C₄²×Q₈	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₁₇	C₂³×C₄	1	6	yes	—
48	C₂³×C₁₇	C₂²×Q₈	1	2	yes	—
49	C₂³×C₁₇	C₂×#41	1	6	yes	—
50	C₂³×C₁₇	C₂⁵	1	1	yes	characteristic
51	Q₈×C₁₇	C₂×C₄²	1	64	yes	direct product factor
52	C₂×C₄×C₁₇	—	2	72	no	—
53	C₂×C₄×C₁₇	—	2	36	no	—
54	C₂×C₄×C₁₇	C₂×C₄²	1	48	yes	—
55	C₂×C₄×C₁₇	C₄×Q₈	1	48	yes	direct product factor
56	C₂×C₄×C₁₇	C₂³×C₄	1	36	yes	—
57	C₂×C₄×C₁₇	C₂³×C₄	1	12	yes	—
58	C₂×C₄×C₁₇	C₂³×C₄	1	36	yes	—
59	C₂×C₄×C₁₇	C₂²×Q₈	1	12	yes	—
60	C₂×C₄×C₁₇	C₂×#41	1	12	yes	—
61	C₂²×C₄×Q₈	C₂×C₁₇	1	3	yes	—
62	C₄²×Q₈	C₂×C₁₇	1	16	yes	direct product factor
63	C₂×C₄×C₄⋊C₄	C₂×C₁₇	1	9	yes	—
64	C₂×C₄³	C₂×C₁₇	1	3	yes	—
65	C₂²×C₁₇	C₂²×C₄²	1	4	yes	—
66	C₂²×C₁₇	C₂×C₄×Q₈	1	12	yes	—
67	C₂²×C₁₇	C₄×#41	1	12	yes	—
68	C₂²×C₁₇	C₂⁴×C₄	1	3	yes	—
69	C₂²×C₁₇	C₂³×Q₈	1	1	yes	characteristic
70	C₂²×C₁₇	C₂²×#41	1	3	yes	—
71	C₄×C₁₇	—	2	36	no	—
72	C₄×C₁₇	C₂²×C₄²	1	24	yes	—
73	C₄×C₁₇	C₂×C₄×Q₈	1	24	yes	direct product factor
74	C₂³×Q₈	C₂²×C₁₇	1	1	yes	characteristic
75	C₂×C₄×Q₈	C₄×C₁₇	1	48	yes	direct product factor
76	C₂×C₄×Q₈	C₂²×C₁₇	1	24	yes	—
77	C₂²×C₄⋊C₄	C₂²×C₁₇	1	9	yes	—
78	C₂²×C₄²	C₂²×C₁₇	1	9	yes	—
79	C₂²×C₄²	C₂²×C₁₇	1	9	yes	—
80	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂²×C₁₇	1	72	yes	—
82	C₂×#95	C₂²×C₁₇	1	6	yes	—
83	C₄³	C₂²×C₁₇	1	24	yes	—
84	C₂×C₁₇	C₄²×Q₈	1	8	yes	direct product factor
85	C₂×C₁₇	C₂³×C₄²	1	1	yes	characteristic
86	C₂×C₁₇	C₂²×C₄×Q₈	1	3	yes	—
87	C₂×C₁₇	C₂×C₄×#41	1	3	yes	—
88	C₂²×Q₈	C₂×C₄×C₁₇	1	24	yes	—
89	C₂²×Q₈	C₂³×C₁₇	1	4	yes	—
90	C₂³×C₄	C₂³×C₁₇	1	9	yes	—
91	C₂³×C₄	C₂³×C₁₇	1	3	yes	—
92	C₂³×C₄	C₂³×C₁₇	1	3	yes	—
93	C₄×Q₈	C₂×C₄×C₁₇	1	192	yes	direct product factor
94	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	72	yes	—
95	C₂×C₄⋊C₄	C₂³×C₁₇	1	36	yes	—
96	C₂×C₄²	—	2	36	no	—
97	C₂×C₄²	C₂×C₄×C₁₇	1	72	yes	—
98	C₂×C₄²	Q₈×C₁₇	1	8	yes	direct product factor
99	C₂×C₄²	C₂³×C₁₇	1	36	yes	—
100	C₂×C₄²	C₂³×C₁₇	1	36	yes	—
101	C₂×C₄²	C₂³×C₁₇	1	4	yes	—
102	#95	C₂³×C₁₇	1	24	yes	—
103	C₁₇	C₂×C₄²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
104	C₂⁴	C₂⁴×C₁₇	1	1	yes	characteristic
105	C₂×Q₈	C₄²×C₁₇	1	64	yes	direct product factor
106	C₂×Q₈	C₂²×C₄×C₁₇	1	48	yes	—
107	C₂²×C₄	—	2	36	no	—
108	C₂²×C₄	C₂²×C₄×C₁₇	1	36	yes	—
109	C₂²×C₄	C₂²×C₄×C₁₇	1	36	yes	—
110	C₂²×C₄	C₂²×C₄×C₁₇	1	12	yes	—
111	C₂²×C₄	C₂×Q₈×C₁₇	1	12	yes	—
