Subgroups — C₂²×#41×Dic₁₇ #17940900

Order 4352.

12878 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₁₇	C₂	1	48	yes	maximal, direct product factor
2	C₂²×Q₈×Dic₁₇	C₂	1	2	yes	maximal, complemented
3	C₂²×D₄×Dic₁₇	C₂	1	6	yes	maximal, complemented
4	C₂²×#10587612	C₂	1	3	yes	maximal, complemented
5	C₂³×C₄×Dic₁₇	C₂	1	3	yes	maximal, complemented
6	C₂³×#41×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₁₇	C₂²	1	6	yes	—
8	#41×Dic₁₇	C₂²	1	256	yes	direct product factor
9	C₂×Q₈×Dic₁₇	C₂²	1	48	yes	complemented
10	C₂×D₄×Dic₁₇	C₂²	1	144	yes	complemented
11	C₂×#10587612	C₂²	1	72	yes	complemented
12	C₂²×C₄⋊Dic₁₇	C₂²	1	6	yes	—
13	C₂²×C₂²⋊Dic₁₇	C₂²	1	6	yes	—
14	C₂²×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₁₇	C₂²	1	3	yes	—
16	C₂²×C₄×Dic₁₇	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₁₇	C₂²	1	72	yes	complemented
18	C₂²×#41×C₁₇	C₄	1	32	yes	complemented
19	C₂²×#41×C₁₇	C₂²	1	24	yes	—
20	C₂³×Q₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₂³×D₄×C₁₇	C₂²	1	3	yes	—
22	C₂⁴×C₄×C₁₇	C₂²	1	3	yes	—
23	C₂⁵×C₁₇	C₂³	1	3	yes	—
24	C₂×#41×C₁₇	C₂×C₄	1	384	yes	complemented
25	C₂×#41×C₁₇	C₂³	1	64	yes	—
26	C₂²×Q₈×C₁₇	C₂×C₄	1	16	yes	complemented
27	C₂²×Q₈×C₁₇	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₇	C₂×C₄	1	48	yes	complemented
29	C₂²×D₄×C₁₇	C₂³	1	36	yes	—
30	C₂³×C₄×C₁₇	C₂×C₄	1	48	yes	complemented
31	C₂³×C₄×C₁₇	C₂³	1	3	yes	—
32	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₇	C₂³	1	36	yes	—
34	C₂³×Dic₁₇	—	2	48	no	—
35	C₂³×Dic₁₇	C₂³	1	6	yes	—
36	C₂³×Dic₁₇	C₂³	1	72	yes	—
37	C₂³×Dic₁₇	C₂³	1	2	yes	—
38	C₂×C₂²⋊Dic₁₇	C₂³	1	72	yes	—
39	Q₈×Dic₁₇	C₂³	1	128	yes	complemented
40	D₄×Dic₁₇	C₂³	1	384	yes	complemented
41	#10587612	C₂³	1	192	yes	complemented
42	C₂×C₄⋊Dic₁₇	C₂³	1	72	yes	—
43	C₂×C₄×Dic₁₇	C₂³	1	12	yes	—
44	C₂×C₄×Dic₁₇	C₂³	1	36	yes	—
45	C₂×C₄×Dic₁₇	C₂³	1	72	yes	—
46	C₂×C₄×Dic₁₇	C₂³	1	192	yes	complemented
47	C₂⁴×C₁₇	—	2	24	no	—
48	C₂⁴×C₁₇	C₂²×C₄	1	24	yes	—
49	C₂⁴×C₁₇	C₂⁴	1	18	yes	—
50	C₂⁴×C₁₇	C₂⁴	1	1	yes	characteristic
51	#41×C₁₇	C₂²×C₄	1	512	yes	complemented
52	C₂×Q₈×C₁₇	C₂²×C₄	1	96	yes	complemented
53	C₂×Q₈×C₁₇	C₂⁴	1	16	yes	—
54	C₂×D₄×C₁₇	C₂²×C₄	1	288	yes	complemented
55	C₂×D₄×C₁₇	C₂⁴	1	48	yes	—
56	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₁₇	C₂²×C₄	1	24	yes	—
58	C₂²×C₄×C₁₇	C₂²×C₄	1	288	yes	complemented
