Subgroups — C₂×D₁₇⋊#368 #17978439

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇⋊#108	C₂	1	2	yes	maximal
2	C₂×#11763738	C₂	1	2	yes	maximal, complemented
3	D₁₇⋊#368	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11763737	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×D₁₇⋊M₁₆	—	2	2	no	—
7	C₂×#10587536	C₂²	1	4	yes	—
8	D₁₇⋊#108	—	2	4	no	—
9	D₁₇⋊#108	C₂²	1	8	yes	—
10	C₂×#10587535	—	2	2	no	—
11	#11763738	C₂²	1	8	yes	complemented
12	#11763737	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
14	C₂²×#63214	C₄	1	1	yes	characteristic
15	C₂×C₁₆×D₁₇	C₄	1	4	yes	—
16	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
17	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
18	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
20	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
22	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
23	C₂×#63247	—	4	2	no	—
24	D₁₇⋊M₁₆	—	2	8	no	—
25	D₁₇⋊M₁₆	—	4	4	no	—
26	C₂×#63243	—	2	4	no	—
27	C₂×#63242	—	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	2	yes	—
31	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
37	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
38	#10587536	—	2	4	no	—
39	#10587536	C₂³	1	8	yes	—
40	#10587535	—	2	8	no	—
41	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
43	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×#63214	C₂×C₄	1	1	yes	characteristic
45	C₂×#63214	C₂×C₄	1	4	yes	—
46	C₂×#63214	C₂×C₄	1	1	yes	characteristic
47	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
48	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
49	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
50	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
51	C₁₆×D₁₇	C₂³	1	2	yes	—
52	C₁₆×D₁₇	C₂³	1	2	yes	—
53	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
56	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
59	C₂×C₄×D₁₇	D₈	1	2	yes	—
60	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
61	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
64	#63247	—	4	8	no	—
65	C₂×#3624	—	4	4	no	—
66	#63243	—	2	8	no	—
67	#63243	—	4	4	no	—
68	#63242	—	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	2	yes	—
72	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#3622	C₄⋊C₄	1	2	yes	—
75	C₂×#3622	C₂×Q₈	1	2	yes	—
76	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
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	2	yes	—
81	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₈×D₁₇	C₂×D₄	1	2	yes	—
83	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
84	C₈×D₁₇	C₂×Q₈	1	2	yes	—
85	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
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	C₂²×#368	—	17	1	no	maximal, Sylow
92	C₂³×C₁₇	#107	1	1	yes	characteristic
93	C₂²×D₁₇	—	2	4	no	—
94	C₂²×D₁₇	#107	1	2	yes	—
95	C₂²×D₁₇	#107	1	1	yes	characteristic
96	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
97	C₂²×D₁₇	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₁₇	#107	1	2	yes	—
99	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×Dic₁₇	#107	1	2	yes	—
103	C₂×Dic₁₇	#107	1	1	yes	characteristic
104	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
105	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
106	C₄×D₁₇	#107	1	2	yes	—
107	C₄×D₁₇	#107	1	2	yes	—
108	C₄×D₁₇	#107	1	2	yes	—
109	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₄×D₁₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₄×D₁₇	C₂×D₈	1	2	yes	—
112	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
113	C₄×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
114	C₄×D₁₇	C₂×Q₁₆	1	2	yes	—
115	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
116	#3624	—	4	8	no	—
117	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
120	#3622	C₂×C₄⋊C₄	1	2	yes	—
121	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
122	#3622	C₂²×Q₈	1	1	yes	characteristic
123	C₂³×C₁₆	—	17	1	no	—
124	C₂²×#108	—	17	2	no	—
125	C₂×#368	—	17	2	no	—
126	C₂×#368	—	17	8	no	—
127	C₂×#368	—	17	2	no	—
128	C₂²×C₁₇	—	2	2	no	—
129	C₂²×C₁₇	C₂×#107	1	2	yes	—
130	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
131	C₂×D₁₇	—	2	2	no	—
132	C₂×D₁₇	—	2	4	no	—
133	C₂×D₁₇	—	2	4	no	—
134	C₂×D₁₇	#368	1	4	yes	complemented
135	C₂×D₁₇	C₂×#107	1	1	yes	characteristic
136	C₂×D₁₇	C₂×#107	1	2	yes	—
137	C₂×D₁₇	C₂²×Q₁₆	1	1	yes	characteristic
138	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
