Subgroups — #16950520

Order 4352.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763227	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762638	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530566	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528839	C₂	1	1	yes	maximal, characteristic
5	#14528835	C₂	1	1	yes	maximal, characteristic
6	#14526899	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587587	C₂²	1	1	yes	characteristic
9	#11763227	—	2	1	no	—
10	#11763227	—	2	1	no	—
11	#11763227	C₄	1	2	yes	complemented
12	#11763227	C₄	1	2	yes	complemented
13	C₂×#10587469	—	2	1	no	—
14	C₂²×#63214	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₁₇	—	2	1	no	—
16	C₂×#10587490	C₂²	1	1	yes	characteristic
17	#11762847	C₂²	1	1	yes	characteristic
18	#11762827	—	2	1	no	—
19	#11762825	—	2	1	no	—
20	#11762638	—	2	1	no	—
21	#11762638	—	2	1	no	—
22	#11762636	—	2	1	no	—
23	#11762634	—	2	1	no	—
24	#11762580	—	2	1	no	—
25	#11762579	—	2	1	no	—
26	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
28	C₁₇×#427	C₂²	1	1	yes	characteristic
29	C₂×#63252	D₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₇	—	2	1	no	—
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×#117	—	2	1	no	—
33	#10587612	—	2	1	no	—
34	C₂×#63223	—	2	1	no	—
35	C₂×#63223	D₄	1	1	yes	characteristic
36	#10587587	—	2	1	no	—
37	#10587587	—	2	1	no	—
38	#10587587	C₂×C₄	1	2	yes	—
39	#10587587	Q₈	1	2	yes	—
40	C₂×#63218	—	2	1	no	—
41	C₂×C₈×D₁₇	—	2	1	no	—
42	C₁₇×M₃₂	—	2	1	no	—
43	C₁₇×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×C₁₆×C₁₇	—	2	1	no	—
45	C₁₇×#108	—	2	1	no	—
46	C₁₇×#108	—	2	1	no	—
47	C₁₇×#107	—	2	1	no	—
48	C₁₇×#106	—	2	1	no	—
49	#10587496	—	4	2	no	—
50	#10587490	—	2	1	no	—
51	#10587490	—	2	1	no	—
52	#10587489	—	2	1	no	—
53	#10587488	—	2	1	no	—
54	C₂×#63214	C₂×C₄	1	1	yes	characteristic
55	C₂×#63214	C₂×C₄	1	1	yes	characteristic
56	C₂×#63214	D₄	1	1	yes	characteristic
57	C₂×#63214	D₄	1	1	yes	characteristic
58	C₂×#63214	D₄	1	1	yes	characteristic
59	C₂×#63214	D₄	1	1	yes	characteristic
60	#10587476	—	4	2	no	—
61	#10587469	—	2	2	no	—
62	#10587469	—	2	2	no	—
63	C₁₆×D₁₇	—	2	2	no	—
64	C₁₆×D₁₇	—	2	2	no	—
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₇	—	2	1	no	—
67	#63252	—	2	1	no	—
68	#63252	—	2	1	no	—
69	#63252	C₄⋊C₄	1	2	yes	—
70	#63252	D₈	1	2	yes	—
71	C₂×D₆₈	Q₁₆	1	1	yes	characteristic
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×Dic₃₄	Q₁₆	1	1	yes	characteristic
74	M₁₆×C₁₇	—	4	1	no	—
75	M₁₆×C₁₇	—	4	1	no	—
76	C₂×C₈×C₁₇	—	4	1	no	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	—	4	1	no	—
84	C₂²⋊C₄×C₁₇	—	4	1	no	—
85	C₄²×C₁₇	—	4	1	no	—
86	C₂²⋊Dic₁₇	—	4	1	no	—
87	C₄⋊Dic₁₇	—	4	1	no	—
88	C₄×Dic₁₇	—	4	1	no	—
89	#63223	—	2	1	no	—
90	#63223	—	4	1	no	—
91	#63223	—	4	1	no	—
92	#63223	C₄⋊C₄	1	2	yes	—
93	C₂×#3622	—	2	1	no	—
94	C₂×#3622	—	4	1	no	—
95	#63218	—	2	2	no	—
96	#63218	—	2	2	no	—
97	C₈×D₁₇	—	2	2	no	—
98	C₈×D₁₇	—	2	2	no	—
99	C₁₆×C₁₇	—	2	2	no	—
100	#63214	C₂²⋊C₄	1	1	yes	characteristic
101	#63214	C₂²⋊C₄	1	1	yes	characteristic