59	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₁₇	C₂⁴	1	18	yes	—
61	C₂²×C₄×C₁₇	C₂⁴	1	48	yes	—
62	C₂²×Dic₁₇	—	2	24	no	—
63	C₂²×Dic₁₇	—	2	288	no	—
64	C₂²×Dic₁₇	#41	1	16	yes	direct product factor
65	C₂²×Dic₁₇	C₂⁴	1	36	yes	—
66	C₂²×Dic₁₇	C₂⁴	1	96	yes	—
67	C₂²×Dic₁₇	C₂⁴	1	12	yes	—
68	C₂²⋊Dic₁₇	C₂⁴	1	96	yes	—
69	C₄⋊Dic₁₇	C₂⁴	1	96	yes	—
70	C₄×Dic₁₇	C₂⁴	1	16	yes	—
71	C₄×Dic₁₇	C₂⁴	1	48	yes	—
72	C₄×Dic₁₇	C₂⁴	1	96	yes	—
73	C₂²×C₄×#41	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	—	2	96	no	—
75	C₂³×C₁₇	—	2	72	no	—
76	C₂³×C₁₇	C₂³×C₄	1	72	yes	—
77	C₂³×C₁₇	C₂³×C₄	1	4	yes	—
78	C₂³×C₁₇	C₂×#41	1	8	yes	—
79	C₂³×C₁₇	C₂⁵	1	12	yes	—
80	C₂³×C₁₇	C₂⁵	1	3	yes	—
81	Q₈×C₁₇	C₂³×C₄	1	64	yes	complemented
82	D₄×C₁₇	C₂³×C₄	1	192	yes	complemented
83	C₂×C₄×C₁₇	C₂³×C₄	1	24	yes	—
84	C₂×C₄×C₁₇	C₂³×C₄	1	72	yes	—
85	C₂×C₄×C₁₇	C₂³×C₄	1	192	yes	complemented
86	C₂×C₄×C₁₇	C₂⁵	1	4	yes	—
87	C₂×C₄×C₁₇	C₂⁵	1	12	yes	—
88	C₂×Dic₁₇	—	2	72	no	—
89	C₂×Dic₁₇	—	2	192	no	—
90	C₂×Dic₁₇	C₂×#41	1	48	yes	direct product factor
91	C₂×Dic₁₇	C₂⁵	1	24	yes	—
92	C₂×Dic₁₇	C₂⁵	1	8	yes	—
93	C₂³×#41	D₁₇	1	1	yes	characteristic
94	C₂×C₄×#41	—	17	48	no	—
95	C₂²×C₄×Q₈	—	17	2	no	—
96	C₂²×C₄×D₄	—	17	6	no	—
97	C₂²×#117	—	17	3	no	—
98	C₂³×C₄²	—	17	3	no	—
99	C₂²×C₁₇	—	2	24	no	—
100	C₂²×C₁₇	—	2	144	no	—
101	C₂²×C₁₇	C₄×#41	1	16	yes	complemented
102	C₂²×C₁₇	C₂⁴×C₄	1	24	yes	—
103	C₂²×C₁₇	C₂⁴×C₄	1	6	yes	—
104	C₂²×C₁₇	C₂²×#41	1	12	yes	—
105	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
106	C₄×C₁₇	C₂⁴×C₄	1	8	yes	—
107	C₄×C₁₇	C₂⁴×C₄	1	24	yes	—
108	Dic₁₇	—	2	24	no	—
109	Dic₁₇	C₂²×#41	1	16	yes	direct product factor
110	C₂²×#41	Dic₁₇	1	32	yes	direct product factor
111	C₂²×#41	C₂×D₁₇	1	24	yes	—
112	C₂³×Q₈	C₂×D₁₇	1	1	yes	characteristic
113	C₂³×D₄	C₂×D₁₇	1	3	yes	—
114	C₂⁴×C₄	—	17	6	no	—
115	C₂⁴×C₄	C₂×D₁₇	1	3	yes	—
116	C₄×#41	—	17	256	no	—
117	C₂×C₄×Q₈	—	17	48	no	—
118	C₂×C₄×D₄	—	17	144	no	—
119	C₂×#117	—	17	72	no	—
120	C₂²×C₄⋊C₄	—	17	6	no	—
121	C₂²×C₂²⋊C₄	—	17	6	no	—
122	C₂²×C₄²	—	17	6	no	—
123	C₂²×C₄²	—	17	72	no	—
124	C₂²×C₄²	—	17	1	no	—
125	C₂²×C₄²	—	17	3	no	—
126	C₂×C₁₇	—	2	24	no	—
127	C₂×C₁₇	C₂×C₄×#41	1	12	yes	complemented
128	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
129	C₂×C₁₇	C₂³×#41	1	2	yes	—
130	C₂⁵	C₂²×D₁₇	1	3	yes	—
