Subgroups — #17960967

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₂×#11763696	C₂	1	2	yes	maximal
3	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
4	#14543821	C₂	1	2	yes	maximal
5	D₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	C₂²	1	2	yes	—
7	C₄×#63242	C₄	1	1	yes	characteristic
8	C₂×C₈×#595	C₄	1	1	yes	characteristic
9	C₂×C₈×#595	C₄	1	1	yes	characteristic
10	C₂×C₈×#595	C₂²	1	1	yes	characteristic
11	C₂×C₈×#595	C₂²	1	1	yes	characteristic
12	C₄²×#595	C₄	1	1	yes	characteristic
13	#11763730	C₄	1	2	yes	complemented
14	C₄⋊#63216	C₄	1	2	yes	complemented
15	#11763696	—	2	4	no	—
16	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
17	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₄⋊#63214	C₄	1	1	yes	characteristic, complemented
19	#11762627	C₄	1	2	yes	complemented
20	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×#595	C₈	1	2	yes	—
22	C₂×C₄×#595	C₈	1	2	yes	—
23	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
25	C₂×#63242	C₂×C₄	1	1	yes	characteristic
26	C₂×#63242	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
29	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×#3624	C₈	1	2	yes	—
31	C₄×#3624	C₈	1	2	yes	—
32	C₂×#63216	C₂×C₄	1	4	yes	—
33	C₈×#595	—	2	1	no	—
34	C₈×#595	—	2	1	no	—
35	C₈×#595	—	2	2	no	—
36	C₈×#595	—	2	2	no	—
37	C₈×#595	D₄	1	2	yes	—
38	C₈×#595	Q₈	1	2	yes	—
39	#10587529	—	2	2	no	—
40	#10587529	—	2	2	no	—
41	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×#63214	C₂×C₄	1	2	yes	—
46	C₄×#3622	C₂×C₄	1	1	yes	characteristic
47	C₁₆×D₁₇	—	2	2	no	—
48	C₁₆×D₁₇	—	2	2	no	—
49	C₂²×#595	C₂×C₈	1	2	yes	—
50	C₂²×#595	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×#3624	C₂×C₈	1	2	yes	—
55	C₂×#3624	C₂×C₈	1	2	yes	—
56	C₄×#595	—	2	4	no	—
57	C₄×#595	—	2	4	no	—
58	C₄×#595	—	2	1	no	—
59	C₄×#595	—	2	1	no	—
60	C₄×#595	C₄⋊C₄	1	2	yes	—
61	C₄×#595	M₁₆	1	2	yes	—
62	#63242	—	2	1	no	—
63	#63242	—	2	1	no	—
64	#63242	C₄⋊C₄	1	2	yes	—
65	#63242	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₄²	1	1	yes	characteristic
69	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
70	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
71	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
72	C₂×#3622	C₄²	1	1	yes	characteristic
73	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
74	C₈×D₁₇	—	2	1	no	—
75	C₈×D₁₇	—	2	1	no	—
76	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
77	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
78	C₈×D₁₇	#41	1	1	yes	characteristic
79	C₈×D₁₇	#41	1	1	yes	characteristic
80	#63216	—	2	4	no	—
81	C₁₆×C₁₇	—	2	2	no	—
82	#63214	—	2	2	no	—
83	#12653	—	17	1	no	maximal, Sylow
84	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×#595	—	2	2	no	—
86	C₂×#595	—	2	2	no	—
87	C₂×#595	—	2	2	no	—
88	C₂×#595	—	2	2	no	—
89	C₂×#595	C₄⋊C₈	1	4	yes	—
90	C₂×#595	M₃₂	1	4	yes	—
91	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
95	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
96	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
97	C₄×D₁₇	—	2	2	no	—
98	C₄×D₁₇	M₃₂	1	4	yes	—
99	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₁₇	#117	1	1	yes	characteristic
101	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
102	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
103	#3624	—	2	2	no	—
104	#3624	C₄⋊C₈	1	4	yes	—
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
107	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
108	#3622	—	2	1	no	—
109	#3622	C₄×D₄	1	1	yes	characteristic
110	#3622	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄⋊C₁₆	—	17	1	no	—
112	C₂×C₄⋊C₁₆	—	17	1	no	—
113	C₂×#346	—	17	2	no	—
114	C₄²×C₈	—	17	1	no	—
115	#1374	—	17	2	no	—
116	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂×D₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂×D₁₇	#431	1	1	yes	characteristic
119	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
120	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
121	#595	—	2	4	no	—
122	#595	—	2	4	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	#346	1	2	yes	complemented
125	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
127	Dic₁₇	—	2	1	no	—
