Subgroups — C₂×C₁₇×#452 #14526007

Order 2176.

760 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#452	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₇	—	2	4	no	—
10	C₂×#41×C₁₇	—	2	4	no	—
11	C₂×#41×C₁₇	—	2	2	no	—
12	C₂×#41×C₁₇	C₂²	1	4	yes	—
13	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₁₇	—	2	1	no	—
16	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₇	—	2	1	no	—
18	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₇	—	2	4	no	—
20	C₂×Q₁₆×C₁₇	C₂²	1	4	yes	complemented
21	C₂×SD₁₆×C₁₇	—	2	4	no	—
22	C₂×SD₁₆×C₁₇	C₂²	1	4	yes	complemented
23	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×C₂²⋊Q₈	—	2	2	no	—
26	C₁₇×C₂²⋊Q₈	C₂²	1	4	yes	complemented
27	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
30	C₁₇×#103	C₂²	1	4	yes	complemented
31	C₁₇×#102	C₂²	1	4	yes	complemented
32	C₁₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
33	C₂⁴×C₁₇	—	2	1	no	—
34	C₂⁴×C₁₇	D₄	1	1	yes	characteristic
35	#41×C₁₇	—	2	16	no	—
36	#41×C₁₇	—	2	8	no	—
37	#41×C₁₇	—	2	8	no	—
38	C₂×Q₈×C₁₇	—	2	2	no	—
39	C₂×Q₈×C₁₇	—	2	1	no	—
40	C₂×Q₈×C₁₇	—	2	1	no	—
41	C₂×Q₈×C₁₇	—	2	2	no	—
42	C₂×Q₈×C₁₇	—	2	2	no	—
43	C₂×Q₈×C₁₇	D₄	1	4	yes	—
44	C₂×Q₈×C₁₇	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₇	C₂³	1	2	yes	—
46	C₂×D₄×C₁₇	—	2	8	no	—
47	C₂×D₄×C₁₇	—	2	4	no	—
48	C₂×D₄×C₁₇	—	2	1	no	—
49	C₂×D₄×C₁₇	—	2	2	no	—
50	C₂×D₄×C₁₇	D₄	1	4	yes	—
51	C₂×D₄×C₁₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₇	—	2	8	no	—
53	C₂²×C₄×C₁₇	—	2	1	no	—
54	C₂²×C₄×C₁₇	—	2	4	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	—	2	2	no	—
57	C₂²×C₄×C₁₇	—	2	1	no	—
58	C₂²×C₄×C₁₇	—	2	1	no	—
59	C₂²×C₄×C₁₇	D₄	1	2	yes	—
60	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	Q₁₆×C₁₇	—	2	8	no	—
64	SD₁₆×C₁₇	—	2	8	no	—
65	C₂×C₈×C₁₇	—	2	2	no	—
66	C₂×C₈×C₁₇	—	2	2	no	—
67	C₂×C₈×C₁₇	C₂³	1	2	yes	—
68	C₂×C₈×C₁₇	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	4	no	—
71	C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₁₇	—	2	4	no	—
73	C₂³×C₁₇	—	2	2	no	—
74	C₂³×C₁₇	—	2	4	no	—
75	C₂³×C₁₇	—	2	2	no	—
76	C₂³×C₁₇	—	2	2	no	—
77	C₂³×C₁₇	—	2	2	no	—
78	C₂³×C₁₇	—	4	2	no	—
79	C₂³×C₁₇	—	4	2	no	—
80	C₂³×C₁₇	C₂×D₄	1	2	yes	—
81	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₁₇	—	2	4	no	—
83	Q₈×C₁₇	—	2	2	no	—
84	Q₈×C₁₇	—	2	2	no	—
85	Q₈×C₁₇	—	2	2	no	—
86	Q₈×C₁₇	—	4	2	no	—
87	Q₈×C₁₇	C₂×D₄	1	4	yes	—
88	D₄×C₁₇	—	2	16	no	—
89	D₄×C₁₇	—	2	4	no	—
90	D₄×C₁₇	—	2	2	no	—
91	D₄×C₁₇	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₁₇	—	2	16	no	—
93	C₂×C₄×C₁₇	—	2	2	no	—
94	C₂×C₄×C₁₇	—	2	4	no	—
95	C₂×C₄×C₁₇	—	2	2	no	—