112	C₂²×C₄	#41×C₁₇	1	12	yes	—
113	C₂²×C₄	C₂⁴×C₁₇	1	18	yes	—
114	C₂²×C₄	C₂⁴×C₁₇	1	6	yes	—
115	C₂²×C₄	C₂⁴×C₁₇	1	6	yes	—
116	C₄⋊C₄	C₂²×C₄×C₁₇	1	144	yes	—
117	C₄²	—	2	72	no	—
118	C₄²	C₂²×C₄×C₁₇	1	144	yes	—
119	C₄²	C₂×Q₈×C₁₇	1	16	yes	direct product factor
120	C₂³	C₂³×C₄×C₁₇	1	6	yes	—
121	C₂³	C₂²×Q₈×C₁₇	1	2	yes	—
122	C₂³	C₂×#41×C₁₇	1	6	yes	—
123	C₂³	C₂⁵×C₁₇	1	1	yes	characteristic
124	Q₈	C₂×C₄²×C₁₇	1	64	yes	direct product factor
125	C₂×C₄	—	2	72	no	—
126	C₂×C₄	—	2	36	no	—
127	C₂×C₄	C₂×C₄²×C₁₇	1	48	yes	—
128	C₂×C₄	C₄×Q₈×C₁₇	1	48	yes	direct product factor
129	C₂×C₄	C₂³×C₄×C₁₇	1	36	yes	—
130	C₂×C₄	C₂³×C₄×C₁₇	1	36	yes	—
131	C₂×C₄	C₂³×C₄×C₁₇	1	12	yes	—
132	C₂×C₄	C₂²×Q₈×C₁₇	1	12	yes	—
133	C₂×C₄	C₂×#41×C₁₇	1	12	yes	—
134	C₂²	C₂²×C₄²×C₁₇	1	4	yes	—
135	C₂²	C₂×C₄×Q₈×C₁₇	1	12	yes	—
136	C₂²	C₄×#41×C₁₇	1	12	yes	—
137	C₂²	C₂⁴×C₄×C₁₇	1	3	yes	—
138	C₂²	C₂³×Q₈×C₁₇	1	1	yes	characteristic
139	C₂²	C₂²×#41×C₁₇	1	3	yes	—
140	C₄	—	2	36	no	—
141	C₄	C₂²×C₄²×C₁₇	1	24	yes	—
142	C₄	C₂×C₄×Q₈×C₁₇	1	24	yes	direct product factor
143	C₂	C₄²×Q₈×C₁₇	1	8	yes	direct product factor
144	C₂	C₂³×C₄²×C₁₇	1	1	yes	characteristic
145	C₂	C₂²×C₄×Q₈×C₁₇	1	3	yes	—
146	C₂	C₂×C₄×#41×C₁₇	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₁₇	C₂²
commutator	C₂	C₂³×C₄²×C₁₇
Frattini, cosocle	C₂³	C₂⁵×C₁₇
socle, prime radical	C₂⁴×C₁₇	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈×C₁₇	C₂	direct (N × H)
6. C₂×C₄×Q₈×C₁₇	C₄	direct (N × H)
20. C₄×Q₈×C₁₇	C₂×C₄	direct (N × H)
25. C₂×C₄²×C₁₇	Q₈	direct (N × H)
31. C₂×Q₈×C₁₇	C₄²	direct (N × H)
45. C₄²×C₁₇	C₂×Q₈	direct (N × H)
46. C₂×C₄²×Q₈	C₁₇	direct (N × H)
51. Q₈×C₁₇	C₂×C₄²	direct (N × H)
55. C₂×C₄×C₁₇	C₄×Q₈	direct (N × H)
62. C₄²×Q₈	C₂×C₁₇	direct (N × H)
73. C₄×C₁₇	C₂×C₄×Q₈	direct (N × H)
75. C₂×C₄×Q₈	C₄×C₁₇	direct (N × H)
84. C₂×C₁₇	C₄²×Q₈	direct (N × H)
93. C₄×Q₈	C₂×C₄×C₁₇	direct (N × H)
98. C₂×C₄²	Q₈×C₁₇	direct (N × H)
103. C₁₇	C₂×C₄²×Q₈	direct (N × H)
105. C₂×Q₈	C₄²×C₁₇	direct (N × H)
119. C₄²	C₂×Q₈×C₁₇	direct (N × H)
124. Q₈	C₂×C₄²×C₁₇	direct (N × H)
128. C₂×C₄	C₄×Q₈×C₁₇	direct (N × H)
142. C₄	C₂×C₄×Q₈×C₁₇	direct (N × H)
143. C₂	C₄²×Q₈×C₁₇	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×Q₈×C₁₇	2
C₄²×Q₈×C₁₇	2
C₂×C₄×C₄⋊C₄×C₁₇	2
C₂×C₄³×C₁₇	2
C₂×C₄²×Q₈	17

Subgroups — C2×C42×Q8×C17 #16891736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×Q₈×C₁₇ #16891736