Subgroups — C₁₇×#10864 #16868608

Order 4352.

806 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₇×#108	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₇×#341	C₂	1	2	yes	maximal
4	C₁₇×#1728	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₁₇	C₂²	1	4	yes	—
6	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₇×#108	—	2	4	no	—
10	C₂×C₁₇×#108	—	2	4	no	—
11	C₂×C₁₇×#108	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₇	—	2	4	no	—
13	C₂×C₄×C₈×C₁₇	C₄	1	4	yes	—
14	C₁₇×#341	—	2	4	no	—
15	C₁₇×#336	—	2	4	no	—
16	C₂³×C₄×C₁₇	Q₈	1	2	yes	—
17	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	—	2	4	no	—
19	C₂×M₁₆×C₁₇	—	2	4	no	—
20	C₂×M₁₆×C₁₇	—	2	4	no	—
21	C₂×M₁₆×C₁₇	—	2	4	no	—
22	C₂×M₁₆×C₁₇	—	2	4	no	—
23	C₂×M₁₆×C₁₇	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	—	2	2	no	—
26	C₂²×C₈×C₁₇	—	2	4	no	—
27	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₇	D₄	1	2	yes	—
31	C₂²×C₈×C₁₇	D₄	1	4	yes	—
32	C₂²×C₈×C₁₇	D₄	1	2	yes	—
33	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₇	—	2	2	no	—
36	C₂×C₄²×C₁₇	—	2	2	no	—
37	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₇	D₄	1	2	yes	—
39	C₁₇×#108	—	2	16	no	—
40	C₄×C₈×C₁₇	—	2	8	no	—
41	C₂⁴×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	—	2	4	no	—
43	C₂²×C₄×C₁₇	—	2	8	no	—
44	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	#41	1	2	yes	—
54	C₂²×C₄×C₁₇	#41	1	2	yes	—
55	M₁₆×C₁₇	—	2	8	no	—
56	M₁₆×C₁₇	—	2	8	no	—
57	M₁₆×C₁₇	—	4	4	no	—
58	M₁₆×C₁₇	—	4	4	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	—	2	4	no	—
62	C₂×C₈×C₁₇	—	2	4	no	—
63	C₂×C₈×C₁₇	—	2	8	no	—
64	C₂×C₈×C₁₇	—	4	4	no	—
65	C₂×C₈×C₁₇	—	4	4	no	—
66	C₂×C₈×C₁₇	C₄²	1	4	yes	—
67	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	8	yes	—
69	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₁₇	C₄⋊C₄	1	8	yes	—
72	C₄²×C₁₇	—	2	2	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	2	4	no	—
75	#10864	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₇	—	2	4	no	—
77	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₇	C₄×Q₈	1	2	yes	—
79	C₂³×C₁₇	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	—	2	16	no	—
83	C₂×C₄×C₁₇	—	2	4	no	—
84	C₂×C₄×C₁₇	—	2	4	no	—
85	C₂×C₄×C₁₇	#108	1	8	yes	complemented
86	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₁₇	#121	1	2	yes	—
98	C₂×C₄×C₁₇	#121	1	2	yes	—
99	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₇	#124	1	2	yes	—
101	C₂×C₄×C₁₇	#125	1	2	yes	—
102	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₈×C₁₇	—	2	4	no	—
105	C₈×C₁₇	—	4	8	no	—
106	C₈×C₁₇	#95	1	8	yes	—
107	C₂²×#108	C₂×C₁₇	1	2	yes	—
108	C₂²×C₄×C₈	C₂×C₁₇	1	1	yes	characteristic
109	C₂×#341	C₂×C₁₇	1	2	yes	—
110	#1728	C₂×C₁₇	1	2	yes	—
111	C₂²×C₁₇	—	2	4	no	—
112	C₂²×C₁₇	—	2	2	no	—
113	C₂²×C₁₇	—	2	2	no	—
114	C₂²×C₁₇	—	2	4	no	—
115	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₇	#380	1	1	yes	characteristic
117	C₂²×C₁₇	#384	1	1	yes	characteristic
118	C₂²×C₁₇	#389	1	2	yes	—
119	C₂²×C₁₇	#391	1	2	yes	—
120	C₂²×C₁₇	C₂×#108	1	4	yes	—
121	C₄×C₁₇	—	2	8	no	—
122	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
123	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₁₇	#380	1	1	yes	characteristic
125	C₄×C₁₇	#382	1	1	yes	characteristic
126	C₄×C₁₇	#389	1	2	yes	—
127	C₄×C₁₇	#390	1	2	yes	—
128	C₂²×M₁₆	C₂²×C₁₇	1	4	yes	—
129	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
130	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
131	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
132	C₂×#108	—	2	4	no	—
