Subgroups — C₂×#14530382 #17186960

Order 4352.

854 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×#10587469	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762763	C₂	1	1	yes	maximal, characteristic
4	#14530382	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11762584	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#63218	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₂×#10587558	C₄	1	1	yes	characteristic
12	C₂×#10587469	C₄	1	8	yes	complemented
13	C₂×#10587469	C₂²	1	4	yes	—
14	C₂²×#63214	C₂²	1	1	yes	characteristic
15	C₂²×#63214	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₁₇	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
18	#11762763	C₄	1	4	yes	—
19	#11762763	C₂²	1	4	yes	—
20	C₂×#10587474	C₄	1	1	yes	characteristic
21	C₂×#10587486	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	#11762584	C₂²	1	4	yes	complemented
24	C₄×#63214	C₂²	1	4	yes	complemented
25	C₁₆×Dic₁₇	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₁₇×#346	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×#63227	C₈	1	2	yes	—
35	C₂×#63225	C₈	1	2	yes	—
36	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
38	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
39	C₂×#63218	C₈	1	8	yes	—
40	C₂×#63218	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₇	C₂³	1	2	yes	—
45	#10587558	C₈	1	4	yes	—
46	#10587558	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
49	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
50	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
51	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₇	C₂³	1	2	yes	—
53	#10587486	C₂×C₄	1	4	yes	—
54	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₇	C₂³	1	2	yes	—
56	C₂×#63214	C₂×C₄	1	4	yes	—
57	C₂×#63214	C₂×C₄	1	4	yes	—
58	C₂×#63214	C₂³	1	2	yes	—
59	C₂×#63214	C₂³	1	2	yes	—
60	#10587474	C₂×C₄	1	4	yes	—
61	#10587469	C₂×C₄	1	16	yes	complemented
62	C₁₆×D₁₇	C₂×C₄	1	16	yes	complemented
63	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
67	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₁₇	C₄²	1	4	yes	—
69	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
78	C₄²×C₁₇	C₂×C₈	1	2	yes	—
79	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
80	#63227	C₂×C₈	1	8	yes	—
81	#63225	C₂×C₈	1	8	yes	—
82	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
83	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
84	C₂×#3622	C₄²	1	4	yes	—
85	C₂×#3622	C₂×C₈	1	4	yes	—
86	C₂×#3622	C₂²×C₄	1	2	yes	—
87	C₂×#3622	C₂²×C₄	1	2	yes	—
88	#63218	C₂×C₈	1	16	yes	—
89	C₈×D₁₇	C₄²	1	8	yes	—
90	C₈×D₁₇	C₂²×C₄	1	4	yes	—
91	C₈×D₁₇	C₂²×C₄	1	4	yes	—
92	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
93	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
94	#63214	C₂²×C₄	1	4	yes	—
95	#63214	C₂²×C₄	1	4	yes	—
96	C₂×#2108	—	17	1	no	maximal, Sylow
97	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₁₇	—	2	4	no	—
99	C₂²×D₁₇	C₄×C₈	1	4	yes	—
