Subgroups — C₂×#14545135 #17978433

Order 4352.

694 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763819	C₂	1	2	yes	maximal
2	#14545135	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11763736	C₂	1	2	yes	maximal, complemented
4	C₂×#11763735	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587677	—	2	2	no	—
7	#11763819	—	2	4	no	—
8	#11763819	C₂²	1	8	yes	—
9	C₂×#10587534	C₂²	1	4	yes	—
10	C₂×#10587533	—	2	2	no	—
11	#11763736	C₂²	1	8	yes	complemented
12	#11763735	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
14	C₂²×#63214	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×D₁₇	C₄	1	4	yes	complemented
16	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
21	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
22	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊#595	—	4	2	no	—
24	#10587677	—	2	8	no	—
25	#10587677	—	4	4	no	—
26	C₂×C₄⋊#595	—	2	4	no	—
27	C₂×C₄×#595	—	4	2	no	—
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	D₄	1	2	yes	—
34	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₁₇	Q₈	1	2	yes	—
36	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₁₇	C₂³	1	2	yes	—
38	#10587534	—	2	4	no	—
39	#10587534	C₂³	1	8	yes	—
40	#10587533	—	2	8	no	—
41	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
44	C₂×#63214	C₂×C₄	1	1	yes	characteristic
45	C₂×#63214	C₂×C₄	1	1	yes	characteristic
46	C₂×#63214	C₂×C₄	1	4	yes	complemented
47	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
48	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
49	C₁₆×D₁₇	C₂×C₄	1	4	yes	complemented
50	C₁₆×D₁₇	C₂×C₄	1	4	yes	complemented
51	C₁₆×D₁₇	C₂³	1	2	yes	—
52	C₁₆×D₁₇	C₂³	1	2	yes	—
53	C₂³×D₁₇	D₈	1	1	yes	characteristic
54	C₂²×#595	—	4	4	no	—
55	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
57	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
60	C₂×C₄×D₁₇	D₈	1	2	yes	—
61	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
62	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
63	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
65	C₂²⋊#595	—	4	8	no	—
66	C₄⋊#595	—	2	8	no	—
67	C₄⋊#595	—	4	4	no	—
68	C₄×#595	—	4	8	no	—
69	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
73	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#3622	C₄⋊C₄	1	2	yes	—
75	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
76	C₂×#3622	C₂×Q₈	1	2	yes	—
77	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
78	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
79	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
80	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₈×D₁₇	C₂²×C₄	1	2	yes	—
82	C₈×D₁₇	C₂×D₄	1	2	yes	—
83	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
84	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
85	C₈×D₁₇	C₂×Q₈	1	2	yes	—
86	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
87	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
88	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
89	#63214	C₂²×C₄	1	2	yes	—
90	#63214	C₂²×C₄	1	2	yes	—
91	C₂×#2158	—	17	1	no	maximal, Sylow
92	C₂³×C₁₇	#107	1	1	yes	characteristic
93	C₂²×D₁₇	—	2	4	no	—
94	C₂²×D₁₇	#107	1	1	yes	characteristic
95	C₂²×D₁₇	#107	1	2	yes	—
96	C₂²×D₁₇	C₂×D₈	1	2	yes	—
97	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
98	C₂×#595	—	4	8	no	—
