Subgroups — #17952321

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544570	C₂	1	2	yes	maximal
2	C₂×#11763670	C₂	1	2	yes	maximal
3	#14543614	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3624	C₂²	1	4	yes	—
6	C₂×#10587542	—	2	2	no	—
7	#11763675	—	2	2	no	—
8	#11763670	C₄	1	8	yes	complemented
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3624	—	2	4	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
28	#10587542	—	4	4	no	—
29	C₄⋊#3624	—	2	4	no	—
30	C₄⋊#3624	C₂×C₄	1	8	yes	—
31	C₄×C₈×C₁₇	—	2	2	no	—
32	C₈×Dic₁₇	—	2	2	no	—
33	C₈×Dic₁₇	—	2	2	no	—
34	C₈×Dic₁₇	—	2	2	no	—
35	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
36	C₄×#3622	—	2	2	no	—
37	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×#3624	—	2	8	no	—
45	C₂×#3624	—	4	4	no	—
46	C₂×#3624	—	4	4	no	—
47	C₂×C₈×C₁₇	—	2	1	no	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	C₄²	1	2	yes	—
50	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
51	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
52	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
53	C₄²×C₁₇	—	2	1	no	—
54	C₄²×C₁₇	—	2	1	no	—
55	C₄×Dic₁₇	—	2	2	no	—
56	C₄×Dic₁₇	—	2	1	no	—
57	C₄×Dic₁₇	—	2	1	no	—
58	C₄×Dic₁₇	—	2	2	no	—
59	C₄×Dic₁₇	—	2	2	no	—
60	C₄×Dic₁₇	SD₁₆	1	4	yes	—
61	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
62	C₄×Dic₁₇	#41	1	2	yes	—
63	C₂×#3622	—	2	1	no	—
64	C₂×#3622	—	2	1	no	—
65	C₂×#3622	—	2	1	no	—
66	C₂×#3622	—	2	1	no	—
67	C₂×#3622	C₄²	1	2	yes	—
68	C₂×#3622	M₁₆	1	2	yes	—
69	#8273	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	#106	1	2	yes	—
76	C₂×C₄×C₁₇	#108	1	2	yes	—
77	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₁₇	—	2	1	no	—
84	C₂×Dic₁₇	—	2	1	no	—
85	C₂×Dic₁₇	—	2	1	no	—
86	C₂×Dic₁₇	—	2	1	no	—
87	C₂×Dic₁₇	—	2	1	no	—
88	C₂×Dic₁₇	—	2	1	no	—
89	C₂×Dic₁₇	#106	1	2	yes	—
90	C₂×Dic₁₇	#108	1	2	yes	—
91	C₂×Dic₁₇	C₄×D₄	1	2	yes	—
92	C₂×Dic₁₇	C₄×Q₈	1	2	yes	—
93	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
94	C₂×Dic₁₇	#135	1	2	yes	—
95	#3624	—	4	8	no	—
96	C₈×C₁₇	—	2	2	no	—
97	C₈×C₁₇	#97	1	4	yes	complemented
98	#3622	—	2	2	no	—
99	#3622	—	2	2	no	—
100	#1767	—	17	2	no	—
101	C₄²×C₈	—	17	1	no	—
102	C₂×#334	—	17	2	no	—
103	#1277	—	17	2	no	—
104	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₇	#423	1	1	yes	characteristic
106	C₂²×C₁₇	#423	1	1	yes	characteristic
107	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
108	C₂²×C₁₇	#427	1	1	yes	characteristic
109	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
110	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
111	C₄×C₁₇	—	2	2	no	—
112	C₄×C₁₇	—	2	2	no	—
113	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
114	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
115	C₄×C₁₇	#432	1	1	yes	characteristic
116	C₄×C₁₇	#432	1	1	yes	characteristic
117	Dic₁₇	—	2	2	no	—
118	Dic₁₇	—	2	2	no	—
119	Dic₁₇	—	2	2	no	—
120	Dic₁₇	C₄×SD₁₆	1	4	yes	—
121	C₂×C₄⋊C₈	—	17	4	no	—
122	C₂×#97	—	34	2	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₂×C₄×C₈	—	17	1	no	—
127	C₂×C₄×C₈	—	17	1	no	—
128	C₂×C₄×C₈	#595	1	1	yes	characteristic
129	C₄³	—	17	1	no	—
130	#336	—	34	2	no	—
131	#334	—	17	8	no	—
132	C₂×C₁₇	#1566	1	1	yes	characteristic
133	C₂×C₁₇	#1566	1	1	yes	characteristic
134	C₂×C₁₇	#1568	1	1	yes	characteristic
135	C₂×C₁₇	#1568	1	1	yes	characteristic
