Subgroups — C₁₇×#2541 #14525547

Order 2176.

948 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#127	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₇×#121	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#392	C₂	1	4	yes	maximal, complemented
6	C₁₇×#390	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
9	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₇	C₂²	1	8	yes	—
11	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₇	C₂²	1	2	yes	—
14	C₁₇×#127	—	2	8	no	—
15	C₁₇×#127	—	2	4	no	—
16	C₁₇×#121	—	2	16	no	—
17	C₄×D₄×C₁₇	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₇	C₂²	1	8	yes	—
22	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×#95	C₂²	1	2	yes	—
25	C₂⁴×C₁₇	C₂³	1	4	yes	—
26	C₂⁴×C₁₇	C₂³	1	2	yes	—
27	C₂×D₄×C₁₇	—	2	16	no	—
28	C₂×D₄×C₁₇	—	2	2	no	—
29	C₂×D₄×C₁₇	—	2	4	no	—
30	C₂×D₄×C₁₇	—	2	4	no	—
31	C₂×D₄×C₁₇	—	2	4	no	—
32	C₂×D₄×C₁₇	—	2	4	no	—
33	C₂×D₄×C₁₇	—	2	16	no	—
34	C₂×D₄×C₁₇	—	2	8	no	—
35	C₂×D₄×C₁₇	—	2	2	no	—
36	C₂×D₄×C₁₇	—	2	4	no	—
37	C₂×D₄×C₁₇	—	4	8	no	—
38	C₂×D₄×C₁₇	—	4	4	no	—
39	C₂×D₄×C₁₇	D₄	1	4	yes	complemented
40	C₂²×C₄×C₁₇	—	2	4	no	—
41	C₂²×C₄×C₁₇	—	2	8	no	—
42	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₇	—	2	2	no	—
48	C₄⋊C₄×C₁₇	D₄	1	8	yes	complemented
49	C₂²⋊C₄×C₁₇	—	2	4	no	—
50	C₂²⋊C₄×C₁₇	—	2	16	no	—
51	C₄²×C₁₇	—	2	4	no	—
52	C₄²×C₁₇	D₄	1	4	yes	complemented
53	C₂³×C₁₇	—	2	2	no	—
54	C₂³×C₁₇	—	2	4	no	—
55	C₂³×C₁₇	—	2	2	no	—
56	C₂³×C₁₇	—	2	4	no	—
57	C₂³×C₁₇	—	2	8	no	—
58	C₂³×C₁₇	—	2	4	no	—
59	C₂³×C₁₇	—	2	4	no	—
60	C₂³×C₁₇	—	2	4	no	—
61	C₂³×C₁₇	—	2	8	no	—
62	C₂³×C₁₇	C₂×D₄	1	4	yes	—
63	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
64	D₄×C₁₇	—	2	8	no	—
65	D₄×C₁₇	—	4	32	no	—
66	D₄×C₁₇	—	4	8	no	—
67	D₄×C₁₇	—	4	8	no	—
68	D₄×C₁₇	—	4	8	no	—
69	C₂×C₄×C₁₇	—	2	4	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	4	no	—
72	C₂×C₄×C₁₇	—	2	4	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	4	8	no	—
75	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×D₄	1	8	yes	—
77	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₁₇	#41	1	2	yes	—
82	#2541	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₁₇	—	2	4	no	—
84	C₂²×C₁₇	—	2	8	no	—
85	C₂²×C₁₇	—	4	2	no	—
86	C₂²×C₁₇	—	4	2	no	—
87	C₂²×C₁₇	—	4	4	no	—
88	C₂²×C₁₇	—	4	4	no	—
89	C₂²×C₁₇	—	4	4	no	—
90	C₂²×C₁₇	—	4	4	no	—
91	C₂²×C₁₇	—	4	8	no	—
92	C₂²×C₁₇	—	4	8	no	—
93	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
94	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₁₇	C₂²×D₄	1	2	yes	—
96	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
97	C₂²×C₁₇	H(5,𝔽₂)	1	2	yes	—
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	2	8	no	—
100	C₄×C₁₇	—	4	2	no	—
101	C₄×C₁₇	#121	1	4	yes	complemented
102	C₄×C₁₇	#127	1	4	yes	complemented
103	C₂×#127	C₂×C₁₇	1	2	yes	complemented
104	C₂×#127	C₂×C₁₇	1	1	yes	characteristic, complemented
105	C₂×#121	C₂×C₁₇	1	4	yes	complemented
106	C₂×C₄×D₄	C₂×C₁₇	1	1	yes	characteristic, complemented
107	#392	C₂×C₁₇	1	4	yes	complemented
108	#390	C₂×C₁₇	1	2	yes	complemented
109	C₄×C₄⋊C₄	C₂×C₁₇	1	1	yes	characteristic, complemented
110	C₂×C₁₇	—	4	4	no	—
111	C₂×C₁₇	—	8	4	no	—
112	C₂×C₁₇	C₂×#121	1	1	yes	characteristic
113	C₂×C₁₇	C₂×#127	1	1	yes	characteristic
114	C₂×C₁₇	#535	1	1	yes	characteristic
115	C₂×C₁₇	D₄²	1	2	yes	—
116	C₂×C₁₇	#550	1	2	yes	—
117	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
118	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