100	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₁₇	—	2	4	no	—
102	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
104	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₁₇	—	2	4	no	—
108	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
109	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
110	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
111	C₄×D₁₇	C₄×C₈	1	8	yes	—
112	C₄×D₁₇	C₂×C₄²	1	4	yes	—
113	C₄×D₁₇	C₂²×C₈	1	4	yes	—
114	C₈×C₁₇	C₂×C₄²	1	2	yes	—
115	C₈×C₁₇	C₂²×C₈	1	4	yes	—
116	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
118	#3622	C₂×C₄²	1	4	yes	—
119	#3622	C₂²×C₈	1	4	yes	—
120	C₂²×M₃₂	—	17	1	no	—
121	C₂³×C₁₆	—	17	1	no	—
122	C₂×#405	—	17	1	no	—
123	#2108	—	17	8	no	—
124	C₂×#346	—	17	1	no	—
125	C₂×#346	D₁₇	1	1	yes	characteristic, complemented
126	C₂×C₄×C₁₆	—	17	1	no	—
127	C₂×C₄×C₁₆	—	17	1	no	—
128	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
129	C₂²×C₁₇	#504	1	4	yes	—
130	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
131	C₂×D₁₇	—	2	8	no	—
132	C₂×D₁₇	—	2	4	no	—
133	C₂×D₁₇	C₂×C₄×C₈	1	4	yes	—
134	C₄×C₁₇	—	2	4	no	—
135	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
136	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
137	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
138	Dic₁₇	—	2	4	no	—
139	Dic₁₇	C₂×C₄×C₈	1	4	yes	—
140	C₂²×M₁₆	—	17	1	no	—
141	C₂³×C₈	—	17	1	no	—
142	C₂×#117	—	17	1	no	—
143	C₂×M₃₂	—	17	8	no	—
144	C₂×M₃₂	—	17	4	no	—
145	C₂²×C₁₆	—	17	1	no	—
146	C₂²×C₁₆	—	17	1	no	—
147	C₂²×C₁₆	—	17	8	no	—
148	C₂²×C₁₆	—	17	4	no	—
149	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
151	#405	—	17	4	no	—
152	#405	—	17	4	no	—
153	C₂×#97	—	17	1	no	—
154	C₂×#97	—	17	1	no	—
155	C₂×C₄×C₈	—	17	1	no	—
156	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
157	#346	—	17	4	no	—
158	#346	C₂×D₁₇	1	4	yes	complemented
159	C₄×C₁₆	—	17	4	no	—
160	C₄×C₁₆	—	17	4	no	—
161	C₂×C₁₇	#2108	1	4	yes	complemented
162	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
163	C₂×C₁₇	C₂×#504	1	2	yes	—
164	D₁₇	—	2	4	no	—
165	C₂³×C₄	—	17	1	no	—
166	C₂×M₁₆	—	17	4	no	—
167	C₂×M₁₆	—	17	8	no	—
168	C₂²×C₈	—	17	1	no	—
169	C₂²×C₈	—	17	1	no	—
170	C₂²×C₈	—	17	2	no	—
171	C₂²×C₈	—	17	4	no	—
172	C₂²×C₈	—	17	4	no	—
173	C₂²×C₈	—	17	2	no	—
174	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
176	#117	—	17	4	no	—
177	#117	—	17	4	no	—
178	C₂×C₄⋊C₄	—	17	2	no	—
179	C₂×C₂²⋊C₄	—	17	2	no	—
180	C₂×C₄²	—	17	1	no	—
181	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
182	M₃₂	—	17	16	no	—
183	C₂×C₁₆	—	17	4	no	—
184	C₂×C₁₆	—	17	4	no	—
185	C₂×C₁₆	—	17	2	no	—
186	C₂×C₁₆	—	17	2	no	—
187	C₂×C₁₆	—	17	16	no	—
188	C₂×C₁₆	C₄×D₁₇	1	4	yes	—
189	C₂×C₁₆	C₄×D₁₇	1	4	yes	—
190	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
191	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
192	#97	—	17	4	no	—
193	#97	—	17	4	no	—
194	C₄×C₈	—	17	2	no	—
195	C₄×C₈	—	17	2	no	—
196	C₄×C₈	C₄×D₁₇	1	2	yes	—
197	C₄×C₈	C₂²×D₁₇	1	2	yes	—
198	C₁₇	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
