Subgroups — #17958756

Order 4352.

382 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545114	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
3	#14543751	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#63216	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	—	2	1	no	—
7	C₂×#10587529	—	2	1	no	—
8	#11763730	—	2	1	no	—
9	C₄⋊#63216	C₂²	1	2	yes	—
10	#11763695	—	2	1	no	—
11	C₄×#63216	C₂²	1	2	yes	—
12	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
13	Q₈×#3622	C₂²	1	1	yes	characteristic
14	C₈×Dic₃₄	—	2	1	no	—
15	C₈×Dic₃₄	C₄	1	1	yes	characteristic
16	#11762933	—	2	1	no	—
17	#11762933	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×D₁₇	—	2	1	no	—
19	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
20	C₄×C₈×D₁₇	—	2	1	no	—
21	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₁₇	C₄	1	1	yes	characteristic
23	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
26	C₂×C₈×D₁₇	—	2	1	no	—
27	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
28	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	—	2	1	no	—
30	C₄⋊C₄×D₁₇	C₈	1	1	yes	characteristic
31	#10587573	—	2	1	no	—
32	#10587573	C₈	1	1	yes	characteristic
33	C₄²×D₁₇	—	2	1	no	—
34	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₃₄	—	2	1	no	—
36	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
37	C₂×#63216	—	2	2	no	—
38	C₂×#63216	—	2	2	no	—
39	#10587529	—	2	4	no	—
40	#10587529	—	4	2	no	—
41	C₁₇×C₄⋊C₈	—	2	1	no	—
42	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	—	2	1	no	—
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	#10587485	—	2	1	no	—
46	#10587485	—	2	1	no	—
47	#10587485	—	2	1	no	—
48	C₈×Dic₁₇	—	2	1	no	—
49	C₈×Dic₁₇	—	2	1	no	—
50	C₈×Dic₁₇	D₄	1	1	yes	characteristic
51	C₈×Dic₁₇	D₄	1	1	yes	characteristic
52	C₄⋊#3622	—	2	1	no	—
53	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
54	C₄×#3622	—	2	1	no	—
55	C₄×#3622	C₂³	1	1	yes	characteristic
56	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
57	Q₈×D₁₇	—	2	2	no	—
58	Q₈×D₁₇	C₁₆	1	4	yes	complemented
59	C₂×C₄×D₁₇	—	2	1	no	—
60	C₂×C₄×D₁₇	—	2	1	no	—
61	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
64	C₂×Dic₃₄	—	2	1	no	—
65	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₁₇	—	2	1	no	—
67	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₁₇	—	2	1	no	—
72	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₄⋊Dic₁₇	—	2	1	no	—
74	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
75	#63225	—	2	1	no	—
76	#63225	—	2	1	no	—
77	#63225	—	2	1	no	—
78	C₄×Dic₁₇	—	2	1	no	—
79	C₄×Dic₁₇	—	2	1	no	—
80	C₄×Dic₁₇	—	2	1	no	—
81	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
83	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
84	C₂×#3622	—	2	1	no	—
85	C₂×#3622	C₂×D₄	1	1	yes	characteristic
86	C₂×#3622	C₂×D₄	1	1	yes	characteristic
87	C₈×D₁₇	—	2	2	no	—
88	C₈×D₁₇	—	4	2	no	—