96	C₂×C₄×C₁₇	—	2	4	no	—
97	C₂×C₄×C₁₇	—	2	2	no	—
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	4	2	no	—
102	C₂×C₄×C₁₇	—	4	2	no	—
103	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₇	—	2	2	no	—
109	C₈×C₁₇	—	2	2	no	—
110	C₂×#452	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₁₇	—	2	2	no	—
112	C₂²×C₁₇	—	2	4	no	—
113	C₂²×C₁₇	—	2	2	no	—
114	C₂²×C₁₇	—	2	2	no	—
115	C₂²×C₁₇	—	4	2	no	—
116	C₂²×C₁₇	—	4	2	no	—
117	C₂²×C₁₇	—	4	4	no	—
118	C₂²×C₁₇	—	4	8	no	—
119	C₂²×C₁₇	#120	1	2	yes	—
120	C₂²×C₁₇	#135	1	2	yes	—
121	C₂²×C₁₇	#137	1	2	yes	—
122	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₁₇	—	2	2	no	—
124	C₄×C₁₇	—	2	4	no	—
125	C₄×C₁₇	—	4	2	no	—
126	C₄×C₁₇	—	4	2	no	—
127	C₄×C₁₇	#120	1	2	yes	—
128	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
130	C₂²×#41	C₂×C₁₇	1	1	yes	characteristic
131	C₂²×Q₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
132	C₂²×SD₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
133	C₂×C₂²⋊Q₈	C₂×C₁₇	1	1	yes	characteristic, complemented
134	#452	C₂×C₁₇	1	8	yes	direct product factor
135	C₂×#103	C₂×C₁₇	1	1	yes	characteristic, complemented
136	C₂×#102	C₂×C₁₇	1	1	yes	characteristic, complemented
137	C₂×C₂²⋊C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
138	C₂×C₁₇	—	4	2	no	—
139	C₂×C₁₇	—	4	4	no	—
140	C₂×C₁₇	#452	1	4	yes	direct product factor
141	C₂×C₁₇	C₂×#120	1	1	yes	characteristic
142	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
143	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
144	C₂×#41	—	2	4	no	—
145	C₂×#41	—	2	4	no	—
146	C₂×#41	—	2	2	no	—
147	C₂×#41	C₂²×C₁₇	1	4	yes	—
148	C₂²×Q₈	C₂²×C₁₇	1	1	yes	characteristic
149	C₂²×Q₈	C₂²×C₁₇	1	1	yes	characteristic
150	C₂²×D₄	—	2	1	no	—
151	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
152	C₂³×C₄	—	2	1	no	—
153	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	C₂²×C₁₇	1	4	yes	complemented
156	C₂×SD₁₆	—	2	4	no	—
157	C₂×SD₁₆	C₂²×C₁₇	1	4	yes	complemented
158	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
160	C₂²⋊Q₈	—	2	2	no	—
161	C₂²⋊Q₈	C₂²×C₁₇	1	4	yes	complemented
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	—	2	1	no	—
165	#103	C₂²×C₁₇	1	4	yes	complemented
166	#102	C₂²×C₁₇	1	4	yes	complemented
167	C₂²⋊C₈	C₂²×C₁₇	1	4	yes	complemented
168	C₁₇	C₂×#452	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	—	2	1	no	—
170	C₂⁴	D₄×C₁₇	1	1	yes	characteristic
171	#41	—	2	16	no	—
172	#41	—	2	8	no	—
173	#41	—	2	8	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	D₄×C₁₇	1	4	yes	—
180	C₂×Q₈	C₂³×C₁₇	1	2	yes	—
