Subgroups — #17952269

Order 4352.

502 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	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763651	C₂	1	2	yes	maximal
4	#14543606	C₂	1	2	yes	maximal
5	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	C₄	1	2	yes	—
7	C₂×C₄⋊#3624	C₄	1	2	yes	—
8	C₂×C₄×#3624	C₂²	1	2	yes	—
9	C₂×C₄×#3624	C₂²	1	2	yes	—
10	#11763675	C₄	1	2	yes	—
11	#11763675	C₄	1	2	yes	—
12	#11763651	—	2	4	no	—
13	C₂×C₄×#3622	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
20	C₂²×#3624	C₂×C₄	1	4	yes	—
21	C₂²×#3624	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
33	C₄⋊#3624	—	2	4	no	—
34	C₄⋊#3624	C₈	1	8	yes	complemented
35	C₄×#3624	—	2	2	no	—
36	C₄×#3624	—	2	2	no	—
37	C₄×#3624	—	2	2	no	—
38	C₄×#3624	—	2	2	no	—
39	C₄×C₈×C₁₇	—	2	2	no	—
40	C₈×Dic₁₇	—	2	1	no	—
41	C₈×Dic₁₇	—	2	1	no	—
42	C₈×Dic₁₇	—	2	2	no	—
43	C₈×Dic₁₇	—	2	2	no	—
44	C₈×Dic₁₇	D₄	1	2	yes	—
45	C₈×Dic₁₇	Q₈	1	2	yes	—
46	C₄×#3622	—	2	2	no	—
47	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
51	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
52	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×#3624	—	2	4	no	—
55	C₂×#3624	—	2	4	no	—
56	C₂×#3624	—	2	4	no	—
57	C₂×#3624	—	2	4	no	—
58	C₂×#3624	—	2	4	no	—
59	C₂×#3624	C₂×C₈	1	8	yes	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	—	2	1	no	—
63	C₂×C₈×C₁₇	—	2	1	no	—
64	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
66	C₄²×C₁₇	M₁₆	1	2	yes	—
67	C₄²×C₁₇	M₁₆	1	2	yes	—
68	C₄×Dic₁₇	—	2	1	no	—
69	C₄×Dic₁₇	—	2	1	no	—
70	C₄×Dic₁₇	—	2	2	no	—
71	C₄×Dic₁₇	—	2	2	no	—
72	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
73	C₄×Dic₁₇	M₁₆	1	4	yes	—
74	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
75	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
76	C₄×Dic₁₇	#41	1	1	yes	characteristic
77	C₄×Dic₁₇	#41	1	1	yes	characteristic
78	C₂×#3622	—	2	1	no	—
79	C₂×#3622	—	2	1	no	—
80	C₂×#3622	—	2	1	no	—
81	C₂×#3622	—	2	1	no	—
82	C₂×#3622	C₄⋊C₄	1	2	yes	—
83	C₂×#3622	M₁₆	1	2	yes	—
84	#8261	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₁₇	#97	1	2	yes	—
90	C₂×C₄×C₁₇	#97	1	2	yes	—
91	C₂×C₄×C₁₇	#97	1	2	yes	—
92	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₁₇	—	2	1	no	—
99	C₂×Dic₁₇	—	2	1	no	—
100	C₂×Dic₁₇	—	2	1	no	—
101	C₂×Dic₁₇	—	2	1	no	—
102	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
103	C₂×Dic₁₇	#97	1	2	yes	—
104	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
105	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
106	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
107	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
108	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
