Subgroups — C₁₇×#3361 #14526367

Order 2176.

362 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#488	C₂	1	2	yes	maximal, complemented
4	C₁₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#478	C₂	1	2	yes	maximal, complemented
6	C₁₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#463	C₂	1	2	yes	maximal, complemented
8	C₁₇×#459	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₇	C₂	1	2	yes	maximal, complemented
12	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₁₇	C₂²	1	2	yes	—
14	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₇×#127	C₂²	1	1	yes	characteristic
16	C₁₇×#125	C₂²	1	2	yes	—
17	C₁₇×#124	—	2	1	no	—
18	C₁₇×#124	C₂²	1	2	yes	—
19	C₁₇×#123	—	2	2	no	—
20	C₁₇×C₂²⋊Q₈	—	2	1	no	—
21	C₁₇×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	—	2	1	no	—
25	C₄×D₄×C₁₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₇	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₁₇×#107	C₂²	1	2	yes	—
30	C₁₇×#106	C₂²	1	1	yes	characteristic
31	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₇×#103	C₂²	1	1	yes	characteristic
34	C₁₇×#103	C₂²	1	2	yes	—
35	C₁₇×#102	C₂²	1	1	yes	characteristic
36	C₁₇×#102	C₂²	1	2	yes	—
37	C₁₇×#102	C₂²	1	2	yes	—
38	C₁₇×#102	C₂²	1	2	yes	—
39	C₄×C₈×C₁₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₇	—	2	1	no	—
42	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₇	—	2	1	no	—
45	C₂×D₄×C₁₇	—	2	1	no	—
46	C₂×D₄×C₁₇	C₂³	1	2	yes	—
47	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	—	2	1	no	—
49	C₂²×C₄×C₁₇	—	2	2	no	—
50	C₂²×C₄×C₁₇	—	2	2	no	—
51	SD₁₆×C₁₇	—	2	2	no	—
52	D₈×C₁₇	—	2	4	no	—
53	C₂×C₈×C₁₇	C₂³	1	2	yes	—
54	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₇	—	2	2	no	—
57	C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₄⋊C₄×C₁₇	—	2	1	no	—
59	C₄⋊C₄×C₁₇	—	2	1	no	—
60	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
61	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₁₇	—	2	1	no	—
66	C₂²⋊C₄×C₁₇	—	2	2	no	—
67	C₂²⋊C₄×C₁₇	—	2	2	no	—
68	C₂²⋊C₄×C₁₇	—	2	2	no	—
69	C₂²⋊C₄×C₁₇	—	2	2	no	—
70	C₄²×C₁₇	—	2	1	no	—
71	C₄²×C₁₇	C₂³	1	2	yes	—
72	C₄²×C₁₇	C₂³	1	1	yes	characteristic
73	C₂³×C₁₇	—	2	1	no	—
74	C₂³×C₁₇	—	2	2	no	—
75	Q₈×C₁₇	—	2	2	no	—
76	Q₈×C₁₇	—	2	1	no	—
77	Q₈×C₁₇	—	2	1	no	—
78	Q₈×C₁₇	—	8	1	no	—
79	D₄×C₁₇	—	2	2	no	—
80	D₄×C₁₇	—	2	1	no	—
81	D₄×C₁₇	—	2	2	no	—
82	D₄×C₁₇	—	4	2	no	—
83	D₄×C₁₇	—	8	1	no	—
84	D₄×C₁₇	#41	1	2	yes	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	4	2	no	—
90	C₂×C₄×C₁₇	—	8	2	no	—
91	C₂×C₄×C₁₇	—	8	2	no	—
92	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	1	no	—
99	C₈×C₁₇	—	2	1	no	—
100	#3361	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₁₇	—	2	2	no	—
102	C₂²×C₁₇	—	4	1	no	—
103	C₂²×C₁₇	—	4	1	no	—
104	C₂²×C₁₇	—	4	2	no	—
105	C₂²×C₁₇	—	4	2	no	—
106	C₂²×C₁₇	—	4	2	no	—
107	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₇	—	2	2	no	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	4	1	no	—
112	C₄×C₁₇	—	4	1	no	—
113	C₄×C₁₇	—	4	1	no	—
114	C₄×C₁₇	—	4	2	no	—
115	C₄×C₁₇	#135	1	2	yes	—
116	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
117	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
118	#559	C₂×C₁₇	1	1	yes	characteristic, complemented
119	#554	C₂×C₁₇	1	1	yes	characteristic, complemented
120	#488	C₂×C₁₇	1	2	yes	complemented
121	#486	C₂×C₁₇	1	1	yes	characteristic, complemented
122	#478	C₂×C₁₇	1	2	yes	complemented
123	#477	C₂×C₁₇	1	1	yes	characteristic, complemented
124	#463	C₂×C₁₇	1	2	yes	complemented
125	#459	C₂×C₁₇	1	1	yes	characteristic