199	C₂⁴	—	17	1	no	—
200	C₂²×C₄	—	17	1	no	—
201	C₂²×C₄	—	17	2	no	—
202	C₂²×C₄	—	17	4	no	—
203	C₂²×C₄	—	17	2	no	—
204	C₂²×C₄	—	17	4	no	—
205	C₂²×C₄	—	17	2	no	—
206	C₂²×C₄	C₈×D₁₇	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
208	M₁₆	—	17	16	no	—
209	C₂×C₈	—	17	2	no	—
210	C₂×C₈	—	17	2	no	—
211	C₂×C₈	—	17	4	no	—
212	C₂×C₈	—	17	4	no	—
213	C₂×C₈	—	17	4	no	—
214	C₂×C₈	—	17	4	no	—
215	C₂×C₈	—	17	8	no	—
216	C₂×C₈	C₈×D₁₇	1	4	yes	—
217	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
221	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
222	C₄⋊C₄	—	17	8	no	—
223	C₂²⋊C₄	—	17	8	no	—
224	C₄²	—	17	2	no	—
225	C₄²	—	17	2	no	—
226	C₄²	C₈×D₁₇	1	2	yes	—
227	C₄²	C₂×C₄×D₁₇	1	2	yes	—
228	C₁₆	—	17	4	no	—
229	C₁₆	—	17	4	no	—
230	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
231	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
232	C₂³	—	17	4	no	—
233	C₂³	—	17	2	no	—
234	C₂³	—	34	4	no	—
235	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	17	4	no	—
238	C₂×C₄	—	17	2	no	—
239	C₂×C₄	—	17	4	no	—
240	C₂×C₄	—	17	8	no	—
241	C₂×C₄	—	17	4	no	—
242	C₂×C₄	—	17	4	no	—
243	C₂×C₄	—	34	4	no	—
244	C₂×C₄	C₄²×D₁₇	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₁₇	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
247	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
248	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
249	C₈	—	17	4	no	—
250	C₈	—	17	4	no	—
251	C₈	C₄²×D₁₇	1	2	yes	—
252	C₈	C₂×C₈×D₁₇	1	4	yes	—
253	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
255	C₂²	—	17	4	no	—
256	C₂²	—	34	4	no	—
257	C₂²	—	34	8	no	—
258	C₂²	C₄×C₈×D₁₇	1	2	yes	—
259	C₂²	#11763227	1	4	yes	—
260	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	17	4	no	—
263	C₄	—	34	4	no	—
264	C₄	C₄×C₈×D₁₇	1	2	yes	—
265	C₄	C₂×C₄²×D₁₇	1	1	yes	characteristic
266	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
267	C₂	—	34	4	no	—
268	C₂	#14530382	1	4	yes	direct product factor
269	C₂	C₂×C₄×C₈×D₁₇	1	1	yes	characteristic
270	C₂	C₂×#11763227	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	C₂×#2108
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
4. #14530382	C₂	direct (N × H)
5. C₂×#11762584	C₂	semidirect
6. C₂×C₄×#63214	C₂	semidirect
7. C₂×C₁₆×Dic₁₇	C₂	semidirect
8. C₂×C₁₇×#346	C₂	semidirect
12. C₂×#10587469	C₄	semidirect
16. C₂×C₁₆×D₁₇	C₄	semidirect
23. #11762584	C₂²	semidirect
24. C₄×#63214	C₂²	semidirect
25. C₁₆×Dic₁₇	C₂²	semidirect
29. C₁₇×#346	C₂²	semidirect
61. #10587469	C₂×C₄	semidirect
62. C₁₆×D₁₇	C₂×C₄	semidirect
125. C₂×#346	D₁₇	semidirect
158. #346	C₂×D₁₇	semidirect
161. C₂×C₁₇	#2108	semidirect
198. C₁₇	C₂×#2108	semidirect
268. C₂	#14530382	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×#10587469	2
C₂²×C₁₆×D₁₇	2
C₂×#11762763	2
#14530382	2
C₂×#11762584	2
C₂×C₄×#63214	2
C₂×C₁₆×Dic₁₇	2
C₂×C₁₇×#346	2
C₂×#2108	17

Subgroups — C2×#14530382 #17186960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530382 #17186960