131	C₂×#41	C₂×Dic₁₇	1	384	yes	direct product factor
132	C₂×#41	C₂²×D₁₇	1	64	yes	—
133	C₂²×Q₈	C₂×Dic₁₇	1	16	yes	complemented
134	C₂²×Q₈	C₂²×D₁₇	1	12	yes	—
135	C₂²×D₄	C₂×Dic₁₇	1	48	yes	complemented
136	C₂²×D₄	C₂²×D₁₇	1	36	yes	—
137	C₂³×C₄	—	17	2	no	—
138	C₂³×C₄	—	17	6	no	—
139	C₂³×C₄	—	17	72	no	—
140	C₂³×C₄	—	34	48	no	—
141	C₂³×C₄	C₂×Dic₁₇	1	48	yes	complemented
142	C₂³×C₄	C₂²×D₁₇	1	3	yes	—
143	C₂³×C₄	C₂²×D₁₇	1	36	yes	—
144	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
145	C₄×Q₈	—	17	128	no	—
146	C₄×D₄	—	17	384	no	—
147	#117	—	17	192	no	—
148	C₂×C₄⋊C₄	—	17	72	no	—
149	C₂×C₂²⋊C₄	—	17	72	no	—
150	C₂×C₄²	—	17	36	no	—
151	C₂×C₄²	—	17	72	no	—
152	C₂×C₄²	—	17	192	no	—
153	C₂×C₄²	—	17	12	no	—
154	C₁₇	C₂²×C₄×#41	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	—	2	24	no	—
156	C₂⁴	C₂²×Dic₁₇	1	24	yes	—
157	C₂⁴	C₂³×D₁₇	1	18	yes	—
158	C₂⁴	C₂³×D₁₇	1	1	yes	characteristic
159	#41	C₂²×Dic₁₇	1	512	yes	direct product factor
160	C₂×Q₈	C₂²×Dic₁₇	1	96	yes	complemented
161	C₂×Q₈	C₂³×D₁₇	1	16	yes	—
162	C₂×D₄	C₂²×Dic₁₇	1	288	yes	complemented
163	C₂×D₄	C₂³×D₁₇	1	48	yes	—
164	C₂²×C₄	—	17	16	no	—
165	C₂²×C₄	—	17	12	no	—
166	C₂²×C₄	—	17	36	no	—
167	C₂²×C₄	—	17	96	no	—
168	C₂²×C₄	—	34	288	no	—
169	C₂²×C₄	—	34	24	no	—
170	C₂²×C₄	C₂²×Dic₁₇	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₁₇	1	24	yes	—
172	C₂²×C₄	C₂²×Dic₁₇	1	288	yes	complemented
173	C₂²×C₄	C₂³×D₁₇	1	6	yes	—
174	C₂²×C₄	C₂³×D₁₇	1	18	yes	—
175	C₂²×C₄	C₂³×D₁₇	1	48	yes	—
176	C₄⋊C₄	—	17	96	no	—
177	C₂²⋊C₄	—	17	96	no	—
178	C₄²	—	17	16	no	—
179	C₄²	—	17	48	no	—
180	C₄²	—	17	96	no	—
181	C₂³	—	2	96	no	—
182	C₂³	—	2	72	no	—
183	C₂³	#41×D₁₇	1	8	yes	—
184	C₂³	C₂³×Dic₁₇	1	72	yes	—
185	C₂³	C₂³×Dic₁₇	1	4	yes	—
186	C₂³	C₂⁴×D₁₇	1	12	yes	—
187	C₂³	C₂⁴×D₁₇	1	3	yes	—
188	Q₈	C₂³×Dic₁₇	1	64	yes	complemented
189	D₄	C₂³×Dic₁₇	1	192	yes	complemented
190	C₂×C₄	—	17	48	no	—
191	C₂×C₄	—	17	8	no	—
192	C₂×C₄	—	17	24	no	—
193	C₂×C₄	—	34	192	no	—
194	C₂×C₄	—	34	72	no	—
195	C₂×C₄	C₂³×Dic₁₇	1	24	yes	—
196	C₂×C₄	C₂³×Dic₁₇	1	72	yes	—
197	C₂×C₄	C₂³×Dic₁₇	1	192	yes	complemented
198	C₂×C₄	C₂⁴×D₁₇	1	4	yes	—
199	C₂×C₄	C₂⁴×D₁₇	1	12	yes	—
200	C₂²	—	2	144	no	—
201	C₂²	—	2	24	no	—
202	C₂²	#41×Dic₁₇	1	16	yes	direct product factor
203	C₂²	C₂⁴×Dic₁₇	1	24	yes	—
204	C₂²	C₂⁴×Dic₁₇	1	6	yes	—
205	C₂²	C₂×#41×D₁₇	1	12	yes	—