126	C₈×Q₈	C₂×C₁₇	1	1	yes	characteristic, complemented
127	C₄×SD₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
128	C₄×D₈	C₂×C₁₇	1	2	yes	complemented
129	C₂×C₁₇	—	4	2	no	—
130	C₂×C₁₇	—	8	2	no	—
131	C₂×C₁₇	#548	1	1	yes	characteristic
132	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
133	C₂×C₁₇	#576	1	1	yes	characteristic
134	C₂×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
135	C₂×D₈	C₂²×C₁₇	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
137	#127	C₂²×C₁₇	1	1	yes	characteristic
138	#125	C₂²×C₁₇	1	2	yes	—
139	#124	—	2	1	no	—
140	#124	C₂²×C₁₇	1	2	yes	—
141	#123	—	2	2	no	—
142	C₂²⋊Q₈	—	2	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
145	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	C₂²×C₁₇	1	2	yes	—
148	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
149	C₄×D₄	C₂²×C₁₇	1	2	yes	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	#107	C₂²×C₁₇	1	2	yes	—
152	#106	C₂²×C₁₇	1	1	yes	characteristic
153	C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
155	#103	C₂²×C₁₇	1	1	yes	characteristic
156	#103	C₂²×C₁₇	1	2	yes	—
157	#102	C₂²×C₁₇	1	1	yes	characteristic
158	#102	C₂²×C₁₇	1	2	yes	—
159	#102	C₂²×C₁₇	1	2	yes	—
160	#102	C₂²×C₁₇	1	2	yes	—
161	C₄×C₈	C₂²×C₁₇	1	2	yes	—
162	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
163	C₁₇	#3361	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	D₄×C₁₇	1	1	yes	characteristic
166	C₂×Q₈	C₂³×C₁₇	1	1	yes	characteristic
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂³×C₁₇	1	2	yes	—
170	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	SD₁₆	—	2	2	no	—
175	D₈	—	2	4	no	—
176	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₁₇	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
184	C₄⋊C₄	D₄×C₁₇	1	2	yes	—
185	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
187	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₁₇	1	2	yes	—
195	C₄²	C₂³×C₁₇	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	8	1	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	1	no	—
204	D₄	—	2	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	1	no	—
207	D₄	#41×C₁₇	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	C₁₇×#135	1	2	yes	—
238	C₄	C₂×#41×C₁₇	1	1	yes	characteristic
239	C₄	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	2	no	—
242	C₂	C₁₇×#548	1	1	yes	characteristic
243	C₂	C₂×C₁₇×#135	1	1	yes	characteristic
244	C₂	C₁₇×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₁₇×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₄×Q₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#559	C₂	semidirect
2. C₁₇×#554	C₂	semidirect
3. C₁₇×#488	C₂	semidirect
4. C₁₇×#486	C₂	semidirect
5. C₁₇×#478	C₂	semidirect
6. C₁₇×#477	C₂	semidirect
7. C₁₇×#463	C₂	semidirect
9. C₈×Q₈×C₁₇	C₂	semidirect
10. C₄×SD₁₆×C₁₇	C₂	semidirect
11. C₄×D₈×C₁₇	C₂	semidirect
100. #3361	C₁₇	direct (N × H)
118. #559	C₂×C₁₇	semidirect
119. #554	C₂×C₁₇	semidirect
120. #488	C₂×C₁₇	semidirect
121. #486	C₂×C₁₇	semidirect
122. #478	C₂×C₁₇	semidirect
123. #477	C₂×C₁₇	semidirect
124. #463	C₂×C₁₇	semidirect
126. C₈×Q₈	C₂×C₁₇	semidirect
127. C₄×SD₁₆	C₂×C₁₇	semidirect
128. C₄×D₈	C₂×C₁₇	semidirect
163. C₁₇	#3361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#559	2
C₁₇×#554	2
C₁₇×#488	2
C₁₇×#486	2
C₁₇×#478	2
C₁₇×#477	2
C₁₇×#463	2
C₁₇×#459	2
C₈×Q₈×C₁₇	2
C₄×SD₁₆×C₁₇	2
C₄×D₈×C₁₇	2
#3361	17

Subgroups — C17×#3361 #14526367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3361 #14526367