136	C₂×C₁₇	#1767	1	1	yes	characteristic
137	C₂×C₁₇	C₄×#106	1	1	yes	characteristic
138	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	34	4	no	—
142	C₂²×C₈	C₂×#595	1	1	yes	characteristic
143	C₂²×C₈	C₂×#595	1	1	yes	characteristic
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	17	1	no	—
146	C₂×C₄²	—	17	1	no	—
147	C₂×C₄²	—	17	1	no	—
148	C₂×C₄²	—	17	1	no	—
149	C₂×C₄²	—	17	1	no	—
150	C₂×C₄²	C₂×#595	1	1	yes	characteristic
151	C₄⋊C₈	—	17	8	no	—
152	C₄⋊C₈	—	34	4	no	—
153	#97	—	68	4	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	17	4	no	—
156	C₄×C₈	—	34	2	no	—
157	C₄×C₈	—	34	2	no	—
158	C₄×C₈	—	34	2	no	—
159	C₄×C₈	—	34	2	no	—
160	C₁₇	#8273	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	17	1	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
167	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	17	2	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	1	no	—
174	C₂×C₈	—	34	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	8	no	—
177	C₂×C₈	—	68	4	no	—
178	C₂×C₈	—	68	4	no	—
179	C₂×C₈	#63243	1	2	yes	—
180	C₂×C₈	#63243	1	2	yes	—
181	C₂×C₈	C₄×#595	1	2	yes	—
182	C₂×C₈	#63247	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	17	2	no	—
186	C₄²	—	17	2	no	—
187	C₄²	—	17	4	no	—
188	C₄²	—	34	2	no	—
189	C₄²	—	34	2	no	—
190	C₄²	—	34	1	no	—
191	C₄²	—	34	1	no	—
192	C₄²	—	34	2	no	—
193	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	17	2	no	—
200	C₂×C₄	—	17	2	no	—
201	C₂×C₄	—	17	2	no	—
202	C₂×C₄	—	17	2	no	—
203	C₂×C₄	—	17	2	no	—
204	C₂×C₄	—	34	1	no	—
205	C₂×C₄	—	34	1	no	—
206	C₂×C₄	—	34	1	no	—
207	C₂×C₄	—	34	1	no	—
208	C₂×C₄	—	34	1	no	—
209	C₂×C₄	—	34	1	no	—
210	C₂×C₄	#10587533	1	2	yes	—
211	C₂×C₄	#10587535	1	2	yes	—
212	C₂×C₄	C₂×#63243	1	1	yes	characteristic
213	C₂×C₄	C₂×#63243	1	1	yes	characteristic
214	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
216	C₂×C₄	#10587677	1	1	yes	characteristic
217	C₂×C₄	C₂×#63247	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	34	2	no	—
220	C₈	—	34	2	no	—
221	C₈	—	68	8	no	—
222	C₈	#10587542	1	4	yes	complemented
223	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
224	C₂²	#11763812	1	1	yes	characteristic
225	C₂²	#11763813	1	1	yes	characteristic
226	C₂²	C₂×#10587533	1	1	yes	characteristic
227	C₂²	#11763819	1	1	yes	characteristic
228	C₂²	C₂×#10587535	1	1	yes	characteristic
229	C₂²	D₁₇⋊#108	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	17	4	no	—
233	C₄	—	34	2	no	—
234	C₄	—	34	2	no	—
235	C₄	—	34	2	no	—
236	C₄	C₂×#10587542	1	1	yes	characteristic
237	C₄	C₄×#63243	1	1	yes	characteristic
238	C₄	#11763830	1	1	yes	characteristic
239	C₄	#11763831	1	1	yes	characteristic
240	C₂	#14544338	1	1	yes	characteristic
241	C₂	#14544339	1	1	yes	characteristic
242	C₂	#14544344	1	1	yes	characteristic
243	C₂	#14544345	1	1	yes	characteristic
244	C₂	#14544570	1	1	yes	characteristic
245	C₂	C₄×#10587533	1	1	yes	characteristic
246	C₂	C₄×#10587535	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8273
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #11763670	C₄	semidirect
97. C₈×C₁₇	#97	semidirect
160. C₁₇	#8273	semidirect
222. C₈	#10587542	semidirect

Maximal subgroups

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

subgroup	index
#14544570	2
C₂×#11763670	2
#14543614	2
C₄×C₈×Dic₁₇	2
#8273	17