206	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
207	C₄	—	17	16	no	—
208	C₄	—	34	24	no	—
209	C₄	C₂⁴×Dic₁₇	1	8	yes	—
210	C₄	C₂⁴×Dic₁₇	1	24	yes	—
211	C₂	—	2	24	no	—
212	C₂	C₂×#41×Dic₁₇	1	12	yes	direct product factor
213	C₂	C₂⁵×Dic₁₇	1	1	yes	characteristic
214	C₂	C₂²×#41×D₁₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting, prime radical	C₂³×#41×C₁₇	C₂
socle	C₂⁴×C₁₇	C₂⁴
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂³×#41	D₁₇
hypocenter	C₁₇	C₂²×C₄×#41
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×Dic₁₇	C₂	direct (N × H)
2. C₂²×Q₈×Dic₁₇	C₂	semidirect
3. C₂²×D₄×Dic₁₇	C₂	semidirect
4. C₂²×#10587612	C₂	semidirect
5. C₂³×C₄×Dic₁₇	C₂	semidirect
8. #41×Dic₁₇	C₂²	direct (N × H)
9. C₂×Q₈×Dic₁₇	C₂²	semidirect
10. C₂×D₄×Dic₁₇	C₂²	semidirect
11. C₂×#10587612	C₂²	semidirect
17. C₂²×C₄×Dic₁₇	C₂²	semidirect
18. C₂²×#41×C₁₇	C₄	semidirect
24. C₂×#41×C₁₇	C₂×C₄	semidirect
26. C₂²×Q₈×C₁₇	C₂×C₄	semidirect
28. C₂²×D₄×C₁₇	C₂×C₄	semidirect
30. C₂³×C₄×C₁₇	C₂×C₄	semidirect
39. Q₈×Dic₁₇	C₂³	semidirect
40. D₄×Dic₁₇	C₂³	semidirect
41. #10587612	C₂³	semidirect
46. C₂×C₄×Dic₁₇	C₂³	semidirect
51. #41×C₁₇	C₂²×C₄	semidirect
52. C₂×Q₈×C₁₇	C₂²×C₄	semidirect
54. C₂×D₄×C₁₇	C₂²×C₄	semidirect
58. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
64. C₂²×Dic₁₇	#41	direct (N × H)
81. Q₈×C₁₇	C₂³×C₄	semidirect
82. D₄×C₁₇	C₂³×C₄	semidirect
85. C₂×C₄×C₁₇	C₂³×C₄	semidirect
90. C₂×Dic₁₇	C₂×#41	direct (N × H)
101. C₂²×C₁₇	C₄×#41	semidirect
109. Dic₁₇	C₂²×#41	direct (N × H)
110. C₂²×#41	Dic₁₇	direct (N × H)
127. C₂×C₁₇	C₂×C₄×#41	semidirect
131. C₂×#41	C₂×Dic₁₇	direct (N × H)
133. C₂²×Q₈	C₂×Dic₁₇	semidirect
135. C₂²×D₄	C₂×Dic₁₇	semidirect
141. C₂³×C₄	C₂×Dic₁₇	semidirect
154. C₁₇	C₂²×C₄×#41	semidirect
159. #41	C₂²×Dic₁₇	direct (N × H)
160. C₂×Q₈	C₂²×Dic₁₇	semidirect
162. C₂×D₄	C₂²×Dic₁₇	semidirect
172. C₂²×C₄	C₂²×Dic₁₇	semidirect
188. Q₈	C₂³×Dic₁₇	semidirect
189. D₄	C₂³×Dic₁₇	semidirect
197. C₂×C₄	C₂³×Dic₁₇	semidirect
202. C₂²	#41×Dic₁₇	direct (N × H)
212. C₂	C₂×#41×Dic₁₇	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×#41×Dic₁₇	2
C₂²×Q₈×Dic₁₇	2
C₂²×D₄×Dic₁₇	2
C₂²×#10587612	2
C₂³×C₄×Dic₁₇	2
C₂³×#41×C₁₇	2
C₂²×C₄×#41	17

Subgroups — C22×#41×Dic17 #17940900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×Dic₁₇ #17940900