102	#63214	C₂²⋊C₄	1	1	yes	characteristic
103	#63214	C₂²⋊C₄	1	1	yes	characteristic
104	#12841	—	17	1	no	maximal, Sylow
105	C₂³×C₁₇	#135	1	1	yes	characteristic
106	C₂²×D₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	8	1	no	—
108	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
114	C₂²⋊D₁₇	—	2	2	no	—
115	C₂²⋊D₁₇	—	2	2	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	8	1	no	—
118	D₆₈	—	2	1	no	—
119	D₆₈	#107	1	2	yes	—
120	C₄×D₁₇	—	2	2	no	—
121	C₄×D₁₇	—	2	2	no	—
122	Dic₃₄	—	2	1	no	—
123	Dic₃₄	#107	1	2	yes	—
124	C₈×C₁₇	—	8	1	no	—
125	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₈×C₁₇	#117	1	1	yes	characteristic
127	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
129	#3622	—	2	2	no	—
130	#3622	—	8	1	no	—
131	C₂×#504	—	17	1	no	—
132	C₂×#368	—	17	1	no	—
133	#2158	—	17	1	no	—
134	#1814	—	17	1	no	—
135	#1814	—	17	1	no	—
136	#1389	—	17	1	no	—
137	#1389	D₁₇	1	1	yes	characteristic, complemented
138	C₂²×C₁₇	—	4	1	no	—
139	C₂²×C₁₇	#427	1	1	yes	characteristic
140	C₂²×C₁₇	#466	1	1	yes	characteristic
141	C₂²×C₁₇	#467	1	1	yes	characteristic
142	C₂×D₁₇	—	2	2	no	—
143	C₂×D₁₇	—	4	2	no	—
144	C₄×C₁₇	—	16	1	no	—
145	C₄×C₁₇	#380	1	1	yes	characteristic
146	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
147	C₄×C₁₇	#466	1	1	yes	characteristic
148	C₄×C₁₇	#467	1	1	yes	characteristic
149	Dic₁₇	—	2	2	no	—
150	Dic₁₇	—	16	1	no	—
151	C₂×#131	—	17	1	no	—
152	#504	—	17	2	no	—
153	#504	—	17	2	no	—
154	#504	—	34	1	no	—
155	#504	—	34	1	no	—
156	C₂×M₃₂	—	34	1	no	—
157	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
158	C₂²×C₁₆	—	17	1	no	—
159	C₂²×C₁₆	—	34	1	no	—
160	C₂×#108	—	17	1	no	—
161	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
162	#427	—	17	1	no	—
163	#427	C₂×D₁₇	1	1	yes	characteristic
164	#421	—	34	1	no	—
165	#421	—	34	1	no	—
166	#368	—	34	1	no	—
167	#368	—	34	1	no	—
168	#367	—	34	1	no	—
169	#366	—	34	1	no	—
170	#344	—	34	1	no	—
171	#344	—	34	1	no	—
172	C₂×C₁₇	—	2	1	no	—
173	C₂×C₁₇	—	2	1	no	—
174	C₂×C₁₇	—	2	1	no	—
175	C₂×C₁₇	#1808	1	1	yes	characteristic
176	D₁₇	—	4	2	no	—
177	C₂×#41	—	17	1	no	—
178	#131	—	17	2	no	—
179	#131	—	17	2	no	—
180	#131	—	34	1	no	—
181	#131	—	34	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	17	1	no	—
184	C₂×M₁₆	—	34	1	no	—
185	C₂×M₁₆	—	34	1	no	—
186	C₂²×C₈	—	34	1	no	—
187	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
188	#117	—	2	1	no	—
189	#117	—	34	1	no	—
190	M₃₂	—	2	1	no	—
191	M₃₂	—	34	2	no	—
192	M₃₂	—	34	2	no	—
193	M₃₂	C₄×D₁₇	1	2	yes	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	17	1	no	—
196	C₂×C₁₆	—	17	1	no	—
197	C₂×C₁₆	—	17	1	no	—
198	C₂×C₁₆	—	17	1	no	—
199	C₂×C₁₆	—	17	1	no	—
200	C₂×C₁₆	—	17	1	no	—
201	C₂×C₁₆	—	34	2	no	—
202	C₂×C₁₆	—	34	2	no	—
203	#108	—	2	1	no	—
204	#108	—	2	1	no	—
205	#108	—	34	1	no	—
206	#108	—	34	1	no	—
207	#107	—	2	1	no	—
208	#107	—	34	1	no	—
209	#106	—	2	1	no	—
210	#106	—	34	1	no	—
211	#104	—	68	2	no	—
212	#104	—	68	2	no	—
213	C₁₇	#12841	1	1	yes	characteristic, complemented, Sylow
214	#41	—	17	2	no	—
215	#41	—	17	2	no	—
216	#41	—	34	1	no	—
217	#41	—	34	1	no	—