99	C₂×#595	—	8	8	no	—
100	C₂×C₄×C₁₇	#107	1	2	yes	—
101	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×Dic₁₇	#107	1	2	yes	—
105	C₂×Dic₁₇	#107	1	1	yes	characteristic
106	C₂×Dic₁₇	C₂×Q₁₆	1	2	yes	—
107	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₄×D₁₇	#107	1	2	yes	—
109	C₄×D₁₇	#107	1	2	yes	—
110	C₄×D₁₇	#107	1	2	yes	—
111	C₄×D₁₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₄×D₁₇	C₂×D₈	1	2	yes	—
114	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
115	C₄×D₁₇	C₂×Q₁₆	1	2	yes	—
116	C₄×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
117	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
121	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
122	#3622	C₂×C₄⋊C₄	1	2	yes	—
123	#3622	C₂²×Q₈	1	1	yes	characteristic
124	C₂³×C₁₆	—	17	1	no	—
125	C₂×#427	—	17	2	no	—
126	#2158	—	17	8	no	—
127	C₂×#367	—	17	2	no	—
128	C₂×#366	—	17	2	no	—
129	C₂²×C₁₇	—	2	2	no	—
130	C₂²×C₁₇	C₂×#107	1	2	yes	—
131	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
132	C₂×D₁₇	—	2	2	no	—
133	C₂×D₁₇	—	2	4	no	—
134	C₂×D₁₇	—	2	4	no	—
135	C₂×D₁₇	C₂×#107	1	2	yes	—
136	C₂×D₁₇	C₂×#107	1	1	yes	characteristic
137	C₂×D₁₇	#508	1	4	yes	—
138	C₂×D₁₇	C₂²×D₈	1	1	yes	characteristic
139	#595	—	8	8	no	—
140	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
141	C₄×C₁₇	C₂×#107	1	2	yes	—
142	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	Dic₁₇	C₂×#107	1	2	yes	—
144	Dic₁₇	C₂×#107	1	1	yes	characteristic
145	Dic₁₇	C₂²×Q₁₆	1	1	yes	characteristic
146	C₂³×C₈	—	17	1	no	—
147	C₂×#117	—	34	2	no	—
148	C₂²×C₁₆	—	17	1	no	—
149	C₂²×C₁₆	—	17	1	no	—
150	C₂²×C₁₆	—	17	1	no	—
151	C₂²×C₁₆	—	17	1	no	—
152	C₂²×C₁₆	—	17	1	no	—
153	C₂²×C₁₆	—	17	4	no	—
154	C₂²×C₁₆	—	17	4	no	—
155	C₂²×C₁₆	#595	1	1	yes	characteristic, complemented
156	#427	—	17	8	no	—
157	#427	—	34	4	no	—
158	C₂×#107	—	17	4	no	—
159	C₂×#106	—	34	2	no	—
160	#367	—	17	8	no	—
161	#366	—	17	8	no	—
162	C₂×C₁₇	—	2	2	no	—
163	C₂×C₁₇	#2158	1	2	yes	complemented
164	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
165	D₁₇	—	2	2	no	—
166	D₁₇	#2158	1	2	yes	complemented
167	D₁₇	C₂×#508	1	2	yes	—
168	C₂³×C₄	—	17	1	no	—
169	C₂²×C₈	—	17	1	no	—
170	C₂²×C₈	—	17	1	no	—
171	C₂²×C₈	—	17	2	no	—
172	C₂²×C₈	—	17	1	no	—
173	C₂²×C₈	—	17	2	no	—
174	C₂²×C₈	—	17	1	no	—
175	C₂²×C₈	—	17	2	no	—
176	C₂²×C₈	—	17	1	no	—
177	C₂²×C₈	—	17	2	no	—
178	C₂²×C₈	C₂×#595	1	1	yes	characteristic
179	#117	—	34	8	no	—
180	#117	—	68	4	no	—
181	C₂×C₄⋊C₄	—	34	4	no	—
182	C₂×C₂²⋊C₄	—	68	2	no	—
183	C₂×C₄²	—	68	2	no	—
184	C₂×C₁₆	—	17	1	no	—
185	C₂×C₁₆	—	17	1	no	—
186	C₂×C₁₆	—	17	4	no	—
187	C₂×C₁₆	—	17	2	no	—
188	C₂×C₁₆	—	17	2	no	—
189	C₂×C₁₆	—	17	4	no	—
190	C₂×C₁₆	—	17	4	no	—
191	C₂×C₁₆	—	17	2	no	—
192	C₂×C₁₆	—	17	2	no	—
193	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
194	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
195	C₂×C₁₆	C₂×#595	1	4	yes	complemented
196	#107	—	17	8	no	—
197	#107	—	34	4	no	—
198	#106	—	34	8	no	—
199	C₁₇	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	—	17	1	no	—
201	C₂²×C₄	—	17	1	no	—
202	C₂²×C₄	—	17	2	no	—
203	C₂²×C₄	—	17	1	no	—
204	C₂²×C₄	—	17	1	no	—
205	C₂²×C₄	—	17	2	no	—
206	C₂²×C₄	—	17	2	no	—
207	C₂²×C₄	—	17	2	no	—
208	C₂²×C₄	—	17	1	no	—
209	C₂²×C₄	—	17	1	no	—
210	C₂²×C₄	—	68	4	no	—