181	C₂×Q₈	C₂³×C₁₇	1	2	yes	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	D₄×C₁₇	1	4	yes	—
187	C₂×D₄	C₂³×C₁₇	1	2	yes	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	D₄×C₁₇	1	1	yes	characteristic
196	C₂²×C₄	D₄×C₁₇	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	C₂³×C₁₇	1	2	yes	—
204	C₂×C₈	C₂³×C₁₇	1	2	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×D₄×C₁₇	1	2	yes	—
217	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	2	4	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	4	2	no	—
223	Q₈	C₂×D₄×C₁₇	1	4	yes	—
224	D₄	—	2	16	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	4	no	—
227	D₄	C₂×D₄×C₁₇	1	4	yes	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
240	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
241	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	C₁₇×#120	1	2	yes	—
255	C₂²	C₁₇×#135	1	2	yes	—
256	C₂²	C₁₇×#137	1	2	yes	—
257	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	C₁₇×#120	1	2	yes	—
263	C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
264	C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	—	4	4	no	—
267	C₂	C₁₇×#452	1	4	yes	direct product factor
268	C₂	C₂×C₁₇×#120	1	1	yes	characteristic
269	C₂	C₂×C₁₇×#135	1	1	yes	characteristic
270	C₂	C₂×C₁₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₂²×#41×C₁₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₁₆×C₁₇	C₂	semidirect
3. C₂²×SD₁₆×C₁₇	C₂	semidirect
4. C₂×C₁₇×C₂²⋊Q₈	C₂	semidirect
5. C₁₇×#452	C₂	direct (N × H)
6. C₂×C₁₇×#103	C₂	semidirect
7. C₂×C₁₇×#102	C₂	semidirect
8. C₂×C₁₇×C₂²⋊C₈	C₂	semidirect
20. C₂×Q₁₆×C₁₇	C₂²	semidirect
22. C₂×SD₁₆×C₁₇	C₂²	semidirect
26. C₁₇×C₂²⋊Q₈	C₂²	semidirect
30. C₁₇×#103	C₂²	semidirect
31. C₁₇×#102	C₂²	semidirect
32. C₁₇×C₂²⋊C₈	C₂²	semidirect
110. C₂×#452	C₁₇	direct (N × H)
131. C₂²×Q₁₆	C₂×C₁₇	semidirect
132. C₂²×SD₁₆	C₂×C₁₇	semidirect
133. C₂×C₂²⋊Q₈	C₂×C₁₇	semidirect
134. #452	C₂×C₁₇	direct (N × H)
135. C₂×#103	C₂×C₁₇	semidirect
136. C₂×#102	C₂×C₁₇	semidirect
137. C₂×C₂²⋊C₈	C₂×C₁₇	semidirect
140. C₂×C₁₇	#452	direct (N × H)
155. C₂×Q₁₆	C₂²×C₁₇	semidirect
157. C₂×SD₁₆	C₂²×C₁₇	semidirect
161. C₂²⋊Q₈	C₂²×C₁₇	semidirect
165. #103	C₂²×C₁₇	semidirect
166. #102	C₂²×C₁₇	semidirect
167. C₂²⋊C₈	C₂²×C₁₇	semidirect
168. C₁₇	C₂×#452	direct (N × H)
267. C₂	C₁₇×#452	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₁₇	2
C₂²×Q₁₆×C₁₇	2
C₂²×SD₁₆×C₁₇	2
C₂×C₁₇×C₂²⋊Q₈	2
C₁₇×#452	2
C₂×C₁₇×#103	2
C₂×C₁₇×#102	2
C₂×C₁₇×C₂²⋊C₈	2
C₂×#452	17

Subgroups — C2×C17×#452 #14526007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#452 #14526007