Subgroups — #17952423

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587536	C₂	1	1	yes	maximal, characteristic
2	#14544334	C₂	1	1	yes	maximal, characteristic
3	#14543994	C₂	1	1	yes	maximal, characteristic
4	C₈×#63243	C₂	1	1	yes	maximal, characteristic
5	#14543616	C₂	1	1	yes	maximal, characteristic
6	#14543616	C₂	1	1	yes	maximal, characteristic
7	D₁₇×#322	C₂	1	1	yes	maximal, characteristic
8	#11763828	C₂²	1	1	yes	characteristic
9	#11763828	C₂²	1	1	yes	characteristic
10	C₂×#10587536	C₄	1	2	yes	—
11	C₄×#63243	C₂²	1	1	yes	characteristic
12	C₄×#63243	C₂²	1	1	yes	characteristic
13	#11763685	C₄	1	2	yes	—
14	#11763672	C₄	1	2	yes	—
15	#11763671	C₄	1	2	yes	—
16	#11763651	—	2	1	no	—
17	#11763651	—	2	1	no	—
18	#11763650	—	2	1	no	—
19	C₈×#3624	—	2	1	no	—
20	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	#11762531	C₄	1	1	yes	characteristic
24	#11762531	C₄	1	1	yes	characteristic
25	#11762530	C₄	1	1	yes	characteristic
26	C₁₇×#322	C₄	1	1	yes	characteristic
27	C₂×#63243	C₂×C₄	1	2	yes	—
28	C₂×#63243	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₁₇	—	2	1	no	—
30	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
33	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
34	C₄²×D₁₇	C₂³	1	1	yes	characteristic
35	#10587543	—	2	1	no	—
36	#10587543	—	2	1	no	—
37	#10587542	—	2	1	no	—
38	#10587542	—	2	1	no	—
39	C₄⋊#3624	C₂×C₄	1	2	yes	—
40	C₄⋊#3624	C₂×C₄	1	2	yes	—
41	C₄×#3624	—	2	1	no	—
42	C₄×#3624	—	2	1	no	—
43	C₄×#3624	—	2	1	no	—
44	C₄×#3624	—	2	1	no	—
45	#10587536	—	2	4	no	—
46	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₁₇	—	2	1	no	—
50	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
52	C₈×Dic₁₇	D₄	1	1	yes	characteristic
53	C₈×Dic₁₇	D₄	1	1	yes	characteristic
54	C₄×#3622	C₂×C₄	1	1	yes	characteristic
55	C₄×#3622	C₂×C₄	1	1	yes	characteristic
56	C₄×#3622	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
60	C₂×#3624	—	2	2	no	—
61	C₂×#3624	—	2	2	no	—
62	C₂×#3624	—	2	2	no	—
63	C₂×#3624	—	2	2	no	—
64	#63243	—	2	2	no	—
65	#63243	—	2	2	no	—
66	#63243	—	2	2	no	—
67	#63243	M₁₆	1	4	yes	—
68	C₂×C₈×C₁₇	—	2	1	no	—
69	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
76	C₄×Dic₁₇	#41	1	1	yes	characteristic
77	C₂×#3622	—	2	1	no	—
78	C₂×#3622	C₄²	1	1	yes	characteristic
79	C₂×#3622	C₄²	1	1	yes	characteristic
80	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
81	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
82	C₈×D₁₇	—	2	2	no	—
83	C₈×D₁₇	—	2	2	no	—
84	C₈×D₁₇	—	2	2	no	—
85	C₈×D₁₇	—	2	2	no	—
86	C₈×D₁₇	—	4	2	no	—
87	#8297	—	17	1	no	maximal, Sylow
88	C₂²×D₁₇	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
92	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
94	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
95	C₄×D₁₇	—	2	1	no	—
96	C₄×D₁₇	—	2	1	no	—
97	C₄×D₁₇	—	2	2	no	—
98	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
99	C₄×D₁₇	#131	1	2	yes	—
100	#3624	—	2	4	no	—
101	#3624	—	4	2	no	—
102	#3624	—	4	2	no	—
103	#3624	—	4	2	no	—
104	C₈×C₁₇	—	2	1	no	—
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	—	2	1	no	—
107	C₈×C₁₇	—	4	1	no	—
108	C₈×C₁₇	#108	1	2	yes	complemented
109	#3622	—	2	1	no	—
110	#3622	—	2	1	no	—
111	#3622	—	2	1	no	—
112	#3622	—	4	1	no	—
113	#3622	#108	1	2	yes	complemented
114	C₄×#108	—	17	1	no	—
115	#1564	—	17	1	no	—
116	#1451	—	17	1	no	—
117	C₈×M₁₆	—	17	1	no	—
118	C₂×#322	—	17	1	no	—
119	#1278	—	17	1	no	—
120	#1278	—	17	1	no	—
121	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₂×D₁₇	—	2	1	no	—
123	C₂×D₁₇	—	2	1	no	—
124	C₂×D₁₇	#446	1	2	yes	—
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
127	C₄×C₁₇	#405	1	1	yes	characteristic
128	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
129	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
130	Dic₁₇	—	2	1	no	—