89	C₈×D₁₇	SD₁₆	1	2	yes	—
90	C₈×D₁₇	Q₁₆	1	2	yes	—
91	#63216	—	2	4	no	—
92	#63216	—	4	2	no	—
93	C₂×Q₈⋊C₁₆	—	17	1	no	maximal, Sylow
94	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
95	Q₈×C₁₇	—	2	1	no	—
96	Q₈×C₁₇	C₂×C₁₆	1	2	yes	complemented
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×Dic₁₇	—	2	1	no	—
103	C₂×Dic₁₇	—	2	1	no	—
104	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
107	C₄×D₁₇	—	2	4	no	—
108	C₄×D₁₇	—	2	2	no	—
109	C₄×D₁₇	—	4	2	no	—
110	C₄×D₁₇	#103	1	2	yes	—
111	C₄×D₁₇	#104	1	2	yes	—
112	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
113	C₄×D₁₇	M₃₂	1	2	yes	—
114	Dic₃₄	—	2	1	no	—
115	Dic₃₄	—	2	4	no	—
116	Dic₃₄	C₂×C₁₆	1	2	yes	complemented
117	C₈×C₁₇	—	2	1	no	—
118	C₈×C₁₇	—	4	1	no	—
119	C₈×C₁₇	#103	1	2	yes	—
120	#3622	—	2	1	no	—
121	#3622	—	4	1	no	—
122	#3622	C₂×SD₁₆	1	1	yes	characteristic
123	#3622	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₈×Q₈	—	17	1	no	—
125	C₂×C₄⋊C₁₆	—	17	1	no	—
126	C₂×C₄×C₁₆	—	17	1	no	—
127	Q₈⋊C₁₆	—	17	2	no	—
128	Q₈⋊C₁₆	—	17	2	no	—
129	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×D₁₇	Q₈⋊C₈	1	2	yes	—
131	C₂×D₁₇	C₂²⋊C₁₆	1	2	yes	—
132	C₂×D₁₇	#350	1	2	yes	—
133	C₄×C₁₇	—	2	2	no	—
134	C₄×C₁₇	—	2	1	no	—
135	C₄×C₁₇	—	4	1	no	—
136	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
137	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
138	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
139	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
140	Dic₁₇	—	2	2	no	—
141	Dic₁₇	—	2	1	no	—
142	Dic₁₇	—	4	1	no	—
143	Dic₁₇	Q₈⋊C₈	1	2	yes	—
144	Dic₁₇	C₂²⋊C₁₆	1	2	yes	—
145	C₂×C₄×Q₈	—	17	1	no	—
146	C₂²×C₁₆	—	34	1	no	—
147	C₂²×C₁₆	—	34	1	no	—
148	C₈×Q₈	—	17	1	no	—
149	C₈×Q₈	—	17	1	no	—
150	C₈×Q₈	—	17	1	no	—
151	C₈×Q₈	—	34	1	no	—
152	C₈×Q₈	—	34	1	no	—
153	C₈×Q₈	#595	1	1	yes	characteristic
154	C₂×C₄⋊C₈	—	17	1	no	—
155	C₂×C₄⋊C₈	—	34	1	no	—
156	C₂×C₄×C₈	—	17	1	no	—
157	C₂×C₄×C₈	—	34	1	no	—
158	C₄⋊C₁₆	—	17	2	no	—
159	C₄⋊C₁₆	—	34	1	no	—
160	C₄×C₁₆	—	17	2	no	—
161	C₄×C₁₆	—	34	1	no	—
162	C₂×C₁₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
163	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
164	C₂×C₁₇	C₂×#350	1	1	yes	characteristic
165	D₁₇	Q₈⋊C₁₆	1	4	yes	complemented
166	C₂²×Q₈	—	17	1	no	—
167	C₂²×C₈	—	17	1	no	—
168	C₂²×C₈	—	17	1	no	—
169	C₂²×C₈	—	34	1	no	—
170	C₄×Q₈	—	17	1	no	—
171	C₄×Q₈	—	17	1	no	—
172	C₄×Q₈	—	17	1	no	—
173	C₄×Q₈	—	34	1	no	—
174	C₄×Q₈	—	34	1	no	—
175	C₄×Q₈	C₂×#595	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	17	1	no	—
177	C₂×C₄⋊C₄	—	34	1	no	—
178	C₂×C₄²	—	17	1	no	—
179	C₂×C₄²	—	34	1	no	—
180	C₂×C₁₆	—	34	2	no	—
181	C₂×C₁₆	—	34	4	no	—
182	C₂×C₁₆	—	34	2	no	—
183	C₂×C₁₆	—	68	2	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	17	1	no	—
186	C₄⋊C₈	—	34	1	no	—
187	C₄⋊C₈	—	34	1	no	—
188	C₄⋊C₈	—	34	1	no	—
189	C₄⋊C₈	—	34	1	no	—