119	C₂²×D₄	C₂²×C₁₇	1	8	yes	—
120	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
121	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
122	C₂³×C₄	C₂²×C₁₇	1	2	yes	—
123	#127	—	2	8	no	—
124	#127	—	2	4	no	—
125	#121	—	2	16	no	—
126	C₄×D₄	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₁₇	1	8	yes	—
131	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
132	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
133	#95	C₂²×C₁₇	1	2	yes	—
134	C₁₇	#2541	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	C₂³×C₁₇	1	4	yes	—
136	C₂⁴	C₂³×C₁₇	1	2	yes	—
137	C₂×D₄	—	2	16	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	16	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	8	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	D₄×C₁₇	1	4	yes	complemented
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₁₇	1	4	yes	—
155	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	D₄×C₁₇	1	8	yes	complemented
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	2	16	no	—
161	C₄²	—	2	4	no	—
162	C₄²	D₄×C₁₇	1	4	yes	complemented
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	8	no	—
172	C₂³	C₂×D₄×C₁₇	1	4	yes	—
173	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
174	D₄	—	2	8	no	—
175	D₄	—	4	32	no	—
176	D₄	—	4	8	no	—
177	D₄	—	4	8	no	—
178	D₄	—	4	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₇	1	8	yes	—
187	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
188	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
189	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
191	C₂×C₄	#41×C₁₇	1	2	yes	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	8	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	4	no	—
198	C₂²	—	4	4	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	4	8	no	—
202	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
203	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
204	C₂²	C₂²×D₄×C₁₇	1	2	yes	—
205	C₂²	C₂×#41×C₁₇	1	1	yes	characteristic
206	C₂²	C₁₇×H(5,𝔽₂)	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	8	no	—
209	C₄	—	4	2	no	—
210	C₄	C₁₇×#121	1	4	yes	complemented
211	C₄	C₁₇×#127	1	4	yes	complemented
212	C₂	—	4	4	no	—
213	C₂	—	8	4	no	—
214	C₂	C₂×C₁₇×#121	1	1	yes	characteristic
215	C₂	C₂×C₁₇×#127	1	1	yes	characteristic
216	C₂	C₁₇×#535	1	1	yes	characteristic
217	C₂	D₄²×C₁₇	1	2	yes	—
218	C₂	C₁₇×#550	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#127	C₂	semidirect
2. C₂×C₁₇×#127	C₂	semidirect
3. C₂×C₁₇×#121	C₂	semidirect
4. C₂×C₄×D₄×C₁₇	C₂	semidirect
5. C₁₇×#392	C₂	semidirect
6. C₁₇×#390	C₂	semidirect
7. C₄×C₄⋊C₄×C₁₇	C₂	semidirect
39. C₂×D₄×C₁₇	D₄	semidirect
48. C₄⋊C₄×C₁₇	D₄	semidirect
52. C₄²×C₁₇	D₄	semidirect
82. #2541	C₁₇	direct (N × H)
101. C₄×C₁₇	#121	semidirect
102. C₄×C₁₇	#127	semidirect
103. C₂×#127	C₂×C₁₇	semidirect
104. C₂×#127	C₂×C₁₇	semidirect
105. C₂×#121	C₂×C₁₇	semidirect
106. C₂×C₄×D₄	C₂×C₁₇	semidirect
107. #392	C₂×C₁₇	semidirect
108. #390	C₂×C₁₇	semidirect
109. C₄×C₄⋊C₄	C₂×C₁₇	semidirect
134. C₁₇	#2541	direct (N × H)
149. C₂×D₄	D₄×C₁₇	semidirect
158. C₄⋊C₄	D₄×C₁₇	semidirect
162. C₄²	D₄×C₁₇	semidirect
210. C₄	C₁₇×#121	semidirect
211. C₄	C₁₇×#127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#127	2
C₂×C₁₇×#121	2
C₂×C₄×D₄×C₁₇	2
C₁₇×#392	2
C₁₇×#390	2
C₄×C₄⋊C₄×C₁₇	2
#2541	17

Subgroups — C17×#2541 #14525547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2541 #14525547