139	C₄×C₁₇	C₂×#107	1	2	yes	—
140	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	Dic₁₇	C₂×#107	1	1	yes	characteristic
142	Dic₁₇	C₂×#107	1	2	yes	—
143	Dic₁₇	C₂²×D₈	1	1	yes	characteristic
144	C₂²×M₁₆	—	34	2	no	—
145	C₂³×C₈	—	17	1	no	—
146	C₂²×C₁₆	—	17	1	no	—
147	C₂²×C₁₆	—	17	1	no	—
148	C₂²×C₁₆	—	17	4	no	—
149	C₂²×C₁₆	—	17	1	no	—
150	C₂²×C₁₆	—	17	1	no	—
151	C₂²×C₁₆	—	17	4	no	—
152	C₂²×C₁₆	—	17	1	no	—
153	C₂²×C₁₆	#595	1	1	yes	characteristic
154	C₂×#108	—	17	8	no	—
155	C₂×#108	—	17	4	no	—
156	C₂×#108	—	34	2	no	—
157	C₂×#108	—	34	4	no	—
158	#368	—	17	8	no	—
159	#368	—	17	8	no	—
160	C₂×C₁₇	—	2	2	no	—
161	C₂×C₁₇	C₂×#368	1	2	yes	complemented
162	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
163	D₁₇	—	2	1	no	—
164	D₁₇	—	2	1	no	—
165	D₁₇	C₂×#368	1	4	yes	complemented
166	C₂³×C₄	—	17	1	no	—
167	C₂×M₁₆	—	34	4	no	—
168	C₂×M₁₆	—	34	8	no	—
169	C₂×M₁₆	—	68	4	no	—
170	C₂×M₁₆	—	68	2	no	—
171	C₂²×C₈	—	17	1	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	2	no	—
177	C₂²×C₈	—	17	1	no	—
178	C₂²×C₈	—	17	1	no	—
179	C₂²×C₈	—	17	2	no	—
180	C₂²×C₈	—	68	2	no	—
181	C₂²×C₈	C₂×#595	1	1	yes	characteristic
182	C₂×C₁₆	—	17	1	no	—
183	C₂×C₁₆	—	17	4	no	—
184	C₂×C₁₆	—	17	1	no	—
185	C₂×C₁₆	—	17	2	no	—
186	C₂×C₁₆	—	17	4	no	—
187	C₂×C₁₆	—	17	4	no	—
188	C₂×C₁₆	—	17	2	no	—
189	C₂×C₁₆	—	17	2	no	—
190	C₂×C₁₆	—	17	2	no	—
191	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
192	C₂×C₁₆	C₂×#595	1	4	yes	—
193	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
194	#108	—	17	8	no	—
195	#108	—	34	8	no	—
196	#108	—	34	4	no	—
197	C₁₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	17	1	no	—
199	C₂²×C₄	—	17	1	no	—
200	C₂²×C₄	—	17	1	no	—
201	C₂²×C₄	—	17	2	no	—
202	C₂²×C₄	—	17	1	no	—
203	C₂²×C₄	—	17	2	no	—
204	C₂²×C₄	—	17	2	no	—
205	C₂²×C₄	—	17	2	no	—
206	C₂²×C₄	—	17	1	no	—
207	C₂²×C₄	—	17	1	no	—
208	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
209	M₁₆	—	34	8	no	—
210	M₁₆	—	68	4	no	—
211	M₁₆	—	68	8	no	—
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	2	no	—
222	C₂×C₈	—	17	1	no	—
223	C₂×C₈	—	17	1	no	—
224	C₂×C₈	—	17	2	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	—	68	8	no	—
227	C₂×C₈	—	68	4	no	—
228	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
229	C₂×C₈	C₄⋊#595	1	2	yes	—
230	C₂×C₈	C₂²×#595	1	1	yes	characteristic
231	C₂×C₈	C₂²×#595	1	2	yes	—
232	C₁₆	—	17	2	no	—
233	C₁₆	—	17	2	no	—
234	C₁₆	C₂²×#595	1	2	yes	—
235	C₁₆	C₂²×#595	1	2	yes	—
236	C₂³	—	17	1	no	—
237	C₂³	—	17	2	no	—
238	C₂³	—	17	2	no	—
239	C₂³	—	17	1	no	—
240	C₂³	—	34	4	no	—
241	C₂³	#10587534	1	1	yes	characteristic
242	C₂×C₄	—	17	1	no	—
243	C₂×C₄	—	17	2	no	—
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	2	no	—
250	C₂×C₄	—	17	2	no	—
251	C₂×C₄	—	17	1	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	#10587534	1	2	yes	—
257	C₂×C₄	#10587534	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
260	C₈	—	17	2	no	—
261	C₈	—	17	1	no	—
262	C₈	—	17	1	no	—
263	C₈	—	68	8	no	—
264	C₈	C₂×C₄⋊#595	1	2	yes	—
265	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
266	C₈	C₂³×#595	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	17	2	no	—
269	C₂²	—	17	4	no	—
270	C₂²	—	17	1	no	—
271	C₂²	—	17	1	no	—
272	C₂²	—	34	4	no	—
273	C₂²	—	34	4	no	—
274	C₂²	—	34	2	no	—
275	C₂²	C₂×#10587534	1	2	yes	—
276	C₂²	C₂×#10587534	1	1	yes	characteristic
277	C₄	—	17	1	no	—
278	C₄	—	17	1	no	—
279	C₄	—	17	2	no	—
280	C₄	C₂×#10587534	1	1	yes	characteristic
281	C₄	C₂×#10587534	1	2	yes	—
282	C₄	C₂²×C₄⋊#595	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	17	4	no	—
285	C₂	—	34	1	no	—
286	C₂	—	34	1	no	—
287	C₂	D₁₇⋊#368	1	2	yes	direct product factor
288	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	Q₁₆
cosocle	C₈×D₁₇	C₂⁴
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	C₂²×#368
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. C₂×#11763738	C₂	semidirect
3. D₁₇⋊#368	C₂	direct (N × H)
4. C₂×#11763737	C₂	semidirect
11. #11763738	C₂²	semidirect
12. #11763737	C₂²	semidirect
134. C₂×D₁₇	#368	semidirect
161. C₂×C₁₇	C₂×#368	semidirect
165. D₁₇	C₂×#368	semidirect
197. C₁₇	C₂²×#368	semidirect
287. C₂	D₁₇⋊#368	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×D₁₇⋊#108	2
C₂×#11763738	2
D₁₇⋊#368	2
C₂×#11763737	2
C₂²×C₁₆×D₁₇	2
C₂²×#368	17

Subgroups — C2×D17⋊#368 #17978439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₇⋊#368 #17978439