Subgroups — C₁₇×#3360 #14526366

Order 2176.

350 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#556	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×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#479	C₂	1	2	yes	maximal, complemented
6	C₁₇×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#463	C₂	1	2	yes	maximal
8	C₁₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₁₇	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₁₇	C₂	1	1	yes	maximal, characteristic
12	C₂×SD₁₆×C₁₇	C₂²	1	2	yes	—
13	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₇×#126	—	2	2	no	—
17	C₁₇×#125	C₂²	1	2	yes	—
18	C₁₇×#124	C₂²	1	2	yes	—
19	C₁₇×C₂²⋊Q₈	—	2	1	no	—
20	C₁₇×C₂²⋊Q₈	—	2	1	no	—
21	C₁₇×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₁₇	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×#117	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₁₇×#107	C₂²	1	1	yes	characteristic
30	C₁₇×#106	C₂²	1	2	yes	—
31	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₇×#103	C₂²	1	2	yes	—
34	C₁₇×#103	C₂²	1	2	yes	—
35	C₁₇×#103	C₂²	1	2	yes	—
36	C₁₇×#102	C₂²	1	1	yes	characteristic
37	C₁₇×#102	C₂²	1	1	yes	characteristic
38	C₁₇×#102	C₂²	1	2	yes	—
39	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₇	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₇	—	2	1	no	—
42	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₇	C₂³	1	2	yes	—
44	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	—	2	1	no	—
48	C₂²×C₄×C₁₇	—	2	2	no	—
49	SD₁₆×C₁₇	—	2	4	no	—
50	D₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂³	1	2	yes	—
53	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₇	—	2	2	no	—
55	C₄⋊C₄×C₁₇	—	2	2	no	—
56	C₄⋊C₄×C₁₇	—	2	2	no	—
57	C₄⋊C₄×C₁₇	—	2	1	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	1	no	—
70	C₄²×C₁₇	—	2	2	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	1	no	—
75	Q₈×C₁₇	—	2	2	no	—
76	Q₈×C₁₇	—	2	2	no	—
77	Q₈×C₁₇	—	2	2	no	—
78	Q₈×C₁₇	—	8	1	no	—
79	D₄×C₁₇	—	2	1	no	—
80	D₄×C₁₇	—	2	1	no	—
81	D₄×C₁₇	—	2	1	no	—
82	D₄×C₁₇	#41	1	2	yes	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	4	2	no	—
88	C₂×C₄×C₁₇	—	4	2	no	—
89	C₂×C₄×C₁₇	—	8	2	no	—
90	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₇	—	2	1	no	—
96	C₈×C₁₇	—	2	2	no	—
97	C₈×C₁₇	—	2	1	no	—
98	#3360	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₁₇	—	2	2	no	—
100	C₂²×C₁₇	—	4	1	no	—
101	C₂²×C₁₇	—	4	1	no	—
102	C₂²×C₁₇	—	4	1	no	—
103	C₂²×C₁₇	—	4	1	no	—
104	C₂²×C₁₇	—	4	1	no	—
105	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₇	—	2	2	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	—	4	2	no	—
110	C₄×C₁₇	—	4	1	no	—
111	C₄×C₁₇	—	4	1	no	—
112	C₄×C₁₇	—	4	2	no	—
113	C₄×C₁₇	#135	1	2	yes	—
114	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
115	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
116	#556	C₂×C₁₇	1	1	yes	characteristic, complemented
117	#554	C₂×C₁₇	1	1	yes	characteristic, complemented
118	#488	C₂×C₁₇	1	2	yes	complemented
119	#487	C₂×C₁₇	1	1	yes	characteristic, complemented
120	#479	C₂×C₁₇	1	2	yes	complemented
121	#476	C₂×C₁₇	1	1	yes	characteristic, complemented
122	#463	C₂×C₁₇	1	2	yes	—
123	#461	C₂×C₁₇	1	1	yes	characteristic, complemented
124	C₈×Q₈	C₂×C₁₇	1	1	yes	characteristic, complemented
125	C₄×SD₁₆	C₂×C₁₇	1	2	yes	complemented
