Subgroups — C₁₇×#3156 #14526162

Order 2176.

532 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#137	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₇×#135	C₂	1	1	yes	maximal, characteristic
3	C₂²×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#536	C₂	1	2	yes	maximal, complemented
5	C₁₇×#497	C₂	1	4	yes	maximal, complemented
6	C₁₇×#495	C₂	1	2	yes	maximal, complemented
7	C₁₇×#494	C₂	1	2	yes	maximal, complemented
8	C₁₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₇	C₂²	1	2	yes	—
10	C₂²×Q₈×C₁₇	C₂²	1	2	yes	—
11	C₁₇×#137	—	2	8	no	—
12	C₁₇×#135	—	2	4	no	—
13	C₂×Q₁₆×C₁₇	—	2	4	no	—
14	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₁₇	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₁₇	C₂²	1	2	yes	—
18	C₂×SD₁₆×C₁₇	C₂²	1	4	yes	—
19	C₂×SD₁₆×C₁₇	C₂²	1	2	yes	—
20	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×C₄⋊Q₈	C₂²	1	4	yes	—
24	C₁₇×#124	C₂²	1	4	yes	—
25	C₁₇×#123	—	2	4	no	—
26	C₁₇×C₂²⋊Q₈	—	2	4	no	—
27	C₁₇×#117	C₂²	1	1	yes	characteristic
28	C₁₇×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₁₇	C₂²	1	2	yes	—
30	#41×C₁₇	—	2	4	no	—
31	#41×C₁₇	—	2	4	no	—
32	C₂×Q₈×C₁₇	—	2	2	no	—
33	C₂×Q₈×C₁₇	—	2	4	no	—
34	C₂×Q₈×C₁₇	—	2	4	no	—
35	C₂×Q₈×C₁₇	C₂³	1	4	yes	—
36	C₂×Q₈×C₁₇	C₂³	1	2	yes	—
37	C₂×D₄×C₁₇	—	2	2	no	—
38	C₂×D₄×C₁₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₇	—	2	2	no	—
40	C₂²×C₄×C₁₇	—	2	2	no	—
41	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
42	Q₁₆×C₁₇	—	2	4	no	—
43	Q₁₆×C₁₇	—	2	4	no	—
44	Q₁₆×C₁₇	—	2	2	no	—
45	Q₁₆×C₁₇	—	2	8	no	—
46	SD₁₆×C₁₇	—	2	4	no	—
47	SD₁₆×C₁₇	—	2	8	no	—
48	D₈×C₁₇	—	2	2	no	—
49	M₁₆×C₁₇	D₄	1	4	yes	—
50	C₂×C₈×C₁₇	D₄	1	4	yes	—
51	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₇	—	2	8	no	—
55	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
56	C₂²⋊C₄×C₁₇	—	2	8	no	—
57	C₂²⋊C₄×C₁₇	D₄	1	2	yes	—
58	C₄²×C₁₇	C₂³	1	2	yes	—
59	C₂³×C₁₇	—	2	2	no	—
60	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₁₇	—	2	4	no	—
62	Q₈×C₁₇	—	2	2	no	—
63	Q₈×C₁₇	—	2	4	no	—
64	Q₈×C₁₇	—	2	2	no	—
65	Q₈×C₁₇	—	4	4	no	—
66	D₄×C₁₇	—	2	2	no	—
67	D₄×C₁₇	—	2	2	no	—
68	D₄×C₁₇	—	4	4	no	—
69	C₂×C₄×C₁₇	—	2	4	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	4	4	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
78	C₈×C₁₇	C₂×D₄	1	2	yes	—
79	C₈×C₁₇	C₂×D₄	1	2	yes	—
80	C₈×C₁₇	C₂×D₄	1	4	yes	—
81	#3156	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₁₇	—	2	2	no	—
83	C₂²×C₁₇	—	4	2	no	—
84	C₂²×C₁₇	—	4	4	no	—
85	C₂²×C₁₇	#127	1	2	yes	—
86	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₁₇	—	2	4	no	—
88	C₄×C₁₇	—	4	2	no	—
89	C₄×C₁₇	—	4	4	no	—
90	C₄×C₁₇	#127	1	2	yes	—
91	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×#137	C₂×C₁₇	1	2	yes	complemented
94	C₂×#135	C₂×C₁₇	1	1	yes	characteristic
95	C₂²×Q₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
96	#536	C₂×C₁₇	1	2	yes	complemented
97	#497	C₂×C₁₇	1	4	yes	complemented
98	#495	C₂×C₁₇	1	2	yes	complemented
99	#494	C₂×C₁₇	1	2	yes	complemented
100	#405	C₂×C₁₇	1	1	yes	characteristic, complemented
101	C₂×C₁₇	—	2	2	no	—
102	C₂×C₁₇	—	8	2	no	—