218	C₂×Q₈	—	17	1	no	—
219	C₂×D₄	—	17	1	no	—
220	C₂×D₄	—	34	1	no	—
221	C₂²×C₄	—	34	1	no	—
222	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	17	2	no	—
226	M₁₆	—	34	1	no	—
227	M₁₆	—	34	2	no	—
228	M₁₆	—	34	2	no	—
229	M₁₆	—	68	1	no	—
230	M₁₆	—	68	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	34	1	no	—
233	C₂×C₈	—	34	2	no	—
234	C₂×C₈	—	34	2	no	—
235	C₂×C₈	—	68	1	no	—
236	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
237	C₂×C₈	#63252	1	1	yes	characteristic
238	C₂×C₈	#63252	1	1	yes	characteristic
239	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
240	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
241	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	68	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	68	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	68	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	17	1	no	—
250	C₁₆	—	17	1	no	—
251	C₁₆	—	17	1	no	—
252	C₁₆	—	17	1	no	—
253	C₂³	—	34	1	no	—
254	C₂³	#10587602	1	1	yes	characteristic
255	Q₈	—	17	2	no	—
256	Q₈	—	34	1	no	—
257	D₄	—	17	2	no	—
258	D₄	—	34	2	no	—
259	D₄	—	34	2	no	—
260	D₄	—	34	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	34	1	no	—
263	C₂×C₄	—	34	2	no	—
264	C₂×C₄	—	34	2	no	—
265	C₂×C₄	—	136	1	no	—
266	C₂×C₄	#10587570	1	1	yes	characteristic
267	C₂×C₄	#10587570	1	1	yes	characteristic
268	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
269	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
270	C₂×C₄	#10587602	1	1	yes	characteristic
271	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
272	C₈	—	8	1	no	—
273	C₈	—	34	2	no	—
274	C₈	—	136	1	no	—
275	C₈	#10587558	1	1	yes	characteristic
276	C₈	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
277	C₈	#10587582	1	1	yes	characteristic
278	C₈	#10587582	1	1	yes	characteristic
279	C₂²	—	4	1	no	—
280	C₂²	—	34	2	no	—
281	C₂²	—	68	2	no	—
282	C₂²	#11762844	1	1	yes	characteristic
283	C₂²	#11763044	1	1	yes	characteristic
284	C₂²	#11763049	1	1	yes	characteristic
285	C₄	—	16	1	no	—
286	C₄	—	272	1	no	—
287	C₄	—	34	2	no	—
288	C₄	#11762668	1	1	yes	characteristic
289	C₄	#107×D₁₇	1	1	yes	characteristic
290	C₄	#11763044	1	1	yes	characteristic
291	C₄	#11763049	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	68	2	no	—
296	C₂	#14528812	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763044
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1389	C₂
socle	C₂×C₁₇	#1808
prime radical	C₂×#63252	D₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1389	D₁₇
hypocenter	C₁₇	#12841
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

Complements

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

N	H ≅ G/N	type
2. C₂×#11762638	C₂	semidirect
3. #14530566	C₂	semidirect
6. #14526899	C₂	semidirect
7. C₁₇×#1389	C₂	semidirect
11. #11763227	C₄	semidirect
12. #11763227	C₄	semidirect
137. #1389	D₁₇	semidirect
213. C₁₇	#12841	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763227	2
C₂×#11762638	2
#14530566	2
#14528839	2
#14528835	2
#14526899	2
C₁₇×#1389	2
#12841	17