Subgroups — #16949439

Order 4352.

420 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	#14530395	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762586	C₂	1	2	yes	maximal, complemented
3	#14528560	C₂	1	1	yes	maximal, characteristic
4	#14526863	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×C₂²⋊#3622	C₄	1	1	yes	characteristic
8	C₂×#10587486	C₂²	1	2	yes	—
9	C₄×C₂²⋊Dic₁₇	C₄	1	1	yes	characteristic
10	C₂²⋊#63214	C₄	1	2	yes	complemented
11	C₂²⋊#63214	C₄	1	2	yes	complemented
12	#11762586	—	2	4	no	—
13	#11762563	C₄	1	2	yes	—
14	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×M₃₂	—	2	4	no	—
16	C₂×C₁₇×M₃₂	C₄	1	2	yes	complemented
17	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₁₇	C₄	1	2	yes	complemented
20	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	—	2	2	no	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₇	—	2	1	no	—
30	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
31	C₂²×#3622	C₂×C₄	1	2	yes	—
32	C₁₇×M₃₂	—	2	4	no	—
33	C₁₇×M₃₂	—	2	2	no	—
34	C₁₇×M₃₂	D₄	1	4	yes	—
35	C₂×C₁₆×C₁₇	—	2	4	no	—
36	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
38	#10587502	—	2	1	no	—
39	C₂²⋊#3622	—	2	1	no	—
40	C₂²⋊#3622	—	2	1	no	—
41	#10587486	—	2	2	no	—
42	#10587486	—	2	2	no	—
43	#10587483	—	2	2	no	—
44	#10587483	—	2	2	no	—
45	C₂×#63214	C₂×C₄	1	4	yes	—
46	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	—	2	2	no	—
48	C₂²×C₄×C₁₇	—	2	2	no	—
49	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	—	2	1	no	—
58	C₂×C₈×C₁₇	—	2	1	no	—
59	C₂×C₈×C₁₇	—	2	1	no	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₇	#41	1	2	yes	—
69	C₂²⋊Dic₁₇	—	4	2	no	—
70	C₄×Dic₁₇	—	2	2	no	—
71	C₄×Dic₁₇	—	2	2	no	—
72	C₂×#3622	—	2	2	no	—
73	C₂×#3622	—	2	2	no	—
74	C₂×#3622	—	2	2	no	—
75	C₁₆×C₁₇	—	2	2	no	—
76	C₁₆×C₁₇	C₂²⋊C₄	1	4	yes	complemented
77	#63214	—	2	4	no	—
78	#12616	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	—	2	2	no	—
80	C₂³×C₁₇	—	2	2	no	—
81	C₂³×C₁₇	—	4	1	no	—
82	C₂³×C₁₇	#97	1	1	yes	characteristic
83	C₂³×C₁₇	#97	1	1	yes	characteristic
84	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	—	2	2	no	—
92	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₇	#131	1	2	yes	—
98	C₂×Dic₁₇	—	4	2	no	—
99	C₂×Dic₁₇	—	4	2	no	—
100	C₂×Dic₁₇	—	4	2	no	—
101	C₈×C₁₇	—	2	1	no	—
102	C₈×C₁₇	—	2	1	no	—
103	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₈×C₁₇	#117	1	1	yes	characteristic
105	C₈×C₁₇	C₄×D₄	1	2	yes	—
106	#3622	—	4	2	no	—
107	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
108	#2112	—	17	1	no	—
109	C₂×#347	—	17	2	no	—
110	#1752	—	17	1	no	—
111	#1375	—	17	2	no	—
112	C₂²×C₁₇	—	2	1	no	—
113	C₂²×C₁₇	—	2	1	no	—
114	C₂²×C₁₇	—	2	1	no	—
115	C₂²×C₁₇	—	2	1	no	—
116	C₂²×C₁₇	—	2	1	no	—
117	C₂²×C₁₇	—	2	1	no	—
118	C₂²×C₁₇	—	2	1	no	—
119	C₂²×C₁₇	—	4	4	no	—
120	C₂²×C₁₇	#347	1	2	yes	complemented
121	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
122	C₂²×C₁₇	#435	1	2	yes	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₁₇	#405	1	1	yes	characteristic
127	C₄×C₁₇	#435	1	2	yes	—
128	Dic₁₇	—	8	2	no	—
129	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
130	C₂×M₃₂	—	2	4	no	—
131	C₂×M₃₂	—	17	2	no	—
132	C₂×M₃₂	Dic₁₇	1	2	yes	complemented
133	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
134	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
135	C₂²×C₁₆	Dic₁₇	1	2	yes	complemented
136	C₂×C₂²⋊C₈	—	17	1	no	—
137	C₂×#97	—	17	2	no	—
138	C₄×C₂²⋊C₄	—	17	1	no	—
139	C₂²⋊C₁₆	—	17	2	no	—
140	C₂²⋊C₁₆	—	17	2	no	—
141	#347	—	34	4	no	—