103	C₂×C₁₇	C₂×#127	1	1	yes	characteristic
104	C₂×C₁₇	#578	1	2	yes	—
105	C₂×#41	C₂²×C₁₇	1	2	yes	—
106	C₂²×Q₈	C₂²×C₁₇	1	2	yes	—
107	#137	—	2	8	no	—
108	#135	—	2	4	no	—
109	C₂×Q₁₆	—	2	4	no	—
110	C₂×Q₁₆	C₂²×C₁₇	1	2	yes	—
111	C₂×Q₁₆	C₂²×C₁₇	1	2	yes	—
112	C₂×Q₁₆	C₂²×C₁₇	1	1	yes	characteristic
113	C₂×Q₁₆	C₂²×C₁₇	1	4	yes	—
114	C₂×SD₁₆	C₂²×C₁₇	1	2	yes	—
115	C₂×SD₁₆	C₂²×C₁₇	1	4	yes	—
116	C₂×D₈	C₂²×C₁₇	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
119	C₄⋊Q₈	C₂²×C₁₇	1	4	yes	—
120	#124	C₂²×C₁₇	1	4	yes	—
121	#123	—	2	4	no	—
122	C₂²⋊Q₈	—	2	4	no	—
123	#117	C₂²×C₁₇	1	1	yes	characteristic
124	#97	C₂²×C₁₇	1	2	yes	—
125	C₄×C₈	C₂²×C₁₇	1	2	yes	—
126	C₁₇	#3156	1	1	yes	characteristic, direct product factor, Sylow
127	#41	—	2	4	no	—
128	#41	—	2	4	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	C₂³×C₁₇	1	4	yes	—
133	C₂×Q₈	C₂³×C₁₇	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂³×C₁₇	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
139	Q₁₆	—	2	4	no	—
140	Q₁₆	—	2	4	no	—
141	Q₁₆	—	2	2	no	—
142	Q₁₆	—	2	8	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	2	8	no	—
145	D₈	—	2	2	no	—
146	M₁₆	D₄×C₁₇	1	4	yes	—
147	C₂×C₈	D₄×C₁₇	1	4	yes	—
148	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
149	C₂×C₈	C₂³×C₁₇	1	2	yes	—
150	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	D₄×C₁₇	1	2	yes	—
153	C₂²⋊C₄	—	2	8	no	—
154	C₂²⋊C₄	D₄×C₁₇	1	2	yes	—
155	C₄²	C₂³×C₁₇	1	2	yes	—
156	C₂³	—	2	2	no	—
157	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
158	Q₈	—	2	4	no	—
159	Q₈	—	2	2	no	—
160	Q₈	—	2	4	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	4	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
171	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
173	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
174	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
175	C₈	C₂×D₄×C₁₇	1	2	yes	—
176	C₈	C₂×D₄×C₁₇	1	2	yes	—
177	C₈	C₂×D₄×C₁₇	1	4	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₁₇×#127	1	2	yes	—
182	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	4	no	—
186	C₄	C₁₇×#127	1	2	yes	—
187	C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
188	C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	8	2	no	—
191	C₂	C₂×C₁₇×#127	1	1	yes	characteristic
192	C₂	C₁₇×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₂×C₁₇×#135	C₂
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
1. C₂×C₁₇×#137	C₂	semidirect
3. C₂²×Q₁₆×C₁₇	C₂	semidirect
4. C₁₇×#536	C₂	semidirect
5. C₁₇×#497	C₂	semidirect
6. C₁₇×#495	C₂	semidirect
7. C₁₇×#494	C₂	semidirect
8. C₁₇×#405	C₂	semidirect
81. #3156	C₁₇	direct (N × H)
93. C₂×#137	C₂×C₁₇	semidirect
95. C₂²×Q₁₆	C₂×C₁₇	semidirect
96. #536	C₂×C₁₇	semidirect
97. #497	C₂×C₁₇	semidirect
98. #495	C₂×C₁₇	semidirect
99. #494	C₂×C₁₇	semidirect
100. #405	C₂×C₁₇	semidirect
126. C₁₇	#3156	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×C₁₇×#137	2
C₂×C₁₇×#135	2
C₂²×Q₁₆×C₁₇	2
C₁₇×#536	2
C₁₇×#497	2
C₁₇×#495	2
C₁₇×#494	2
C₁₇×#405	2
#3156	17

Subgroups — C17×#3156 #14526162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3156 #14526162