126	C₄×D₈	C₂×C₁₇	1	1	yes	characteristic
127	C₂×C₁₇	—	4	2	no	—
128	C₂×C₁₇	—	8	1	no	—
129	C₂×C₁₇	#548	1	1	yes	characteristic
130	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
131	C₂×C₁₇	#577	1	1	yes	characteristic
132	C₂×SD₁₆	C₂²×C₁₇	1	2	yes	—
133	C₂×D₈	C₂²×C₁₇	1	1	yes	characteristic
134	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
135	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
136	#126	—	2	2	no	—
137	#125	C₂²×C₁₇	1	2	yes	—
138	#124	C₂²×C₁₇	1	2	yes	—
139	C₂²⋊Q₈	—	2	1	no	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₄×Q₈	C₂²×C₁₇	1	2	yes	—
143	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₁₇	1	2	yes	—
146	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
147	#117	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	#107	C₂²×C₁₇	1	1	yes	characteristic
150	#106	C₂²×C₁₇	1	2	yes	—
151	C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
152	C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
153	#103	C₂²×C₁₇	1	2	yes	—
154	#103	C₂²×C₁₇	1	2	yes	—
155	#103	C₂²×C₁₇	1	2	yes	—
156	#102	C₂²×C₁₇	1	1	yes	characteristic
157	#102	C₂²×C₁₇	1	1	yes	characteristic
158	#102	C₂²×C₁₇	1	2	yes	—
159	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₁₇	1	2	yes	—
161	C₁₇	#3360	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	D₄×C₁₇	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₇	1	2	yes	—
165	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
166	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	SD₁₆	—	2	4	no	—
171	D₈	—	2	2	no	—
172	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₇	1	2	yes	—
174	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
182	C₄⋊C₄	D₄×C₁₇	1	2	yes	—
183	C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
184	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	C₂³×C₁₇	1	1	yes	characteristic
193	C₄²	C₂³×C₁₇	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	8	1	no	—
200	D₄	—	2	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	#41×C₁₇	1	2	yes	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	C₁₇×#135	1	2	yes	—
234	C₄	C₂×#41×C₁₇	1	1	yes	characteristic
235	C₄	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
236	C₂	—	4	2	no	—
237	C₂	—	8	1	no	—
238	C₂	C₁₇×#548	1	1	yes	characteristic
239	C₂	C₂×C₁₇×#135	1	1	yes	characteristic
240	C₂	C₁₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₂×D₈×C₁₇	C₂²
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×#556	C₂	semidirect
2. C₁₇×#554	C₂	semidirect
3. C₁₇×#488	C₂	semidirect
4. C₁₇×#487	C₂	semidirect
5. C₁₇×#479	C₂	semidirect
6. C₁₇×#476	C₂	semidirect
8. C₁₇×#461	C₂	semidirect
9. C₈×Q₈×C₁₇	C₂	semidirect
10. C₄×SD₁₆×C₁₇	C₂	semidirect
98. #3360	C₁₇	direct (N × H)
116. #556	C₂×C₁₇	semidirect
117. #554	C₂×C₁₇	semidirect
118. #488	C₂×C₁₇	semidirect
119. #487	C₂×C₁₇	semidirect
120. #479	C₂×C₁₇	semidirect
121. #476	C₂×C₁₇	semidirect
123. #461	C₂×C₁₇	semidirect
124. C₈×Q₈	C₂×C₁₇	semidirect
125. C₄×SD₁₆	C₂×C₁₇	semidirect
161. C₁₇	#3360	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×#556	2
C₁₇×#554	2
C₁₇×#488	2
C₁₇×#487	2
C₁₇×#479	2
C₁₇×#476	2
C₁₇×#463	2
C₁₇×#461	2
C₈×Q₈×C₁₇	2
C₄×SD₁₆×C₁₇	2
C₄×D₈×C₁₇	2
#3360	17

Subgroups — C17×#3360 #14526366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3360 #14526366