109	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
110	C₂×Dic₁₇	#131	1	2	yes	—
111	C₂×Dic₁₇	#131	1	2	yes	—
112	#3624	—	2	8	no	—
113	#3624	—	4	4	no	—
114	C₈×C₁₇	—	2	2	no	—
115	C₈×C₁₇	—	2	2	no	—
116	#3622	—	2	2	no	—
117	#3622	—	2	2	no	—
118	C₄×C₄⋊C₈	—	17	1	no	—
119	C₄×C₄⋊C₈	—	17	1	no	—
120	C₄²×C₈	—	17	1	no	—
121	C₂×#322	—	17	2	no	—
122	#1273	—	17	2	no	—
123	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
126	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
127	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
128	C₂²×C₁₇	#405	1	1	yes	characteristic
129	C₂²×C₁₇	#405	1	1	yes	characteristic
130	C₄×C₁₇	—	2	2	no	—
131	C₄×C₁₇	#322	1	4	yes	complemented
132	C₄×C₁₇	#423	1	1	yes	characteristic
133	C₄×C₁₇	#423	1	1	yes	characteristic
134	C₄×C₁₇	#432	1	1	yes	characteristic
135	C₄×C₁₇	#432	1	1	yes	characteristic
136	Dic₁₇	—	2	2	no	—
137	Dic₁₇	—	2	2	no	—
138	Dic₁₇	C₈×D₄	1	2	yes	—
139	Dic₁₇	#436	1	2	yes	—
140	Dic₁₇	C₈×Q₈	1	2	yes	—
141	Dic₁₇	#446	1	2	yes	—
142	C₂×C₄⋊C₈	—	17	2	no	—
143	C₂×C₄⋊C₈	—	17	2	no	—
144	C₂×C₄×C₈	—	17	1	no	—
145	C₂×C₄×C₈	—	17	2	no	—
146	C₂×C₄×C₈	—	17	2	no	—
147	C₂×C₄×C₈	—	17	1	no	—
148	C₂×C₄×C₈	—	17	1	no	—
149	C₂×C₄×C₈	—	17	1	no	—
150	C₂×C₄×C₈	—	17	1	no	—
151	C₂×C₄×C₈	#595	1	1	yes	characteristic
152	C₄³	—	17	1	no	—
153	#336	—	17	2	no	—
154	#336	—	17	2	no	—
155	#322	—	34	4	no	—
156	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
157	C₂×C₁₇	C₈×M₁₆	1	1	yes	characteristic
158	C₂×C₁₇	#1451	1	1	yes	characteristic
159	C₂×C₁₇	#1452	1	1	yes	characteristic
160	C₂×C₁₇	#1767	1	1	yes	characteristic
161	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₁₇	#1770	1	1	yes	characteristic
163	C₂²×C₈	—	17	4	no	—
164	C₂²×C₈	—	17	1	no	—
165	C₂²×C₈	—	17	1	no	—
166	C₂²×C₈	—	17	4	no	—
167	C₂²×C₈	C₂×#595	1	1	yes	characteristic
168	C₂²×C₈	C₂×#595	1	1	yes	characteristic
169	C₂×C₄²	—	17	1	no	—
170	C₂×C₄²	—	17	1	no	—
171	C₂×C₄²	—	17	1	no	—
172	C₂×C₄²	—	17	1	no	—
173	C₂×C₄²	—	17	1	no	—
174	C₂×C₄²	—	17	1	no	—
175	C₂×C₄²	C₂×#595	1	1	yes	characteristic
176	C₄⋊C₈	—	17	8	no	—
177	C₄⋊C₈	—	34	4	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	17	2	no	—
180	C₄×C₈	—	17	2	no	—
181	C₄×C₈	—	34	1	no	—
182	C₄×C₈	—	34	1	no	—
183	C₄×C₈	—	34	2	no	—
184	C₄×C₈	—	34	2	no	—
185	C₄×C₈	—	34	2	no	—
186	C₄×C₈	—	34	2	no	—
187	C₄×C₈	—	34	2	no	—
188	C₄×C₈	—	34	2	no	—
189	C₄×C₈	—	34	2	no	—
190	C₁₇	#8261	1	1	yes	characteristic, complemented, Sylow
191	C₂²×C₄	—	17	1	no	—
192	C₂²×C₄	—	17	1	no	—
193	C₂²×C₄	—	17	1	no	—
194	C₂²×C₄	—	17	1	no	—
195	C₂²×C₄	C₄×#595	1	1	yes	characteristic
196	C₂²×C₄	C₄×#595	1	1	yes	characteristic
197	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	17	8	no	—
