Subgroups — C₁₇×#3164 #14526170

Order 2176.

340 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#544	C₂	1	2	yes	maximal
2	C₁₇×#501	C₂	1	4	yes	maximal, complemented
3	C₁₇×#500	C₂	1	2	yes	maximal, complemented
4	C₁₇×#499	C₂	1	2	yes	maximal, complemented
5	C₁₇×#428	C₂	1	2	yes	maximal
6	C₁₇×#427	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₇×#107	C₂	1	1	yes	maximal, characteristic
8	C₁₇×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₁₇×C₄⋊Q₈	—	2	2	no	—
12	C₁₇×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₁₇×#125	—	2	2	no	—
14	C₁₇×#125	C₂²	1	4	yes	—
15	C₁₇×C₂²⋊Q₈	—	2	4	no	—
16	C₁₇×#117	C₂²	1	2	yes	—
17	C₁₇×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
19	C₁₇×#107	C₂²	1	1	yes	characteristic
20	C₁₇×#107	C₂²	1	2	yes	—
21	C₁₇×#107	C₂²	1	1	yes	characteristic
22	C₁₇×#107	C₂²	1	4	yes	—
23	C₁₇×#107	C₂²	1	2	yes	—
24	C₁₇×#106	C₂²	1	4	yes	—
25	C₁₇×#106	C₂²	1	2	yes	—
26	C₁₇×#97	C₂²	1	2	yes	—
27	C₄×C₈×C₁₇	C₂²	1	2	yes	—
28	C₂×Q₈×C₁₇	—	2	4	no	—
29	C₂²×C₄×C₁₇	—	2	2	no	—
30	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₇	Q₈	1	4	yes	complemented
32	C₂×C₈×C₁₇	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂³	1	2	yes	—
36	C₄⋊C₄×C₁₇	—	2	4	no	—
37	C₄⋊C₄×C₁₇	—	2	4	no	—
38	C₄⋊C₄×C₁₇	—	2	4	no	—
39	C₄⋊C₄×C₁₇	—	2	2	no	—
40	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
44	C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₁₇	—	2	2	no	—
46	C₂²⋊C₄×C₁₇	D₄	1	2	yes	—
47	C₄²×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	C₂³	1	2	yes	—
49	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₇	—	8	4	no	—
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	4	no	—
54	C₂×C₄×C₁₇	—	8	2	no	—
55	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₇	C₂×Q₈	1	2	yes	—
62	C₈×C₁₇	C₂×Q₈	1	2	yes	—
63	C₈×C₁₇	C₂×Q₈	1	4	yes	—
64	#3164	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₁₇	—	2	1	no	—
66	C₂²×C₁₇	—	2	1	no	—
67	C₂²×C₁₇	C₄⋊Q₈	1	2	yes	—
68	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	—	4	4	no	—
72	C₄×C₁₇	—	4	2	no	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
75	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
77	#544	C₂×C₁₇	1	2	yes	—
78	#501	C₂×C₁₇	1	4	yes	complemented
79	#500	C₂×C₁₇	1	2	yes	complemented
80	#499	C₂×C₁₇	1	2	yes	complemented
81	#428	C₂×C₁₇	1	2	yes	—
82	#427	C₂×C₁₇	1	1	yes	characteristic
83	C₂×#107	C₂×C₁₇	1	1	yes	characteristic
84	#405	C₂×C₁₇	1	1	yes	characteristic
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₁₇	#576	1	1	yes	characteristic
88	C₂×C₁₇	#578	1	1	yes	characteristic
89	C₂×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
90	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
91	C₄⋊Q₈	—	2	2	no	—
92	C₄⋊Q₈	C₂²×C₁₇	1	4	yes	—
93	#125	—	2	2	no	—
94	#125	C₂²×C₁₇	1	4	yes	—
95	C₂²⋊Q₈	—	2	4	no	—
96	#117	C₂²×C₁₇	1	2	yes	—
97	#117	C₂²×C₁₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
99	#107	C₂²×C₁₇	1	2	yes	—
100	#107	C₂²×C₁₇	1	2	yes	—
101	#107	C₂²×C₁₇	1	1	yes	characteristic
102	#107	C₂²×C₁₇	1	1	yes	characteristic
103	#107	C₂²×C₁₇	1	4	yes	—
104	#106	C₂²×C₁₇	1	2	yes	—
105	#106	C₂²×C₁₇	1	4	yes	—
106	#97	C₂²×C₁₇	1	2	yes	—
107	C₄×C₈	C₂²×C₁₇	1	2	yes	—
108	C₁₇	#3164	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×Q₈	—	2	4	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
112	M₁₆	Q₈×C₁₇	1	4	yes	complemented
113	C₂×C₈	Q₈×C₁₇	1	4	yes	complemented
114	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₁₇	1	2	yes	—
116	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
122	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
123	C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
124	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
125	C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	D₄×C₁₇	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂³×C₁₇	1	2	yes	—
130	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
131	Q₈	—	8	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
137	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
139	C₂×C₄	C₂×Q₈×C₁₇	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₁₇	1	2	yes	—
141	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
142	C₈	C₂×Q₈×C₁₇	1	2	yes	—
143	C₈	C₂×Q₈×C₁₇	1	2	yes	—
144	C₈	C₂×Q₈×C₁₇	1	4	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	C₁₇×C₄⋊Q₈	1	2	yes	—
148	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	C₁₇×C₄⋊Q₈	1	2	yes	—
155	C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
156	C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×C₁₇×C₄⋊Q₈	1	1	yes	characteristic
159	C₂	C₁₇×#576	1	1	yes	characteristic
160	C₂	C₁₇×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₂³×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₁₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#501	C₂	semidirect
3. C₁₇×#500	C₂	semidirect
4. C₁₇×#499	C₂	semidirect
31. M₁₆×C₁₇	Q₈	semidirect
32. C₂×C₈×C₁₇	Q₈	semidirect
64. #3164	C₁₇	direct (N × H)
78. #501	C₂×C₁₇	semidirect
79. #500	C₂×C₁₇	semidirect
80. #499	C₂×C₁₇	semidirect
108. C₁₇	#3164	direct (N × H)
112. M₁₆	Q₈×C₁₇	semidirect
113. C₂×C₈	Q₈×C₁₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₇×#544	2
C₁₇×#501	2
C₁₇×#500	2
C₁₇×#499	2
C₁₇×#428	2
C₁₇×#427	2
C₂×C₁₇×#107	2
C₁₇×#405	2
#3164	17

Subgroups — C17×#3164 #14526170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3164 #14526170