131	Dic₁₇	#435	1	2	yes	—
132	Dic₁₇	#444	1	2	yes	—
133	#431	—	17	1	no	—
134	#431	—	17	1	no	—
135	C₂×#108	—	17	2	no	—
136	C₄×M₁₆	—	17	1	no	—
137	C₄×M₁₆	—	17	1	no	—
138	C₂×C₄×C₈	—	17	1	no	—
139	C₂×C₄×C₈	—	17	1	no	—
140	C₂×C₄×C₈	—	17	1	no	—
141	#341	—	17	2	no	—
142	#335	—	17	2	no	—
143	#334	—	17	2	no	—
144	#322	—	17	1	no	—
145	#322	—	17	1	no	—
146	#322	—	17	1	no	—
147	#322	—	34	1	no	—
148	#322	—	34	1	no	—
149	#322	—	34	1	no	—
150	#322	#595	1	1	yes	characteristic
151	C₈²	—	34	1	no	—
152	C₂×C₁₇	#1771	1	1	yes	characteristic
153	C₂×C₁₇	#1773	1	1	yes	characteristic
154	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
155	D₁₇	—	2	2	no	—
156	C₂×M₁₆	—	17	2	no	—
157	C₂×M₁₆	—	17	2	no	—
158	C₂²×C₈	—	17	1	no	—
159	C₂²×C₈	—	17	1	no	—
160	C₂²×C₈	—	17	1	no	—
161	C₂²×C₈	—	17	1	no	—
162	C₂²×C₈	—	34	1	no	—
163	C₂×C₄²	—	17	1	no	—
164	#108	—	34	4	no	—
165	C₄⋊C₈	—	17	2	no	—
166	C₄⋊C₈	—	17	2	no	—
167	C₂²⋊C₈	—	34	1	no	—
168	C₂²⋊C₈	—	34	1	no	—
169	#97	—	34	1	no	—
170	#97	—	34	1	no	—
171	C₄×C₈	—	17	1	no	—
172	C₄×C₈	—	17	1	no	—
173	C₄×C₈	—	17	1	no	—
174	C₄×C₈	—	17	1	no	—
175	C₄×C₈	—	17	1	no	—
176	C₄×C₈	—	17	1	no	—
177	C₄×C₈	—	17	1	no	—
178	C₄×C₈	—	34	1	no	—
179	C₄×C₈	—	34	1	no	—
180	C₄×C₈	—	34	1	no	—
181	C₄×C₈	—	34	1	no	—
182	C₄×C₈	—	34	1	no	—
183	C₄×C₈	C₂×#595	1	1	yes	characteristic
184	C₄×C₈	C₂×#595	1	1	yes	characteristic
185	C₄×C₈	C₂×#595	1	1	yes	characteristic
186	C₁₇	#8297	1	1	yes	characteristic, complemented, Sylow
187	C₂²×C₄	—	17	1	no	—
188	C₂²×C₄	—	17	1	no	—
189	C₂²×C₄	—	17	1	no	—
190	M₁₆	—	17	4	no	—
191	M₁₆	—	34	2	no	—
192	M₁₆	—	34	2	no	—
193	M₁₆	—	34	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	17	1	no	—
196	C₂×C₈	—	17	1	no	—
197	C₂×C₈	—	17	1	no	—
198	C₂×C₈	—	17	1	no	—
199	C₂×C₈	—	34	2	no	—
200	C₂×C₈	—	34	2	no	—
201	C₂×C₈	—	34	2	no	—
202	C₂×C₈	—	34	2	no	—
203	C₂×C₈	—	34	2	no	—
204	C₂×C₈	—	34	2	no	—
205	C₂×C₈	—	34	2	no	—
206	C₂×C₈	—	34	2	no	—
207	C₂×C₈	—	34	1	no	—
208	C₂×C₈	—	68	2	no	—
209	C₂×C₈	C₄×#595	1	1	yes	characteristic
210	C₂×C₈	C₄×#595	1	1	yes	characteristic
211	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
212	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
213	C₄²	—	17	1	no	—
214	C₄²	—	17	1	no	—
215	C₄²	—	17	1	no	—
216	C₄²	C₂²×#595	1	1	yes	characteristic
217	C₂³	—	17	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	2	no	—
221	C₂×C₄	—	17	2	no	—
222	C₂×C₄	—	17	1	no	—
223	C₂×C₄	—	34	2	no	—
224	C₂×C₄	—	34	1	no	—
225	C₂×C₄	—	34	1	no	—
226	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
228	C₂×C₄	#10587677	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	4	1	no	—
233	C₈	—	17	2	no	—
234	C₈	—	34	4	no	—
235	C₈	—	34	1	no	—
236	C₈	—	34	1	no	—
237	C₈	—	34	1	no	—
238	C₈	—	68	1	no	—
239	C₈	—	68	2	no	—
240	C₈	—	68	2	no	—
241	C₈	—	68	2	no	—
242	C₈	#10587536	1	2	yes	complemented
243	C₂²	—	17	2	no	—
244	C₂²	—	34	1	no	—
245	C₂²	—	34	1	no	—
246	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	17	2	no	—
249	C₄	—	17	2	no	—
250	C₄	—	34	1	no	—
251	C₄	M₁₆×#595	1	1	yes	characteristic
252	C₄	#11763775	1	1	yes	characteristic
253	C₄	D₁₇⋊#108	1	1	yes	characteristic
254	C₄	C₂×#10587536	1	1	yes	characteristic
255	C₂	—	34	2	no	—
256	C₂	#14544582	1	1	yes	characteristic
257	C₂	#14544590	1	1	yes	characteristic
258	C₂	C₄×#10587536	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	C₄×Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#8297
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

Complements

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

N	H ≅ G/N	type
108. C₈×C₁₇	#108	semidirect
113. #3622	#108	semidirect
186. C₁₇	#8297	semidirect
242. C₈	#10587536	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587536	2
#14544334	2
#14543994	2
C₈×#63243	2
#14543616	2
D₁₇×#322	2
#8297	17