203	C₂×C₈	—	17	2	no	—
204	C₂×C₈	—	17	2	no	—
205	C₂×C₈	—	34	4	no	—
206	C₂×C₈	—	34	4	no	—
207	C₂×C₈	—	34	1	no	—
208	C₂×C₈	—	34	1	no	—
209	C₂×C₈	—	34	1	no	—
210	C₂×C₈	—	34	1	no	—
211	C₂×C₈	—	34	4	no	—
212	C₂×C₈	—	34	4	no	—
213	C₂×C₈	—	34	4	no	—
214	C₂×C₈	#63243	1	2	yes	—
215	C₂×C₈	C₄⋊#595	1	2	yes	—
216	C₄²	—	17	1	no	—
217	C₄²	—	17	1	no	—
218	C₄²	—	17	1	no	—
219	C₄²	—	17	1	no	—
220	C₄²	—	17	4	no	—
221	C₄²	—	17	4	no	—
222	C₄²	—	34	2	no	—
223	C₄²	—	34	2	no	—
224	C₄²	—	34	1	no	—
225	C₄²	—	34	1	no	—
226	C₄²	#63243	1	2	yes	—
227	C₄²	#63247	1	2	yes	—
228	C₂³	C₂×C₄×#595	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	17	1	no	—
232	C₂×C₄	—	17	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	1	no	—
235	C₂×C₄	—	17	2	no	—
236	C₂×C₄	—	17	2	no	—
237	C₂×C₄	—	17	2	no	—
238	C₂×C₄	—	17	2	no	—
239	C₂×C₄	—	17	2	no	—
240	C₂×C₄	—	17	2	no	—
241	C₂×C₄	—	34	1	no	—
242	C₂×C₄	—	34	1	no	—
243	C₂×C₄	—	34	1	no	—
244	C₂×C₄	—	34	1	no	—
245	C₂×C₄	C₈×#595	1	2	yes	—
246	C₂×C₄	#10587532	1	2	yes	—
247	C₂×C₄	#10587542	1	2	yes	—
248	C₂×C₄	#10587542	1	2	yes	—
249	C₂×C₄	C₂×#63243	1	1	yes	characteristic
250	C₂×C₄	C₂×#63243	1	1	yes	characteristic
251	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
253	C₂×C₄	#10587677	1	1	yes	characteristic
254	C₂×C₄	C₂×#63247	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	34	8	no	—
258	C₈	—	34	2	no	—
259	C₈	—	34	2	no	—
260	C₈	—	68	4	no	—
261	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
262	C₂²	C₂×C₈×#595	1	1	yes	characteristic
263	C₂²	C₂×#10587532	1	1	yes	characteristic
264	C₂²	C₂×#10587542	1	1	yes	characteristic
265	C₂²	C₄×#63243	1	1	yes	characteristic
266	C₂²	#11763772	1	1	yes	characteristic
267	C₂²	#11763772	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	17	2	no	—
270	C₄	—	17	2	no	—
271	C₄	—	17	2	no	—
272	C₄	—	17	2	no	—
273	C₄	—	34	2	no	—
274	C₄	—	34	2	no	—
275	C₄	#11763651	1	4	yes	complemented
276	C₄	#11763812	1	1	yes	characteristic
277	C₄	#11763813	1	1	yes	characteristic
278	C₄	#11763830	1	1	yes	characteristic
279	C₄	#11763831	1	1	yes	characteristic
280	C₂	C₂×#11763651	1	1	yes	characteristic
281	C₂	C₈×#63243	1	1	yes	characteristic
282	C₂	#14543994	1	1	yes	characteristic
283	C₂	#14543997	1	1	yes	characteristic
284	C₂	#14544570	1	1	yes	characteristic
285	C₂	C₈×C₄⋊#595	1	1	yes	characteristic
286	C₂	#14544578	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8261
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
34. C₄⋊#3624	C₈	semidirect
131. C₄×C₁₇	#322	semidirect
190. C₁₇	#8261	semidirect
275. C₄	#11763651	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3624	2
C₂×#11763651	2
#14543606	2
C₄×C₈×Dic₁₇	2
#8261	17