133	C₂×#108	—	2	4	no	—
134	C₂×#108	C₄×C₁₇	1	8	yes	complemented
135	C₂×C₄×C₈	—	2	4	no	—
136	C₂×C₄×C₈	C₄×C₁₇	1	4	yes	—
137	#341	—	2	4	no	—
138	#336	—	2	4	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	—	2	2	no	—
141	C₂×C₁₇	C₄⋊#95	1	1	yes	characteristic
142	C₂×C₁₇	C₄×#108	1	2	yes	—
143	C₂×C₁₇	#1810	1	2	yes	—
144	C₂×C₁₇	#1848	1	2	yes	—
145	C₂³×C₄	Q₈×C₁₇	1	2	yes	—
146	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	C₂×C₄×C₁₇	1	8	yes	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂×C₄×C₁₇	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₁₇	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₁₇	1	4	yes	—
159	C₂²×C₈	D₄×C₁₇	1	2	yes	—
160	C₂²×C₈	D₄×C₁₇	1	4	yes	—
161	C₂²×C₈	D₄×C₁₇	1	2	yes	—
162	C₂²×C₈	Q₈×C₁₇	1	2	yes	—
163	C₂²×C₈	Q₈×C₁₇	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
167	C₂×C₄²	D₄×C₁₇	1	2	yes	—
168	#108	—	2	16	no	—
169	C₄×C₈	—	2	8	no	—
170	C₁₇	#10864	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁴	C₂×Q₈×C₁₇	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
183	C₂²×C₄	#41×C₁₇	1	2	yes	—
184	C₂²×C₄	#41×C₁₇	1	2	yes	—
185	M₁₆	—	2	8	no	—
186	M₁₆	—	2	8	no	—
187	M₁₆	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	C₄²×C₁₇	1	4	yes	—
197	C₂×C₈	C₂²⋊C₄×C₁₇	1	4	yes	—
198	C₂×C₈	C₂²⋊C₄×C₁₇	1	8	yes	—
199	C₂×C₈	C₄⋊C₄×C₁₇	1	4	yes	—
200	C₂×C₈	C₄⋊C₄×C₁₇	1	4	yes	—
201	C₂×C₈	C₄⋊C₄×C₁₇	1	8	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	4	no	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
207	C₂³	C₄×Q₈×C₁₇	1	2	yes	—
208	C₂³	C₁₇×C₂²⋊Q₈	1	2	yes	—
209	C₂³	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
210	C₂³	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	C₁₇×#108	1	8	yes	complemented
215	C₂×C₄	C₂×C₄²×C₁₇	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
221	C₂×C₄	C₁₇×#117	1	1	yes	characteristic
222	C₂×C₄	C₁₇×#117	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
224	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
225	C₂×C₄	C₄×Q₈×C₁₇	1	2	yes	—
226	C₂×C₄	C₁₇×#121	1	2	yes	—
227	C₂×C₄	C₁₇×#121	1	2	yes	—
228	C₂×C₄	C₁₇×C₂²⋊Q₈	1	2	yes	—
229	C₂×C₄	C₁₇×#124	1	2	yes	—
230	C₂×C₄	C₁₇×#125	1	2	yes	—
231	C₂×C₄	C₁₇×#127	1	1	yes	characteristic
232	C₂×C₄	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	4	8	no	—
235	C₈	C₁₇×#95	1	8	yes	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	C₄×C₄⋊C₄×C₁₇	1	1	yes	characteristic
241	C₂²	C₁₇×#380	1	1	yes	characteristic
242	C₂²	C₁₇×#384	1	1	yes	characteristic
243	C₂²	C₁₇×#389	1	2	yes	—
244	C₂²	C₁₇×#391	1	2	yes	—
245	C₂²	C₂×C₁₇×#108	1	4	yes	—
246	C₄	—	2	8	no	—
247	C₄	C₂×C₁₇×#95	1	1	yes	characteristic
248	C₄	C₄×C₄⋊C₄×C₁₇	1	1	yes	characteristic
249	C₄	C₁₇×#380	1	1	yes	characteristic
250	C₄	C₁₇×#382	1	1	yes	characteristic
251	C₄	C₁₇×#389	1	2	yes	—
252	C₄	C₁₇×#390	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	C₁₇×C₄⋊#95	1	1	yes	characteristic
256	C₂	C₄×C₁₇×#108	1	2	yes	—
257	C₂	C₁₇×#1810	1	2	yes	—
258	C₂	C₁₇×#1848	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₇
socle	C₂³×C₁₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₇ ◁ C₂²×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₁₇×#108	C₄	semidirect
75. #10864	C₁₇	direct (N × H)
85. C₂×C₄×C₁₇	#108	semidirect
134. C₂×#108	C₄×C₁₇	semidirect
170. C₁₇	#10864	direct (N × H)
214. C₂×C₄	C₁₇×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₇×#108	2
C₂²×C₄×C₈×C₁₇	2
C₂×C₁₇×#341	2
C₁₇×#1728	2
#10864	17

Subgroups — C17×#10864 #16868608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10864 #16868608