142	#336	—	17	2	no	—
143	C₂×C₁₇	—	2	2	no	—
144	C₂×C₁₇	—	2	2	no	—
145	C₂×C₁₇	—	4	1	no	—
146	C₂×C₁₇	#1752	1	1	yes	characteristic
147	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
148	C₂×C₁₇	#2110	1	1	yes	characteristic
149	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	17	2	no	—
153	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
154	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
155	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
156	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	—	34	1	no	—
160	C₂×C₄²	—	17	2	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	2	4	no	—
163	M₃₂	—	34	2	no	—
164	M₃₂	—	34	2	no	—
165	M₃₂	C₂²⋊D₁₇	1	4	yes	—
166	C₂×C₁₆	—	2	4	no	—
167	C₂×C₁₆	—	17	4	no	—
168	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
169	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
170	C₂²⋊C₈	—	34	1	no	—
171	C₂²⋊C₈	—	34	1	no	—
172	#97	—	34	2	no	—
173	#97	—	34	2	no	—
174	#95	—	34	1	no	—
175	C₁₇	#12616	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	#63218	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	34	2	no	—
180	C₂²×C₄	#63218	1	1	yes	characteristic
181	C₂²×C₄	#63218	1	1	yes	characteristic
182	C₂²×C₄	#63218	1	1	yes	characteristic
183	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
184	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	34	2	no	—
194	C₂×C₈	—	34	2	no	—
195	C₂×C₈	—	34	2	no	—
196	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
197	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
199	C₂×C₈	#63252	1	2	yes	—
200	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
202	C₂²⋊C₄	—	68	2	no	—
203	C₄²	—	34	2	no	—
204	C₄²	—	34	2	no	—
205	C₁₆	—	2	2	no	—
206	C₁₆	—	34	4	no	—
207	C₁₆	C₂²⋊Dic₁₇	1	4	yes	complemented
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	#10587486	1	1	yes	characteristic
212	C₂³	#10587486	1	1	yes	characteristic
213	C₂³	C₂×#63218	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	68	2	no	—
222	C₂×C₄	—	68	2	no	—
223	C₂×C₄	—	68	2	no	—
224	C₂×C₄	#10587486	1	1	yes	characteristic
225	C₂×C₄	#10587486	1	1	yes	characteristic
226	C₂×C₄	C₂×#63218	1	1	yes	characteristic
227	C₂×C₄	#10587587	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
229	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	68	2	no	—
233	C₈	#10587612	1	1	yes	characteristic
234	C₈	C₄×C₂²⋊D₁₇	1	2	yes	—
235	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#11762586	1	2	yes	complemented
245	C₂²	C₂×#10587486	1	1	yes	characteristic
246	C₂²	#11762880	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	136	2	no	—
250	C₄	C₄×C₂²⋊Dic₁₇	1	1	yes	characteristic
251	C₄	#11762764	1	1	yes	characteristic
252	C₄	#11762880	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#14528560	1	1	yes	characteristic
257	C₂	C₂×#11762586	1	1	yes	characteristic
258	C₂	#14530388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	#12616
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11762586	C₂	semidirect
4. #14526863	C₂	semidirect
10. C₂²⋊#63214	C₄	semidirect
11. C₂²⋊#63214	C₄	semidirect
16. C₂×C₁₇×M₃₂	C₄	semidirect
19. C₂²×C₁₆×C₁₇	C₄	semidirect
76. C₁₆×C₁₇	C₂²⋊C₄	semidirect
120. C₂²×C₁₇	#347	semidirect
132. C₂×M₃₂	Dic₁₇	semidirect
135. C₂²×C₁₆	Dic₁₇	semidirect
175. C₁₇	#12616	semidirect
207. C₁₆	C₂²⋊Dic₁₇	semidirect
244. C₂²	#11762586	semidirect

Maximal subgroups

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

subgroup	index
#14530395	2
C₂×#11762586	2
#14528560	2
#14526863	2
C₂²×C₁₇×M₃₂	2
#12616	17