211	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
212	C₂×C₈	—	17	1	no	—
213	C₂×C₈	—	17	2	no	—
214	C₂×C₈	—	17	1	no	—
215	C₂×C₈	—	17	2	no	—
216	C₂×C₈	—	17	2	no	—
217	C₂×C₈	—	17	1	no	—
218	C₂×C₈	—	17	2	no	—
219	C₂×C₈	—	17	2	no	—
220	C₂×C₈	—	17	2	no	—
221	C₂×C₈	—	17	1	no	—
222	C₂×C₈	—	17	2	no	—
223	C₂×C₈	—	17	1	no	—
224	C₂×C₈	—	17	1	no	—
225	C₂×C₈	—	17	2	no	—
226	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
227	C₂×C₈	C₄⋊#595	1	2	yes	—
228	C₂×C₈	C₂²×#595	1	1	yes	characteristic
229	C₂×C₈	C₂²×#595	1	2	yes	—
230	C₄⋊C₄	—	34	8	no	—
231	C₄⋊C₄	—	68	4	no	—
232	C₂²⋊C₄	—	68	8	no	—
233	C₄²	—	68	8	no	—
234	C₁₆	—	17	2	no	—
235	C₁₆	—	17	2	no	—
236	C₁₆	C₂²×#595	1	2	yes	—
237	C₁₆	C₂²×#595	1	2	yes	—
238	C₂³	—	17	1	no	—
239	C₂³	—	17	2	no	—
240	C₂³	—	17	2	no	—
241	C₂³	—	17	1	no	—
242	C₂³	—	34	4	no	—
243	C₂³	#10587534	1	1	yes	characteristic
244	C₂×C₄	—	17	2	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	2	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	2	no	—
249	C₂×C₄	—	17	1	no	—
250	C₂×C₄	—	17	2	no	—
251	C₂×C₄	—	17	2	no	—
252	C₂×C₄	—	17	2	no	—
253	C₂×C₄	—	17	1	no	—
254	C₂×C₄	—	17	2	no	—
255	C₂×C₄	—	17	1	no	—
256	C₂×C₄	—	17	1	no	—
257	C₂×C₄	—	17	2	no	—
258	C₂×C₄	—	68	8	no	—
259	C₂×C₄	—	136	8	no	—
260	C₂×C₄	#10587534	1	2	yes	—
261	C₂×C₄	#10587534	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
264	C₈	—	17	2	no	—
265	C₈	—	17	1	no	—
266	C₈	—	17	1	no	—
267	C₈	C₂×C₄⋊#595	1	2	yes	—
268	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
269	C₈	C₂³×#595	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	17	1	no	—
272	C₂²	—	17	1	no	—
273	C₂²	—	17	4	no	—
274	C₂²	—	17	2	no	—
275	C₂²	—	34	4	no	—
276	C₂²	—	34	4	no	—
277	C₂²	—	34	2	no	—
278	C₂²	C₂×#10587534	1	2	yes	—
279	C₂²	C₂×#10587534	1	1	yes	characteristic
280	C₄	—	17	1	no	—
281	C₄	—	17	2	no	—
282	C₄	—	17	1	no	—
283	C₄	—	136	8	no	—
284	C₄	C₂×#10587534	1	1	yes	characteristic
285	C₄	C₂×#10587534	1	2	yes	—
286	C₄	C₂²×C₄⋊#595	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	17	2	no	—
289	C₂	—	17	2	no	—
290	C₂	—	34	2	no	—
291	C₂	#14545135	1	2	yes	direct product factor
292	C₂	C₂²×#10587534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587534
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₈	C₂³×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×D₁₇	D₈
cosocle	C₈×D₁₇	C₂⁴
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	C₂×#2158
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14545135	C₂	direct (N × H)
3. C₂×#11763736	C₂	semidirect
4. C₂×#11763735	C₂	semidirect
11. #11763736	C₂²	semidirect
12. #11763735	C₂²	semidirect
14. C₂²×#63214	C₄	semidirect
15. C₂×C₁₆×D₁₇	C₄	semidirect
16. C₂×C₁₆×D₁₇	C₄	semidirect
17. C₂×C₁₆×D₁₇	C₄	semidirect
21. C₂²×C₁₆×C₁₇	C₄	semidirect
43. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
46. C₂×#63214	C₂×C₄	semidirect
49. C₁₆×D₁₇	C₂×C₄	semidirect
50. C₁₆×D₁₇	C₂×C₄	semidirect
155. C₂²×C₁₆	#595	semidirect
163. C₂×C₁₇	#2158	semidirect
166. D₁₇	#2158	semidirect
195. C₂×C₁₆	C₂×#595	semidirect
199. C₁₇	C₂×#2158	semidirect
291. C₂	#14545135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763819	2
#14545135	2
C₂×#11763736	2
C₂×#11763735	2
C₂²×C₁₆×D₁₇	2
C₂×#2158	17

Subgroups — C2×#14545135 #17978433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14545135 #17978433