190	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	17	1	no	—
193	C₄×C₈	—	17	1	no	—
194	C₄×C₈	—	17	1	no	—
195	C₄×C₈	—	34	1	no	—
196	C₄×C₈	—	34	1	no	—
197	C₄×C₈	—	34	1	no	—
198	C₄×C₈	C₂×#595	1	1	yes	characteristic
199	C₁₇	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
200	C₂×Q₈	—	17	4	no	—
201	C₂×Q₈	—	17	1	no	—
202	C₂×Q₈	—	34	1	no	—
203	C₂×Q₈	—	34	2	no	—
204	C₂×Q₈	#63242	1	1	yes	characteristic
205	C₂²×C₄	—	17	1	no	—
206	C₂²×C₄	—	17	1	no	—
207	C₂²×C₄	—	17	1	no	—
208	C₂²×C₄	—	34	1	no	—
209	C₂²×C₄	—	34	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	17	1	no	—
212	C₂×C₈	—	17	2	no	—
213	C₂×C₈	—	17	2	no	—
214	C₂×C₈	—	17	1	no	—
215	C₂×C₈	—	34	2	no	—
216	C₂×C₈	—	34	1	no	—
217	C₂×C₈	—	68	2	no	—
218	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	17	1	no	—
222	C₄⋊C₄	—	34	1	no	—
223	C₄⋊C₄	—	34	1	no	—
224	C₄⋊C₄	—	34	1	no	—
225	C₄⋊C₄	—	34	1	no	—
226	C₄⋊C₄	#63242	1	1	yes	characteristic
227	C₄²	—	2	1	no	—
228	C₄²	—	17	1	no	—
229	C₄²	—	17	1	no	—
230	C₄²	—	17	1	no	—
231	C₄²	—	34	1	no	—
232	C₄²	—	34	1	no	—
233	C₄²	—	34	1	no	—
234	C₄²	C₂²×#595	1	1	yes	characteristic
235	C₁₆	—	34	4	no	—
236	C₁₆	—	68	2	no	—
237	C₂³	—	17	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	17	2	no	—
240	Q₈	—	34	4	no	—
241	Q₈	—	34	1	no	—
242	Q₈	#10587529	1	2	yes	complemented
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	1	no	—
247	C₂×C₄	—	17	2	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	—	34	2	no	—
253	C₂×C₄	—	34	4	no	—
254	C₂×C₄	—	34	1	no	—
255	C₂×C₄	—	34	1	no	—
256	C₂×C₄	—	68	2	no	—
257	C₂×C₄	C₂×#63242	1	1	yes	characteristic
258	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	17	1	no	—
263	C₈	—	17	1	no	—
264	C₈	—	34	1	no	—
265	C₈	—	68	1	no	—
266	C₈	#10587549	1	2	yes	—
267	C₂²	—	17	2	no	—
268	C₂²	—	17	2	no	—
269	C₂²	—	17	2	no	—
270	C₂²	#11763777	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	17	2	no	—
275	C₄	—	17	2	no	—
276	C₄	—	34	2	no	—
277	C₄	—	34	1	no	—
278	C₄	—	68	1	no	—
279	C₄	C₂×#10587549	1	1	yes	characteristic
280	C₄	D₁₇⋊#104	1	1	yes	characteristic
281	C₄	C₂×#10587529	1	1	yes	characteristic
282	C₄	#11763865	1	1	yes	characteristic
283	C₂	—	17	4	no	—
284	C₂	#14544073	1	1	yes	characteristic
285	C₂	#14545009	1	1	yes	characteristic
286	C₂	#14545024	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₈×Q₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₈×Q₈	#595
hypocenter	C₁₇	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #14543751	C₂	semidirect
4. Q₈⋊#63216	C₂	semidirect
58. Q₈×D₁₇	C₁₆	semidirect
96. Q₈×C₁₇	C₂×C₁₆	semidirect
116. Dic₃₄	C₂×C₁₆	semidirect
165. D₁₇	Q₈⋊C₁₆	semidirect
199. C₁₇	C₂×Q₈⋊C₁₆	semidirect
242. Q₈	#10587529	semidirect

Maximal subgroups

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

subgroup	index
#14545114	2
C₄×#10587529	2
#14543751	2
Q₈⋊#63216	2
C₈×Q₈×D₁₇	2
C₂×Q₈⋊C₁₆	17