128	Dic₁₇	#346	1	2	yes	complemented
129	Dic₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
130	Dic₁₇	#431	1	1	yes	characteristic
131	C₂²×C₁₆	—	17	2	no	—
132	C₂²×C₁₆	—	17	2	no	—
133	C₂×C₄×C₈	—	17	1	no	—
134	C₂×C₄×C₈	—	17	1	no	—
135	C₂×C₄×C₈	—	17	1	no	—
136	C₂×C₄×C₈	—	17	1	no	—
137	C₂×C₄×C₈	—	17	1	no	—
138	C₂×C₄×C₈	—	17	1	no	—
139	C₄³	—	17	1	no	—
140	C₄⋊C₁₆	—	17	2	no	—
141	C₄⋊C₁₆	—	17	2	no	—
142	C₄⋊C₁₆	—	17	2	no	—
143	C₄⋊C₁₆	—	17	1	no	—
144	C₄⋊C₁₆	#595	1	1	yes	characteristic, complemented
145	#346	—	34	4	no	—
146	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
147	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
148	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
149	D₁₇	#2150	1	2	yes	—
150	D₁₇	#2163	1	2	yes	—
151	C₂²×C₈	—	17	1	no	—
152	C₂²×C₈	—	17	1	no	—
153	C₂²×C₈	—	17	1	no	—
154	C₂²×C₈	—	17	1	no	—
155	C₂×C₄²	—	17	1	no	—
156	C₂×C₄²	—	17	1	no	—
157	C₂×C₄²	—	17	2	no	—
158	C₂×C₄²	—	17	2	no	—
159	C₂×C₄²	—	17	1	no	—
160	C₂×C₁₆	—	17	4	no	—
161	C₂×C₁₆	—	17	2	no	—
162	C₂×C₁₆	—	34	2	no	—
163	C₂×C₁₆	—	34	2	no	—
164	C₂×C₁₆	—	34	2	no	—
165	C₂×C₁₆	—	34	2	no	—
166	C₂×C₁₆	C₂×#595	1	2	yes	—
167	C₄×C₈	—	17	2	no	—
168	C₄×C₈	—	17	2	no	—
169	C₄×C₈	—	17	2	no	—
170	C₄×C₈	—	17	2	no	—
171	C₄×C₈	—	17	1	no	—
172	C₄×C₈	—	17	1	no	—
173	C₄×C₈	—	17	1	no	—
174	C₄×C₈	—	34	2	no	—
175	C₄×C₈	—	34	2	no	—
176	C₄×C₈	—	34	1	no	—
177	C₄×C₈	—	34	1	no	—
178	C₄×C₈	C₂×#595	1	1	yes	characteristic
179	C₁₇	#12653	1	1	yes	characteristic, complemented, Sylow
180	C₂²×C₄	—	17	1	no	—
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	2	no	—
183	C₂²×C₄	—	17	2	no	—
184	C₂²×C₄	—	17	1	no	—
185	C₂×C₈	—	17	2	no	—
186	C₂×C₈	—	17	2	no	—
187	C₂×C₈	—	17	2	no	—
188	C₂×C₈	—	17	1	no	—
189	C₂×C₈	—	17	1	no	—
190	C₂×C₈	—	17	1	no	—
191	C₂×C₈	—	17	1	no	—
192	C₂×C₈	—	17	1	no	—
193	C₂×C₈	—	17	2	no	—
194	C₂×C₈	—	17	1	no	—
195	C₂×C₈	—	34	1	no	—
196	C₂×C₈	—	34	1	no	—
197	C₂×C₈	—	34	1	no	—
198	C₂×C₈	—	34	1	no	—
199	C₂×C₈	C₄×#595	1	1	yes	characteristic
200	C₂×C₈	C₂²×#595	1	1	yes	characteristic
201	C₄²	—	17	2	no	—
202	C₄²	—	17	2	no	—
203	C₄²	—	17	1	no	—
204	C₄²	—	17	1	no	—
205	C₄²	—	17	1	no	—
206	C₄²	—	34	1	no	—
207	C₄²	—	34	1	no	—
208	C₄²	—	34	4	no	—
209	C₄²	—	34	4	no	—
210	C₄²	C₄×#595	1	1	yes	characteristic
211	C₁₆	—	2	2	no	—
212	C₁₆	—	34	4	no	—
213	C₁₆	—	34	2	no	—
214	C₂³	—	17	1	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	17	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	1	no	—
221	C₂×C₄	—	17	4	no	—
222	C₂×C₄	—	17	4	no	—
223	C₂×C₄	—	17	4	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	34	2	no	—
226	C₂×C₄	—	34	2	no	—
227	C₂×C₄	—	34	2	no	—
228	C₂×C₄	—	34	2	no	—
229	C₂×C₄	—	34	2	no	—
230	C₂×C₄	C₈×#595	1	1	yes	characteristic
231	C₂×C₄	C₈×#595	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	17	4	no	—
235	C₈	—	17	1	no	—
236	C₈	—	17	1	no	—
237	C₈	—	34	2	no	—
238	C₈	—	34	1	no	—
239	C₈	D₄×#595	1	1	yes	characteristic
240	C₈	Q₈×#595	1	1	yes	characteristic
241	C₂²	—	17	2	no	—
242	C₂²	—	17	2	no	—
243	C₂²	—	17	1	no	—
244	C₂²	—	17	1	no	—
245	C₂²	C₂×C₈×#595	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	17	2	no	—
248	C₄	—	17	1	no	—
249	C₄	—	17	1	no	—
250	C₄	—	34	4	no	—
251	C₄	—	34	4	no	—
252	C₄	—	34	1	no	—
253	C₄	#11763696	1	2	yes	complemented
254	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
255	C₄	M₁₆×#595	1	1	yes	characteristic
256	C₂	—	17	2	no	—
257	C₂	—	17	2	no	—
258	C₂	C₄⋊C₈×#595	1	1	yes	characteristic
259	C₂	C₂×#11763696	1	1	yes	characteristic
260	C₂	M₃₂×#595	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×C₄⋊C₁₆	C₄
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#12653
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. #11763730	C₄	semidirect
14. C₄⋊#63216	C₄	semidirect
18. C₄⋊#63214	C₄	semidirect
19. #11762627	C₄	semidirect
20. C₁₇×C₄⋊C₁₆	C₄	semidirect
124. C₄×C₁₇	#346	semidirect
128. Dic₁₇	#346	semidirect
144. C₄⋊C₁₆	#595	semidirect
179. C₁₇	#12653	semidirect
253. C₄	#11763696	semidirect

Maximal subgroups

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

subgroup	index
#14545114	2
C₂×#11763696	2
C₄×C₈×#595	2
#14543821	2
D₁₇×C₄⋊C₁₆	2
#12653	17