Subgroups — #16950278

Order 4352.

482 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762630	C₂	1	2	yes	maximal, complemented
2	C₂×#11762591	C₂	1	2	yes	maximal
3	#14526864	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₂²	1	4	yes	—
6	#11762630	—	2	4	no	—
7	#11762591	—	2	4	no	—
8	C₂²×M₁₆×C₁₇	C₄	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₁₇	—	2	4	no	—
12	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
13	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₇	—	2	1	no	—
17	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	—	2	2	no	—
19	C₂×M₁₆×C₁₇	—	2	2	no	—
20	C₂×M₁₆×C₁₇	—	2	2	no	—
21	C₂×M₁₆×C₁₇	—	2	2	no	—
22	C₂×M₁₆×C₁₇	C₈	1	4	yes	—
23	C₂×M₁₆×C₁₇	D₄	1	2	yes	—
24	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
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×C₄²×C₁₇	—	2	1	no	—
30	C₂×C₄²×C₁₇	—	2	2	no	—
31	C₂×C₄²×C₁₇	—	2	2	no	—
32	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₁₇×#97	—	2	2	no	—
35	C₁₇×#97	—	2	2	no	—
36	C₄×C₈×C₁₇	—	2	2	no	—
37	C₄×C₈×C₁₇	D₄	1	4	yes	—
38	#10587483	—	2	4	no	—
39	#10587483	—	2	4	no	—
40	C₂×#63214	—	2	4	no	—
41	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	—	2	1	no	—
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	2	1	no	—
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	—	2	4	no	—
48	C₂²×C₄×C₁₇	—	2	1	no	—
49	C₂²×C₄×C₁₇	—	4	4	no	—
50	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₁₇	—	2	8	no	—
55	M₁₆×C₁₇	—	2	4	no	—
56	M₁₆×C₁₇	—	2	4	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	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₁₇	#41	1	2	yes	—
70	C₄²×C₁₇	—	2	4	no	—
71	C₄²×C₁₇	—	2	1	no	—
72	C₄²×C₁₇	—	2	1	no	—
73	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₇	M₁₆	1	2	yes	—
75	#63214	—	4	4	no	—
76	#12791	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	—	2	1	no	—
78	C₂³×C₁₇	—	2	1	no	—
79	C₂³×C₁₇	—	4	2	no	—
80	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
81	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	—	2	4	no	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	—	4	8	no	—
92	C₂×C₄×C₁₇	—	4	2	no	—
93	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
100	C₈×C₁₇	—	2	2	no	—
101	C₈×C₁₇	—	2	2	no	—
102	C₈×C₁₇	C₂²⋊C₈	1	4	yes	—
103	C₈×C₁₇	#121	1	2	yes	—
104	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×M₁₆	D₁₇	1	1	yes	characteristic
106	C₂×#364	—	17	2	no	—
107	C₂×#349	—	17	2	no	—
108	#1375	—	17	2	no	—
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	—	2	2	no	—
111	C₂²×C₁₇	—	4	2	no	—
112	C₂²×C₁₇	—	4	2	no	—
113	C₂²×C₁₇	—	4	2	no	—
114	C₂²×C₁₇	#407	1	1	yes	characteristic
115	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₁₇	#432	1	1	yes	characteristic
117	C₄×C₁₇	—	2	4	no	—
118	C₄×C₁₇	—	2	2	no	—
119	C₄×C₁₇	—	4	2	no	—
120	C₄×C₁₇	#380	1	1	yes	characteristic
121	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₁₇	#423	1	1	yes	characteristic
123	C₄×C₁₇	#431	1	1	yes	characteristic
124	C₂²×M₁₆	Dic₁₇	1	1	yes	characteristic
125	C₂²×M₁₆	C₂×D₁₇	1	1	yes	characteristic
126	C₂²×C₄²	Dic₁₇	1	1	yes	characteristic
127	C₂×M₃₂	—	17	4	no	—
128	C₄×M₁₆	—	2	4	no	—
129	C₂×#97	Dic₁₇	1	1	yes	characteristic
130	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
131	C₂×C₄×C₈	Dic₁₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
133	#364	—	34	4	no	—
134	#349	—	34	4	no	—
135	C₂×C₁₇	—	2	1	no	—
136	C₂×C₁₇	—	2	1	no	—
137	C₂×C₁₇	—	4	2	no	—
138	C₂×C₁₇	#1797	1	1	yes	characteristic
139	C₂×C₁₇	#2165	1	2	yes	—
140	C₂³×C₄	—	2	1	no	—
141	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	#3622	1	4	yes	—
147	C₂×M₁₆	Dic₃₄	1	2	yes	—
148	C₂×M₁₆	D₆₈	1	2	yes	—
149	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
158	M₃₂	—	34	4	no	—
159	M₃₂	—	34	4	no	—
160	C₂×C₁₆	—	34	4	no	—
161	#97	—	2	2	no	—
162	#97	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	C₂²⋊D₁₇	1	4	yes	—
165	C₁₇	#12791	1	1	yes	characteristic, complemented, Sylow
166	C₂⁴	#63223	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	C₂×#3622	1	2	yes	—
176	C₂²×C₄	#63223	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
178	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
179	M₁₆	—	2	8	no	—
180	M₁₆	—	2	4	no	—
181	M₁₆	—	2	4	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	C₂×#3622	1	2	yes	—
187	C₂×C₈	C₂×#3622	1	2	yes	—
188	C₂×C₈	#63223	1	2	yes	—
189	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
190	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
191	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
192	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
193	C₂×C₈	#63252	1	2	yes	—
194	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	#63223	1	2	yes	—
199	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
200	C₁₆	—	68	4	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	4	2	no	—
204	C₂³	C₄⋊#3622	1	2	yes	—
205	C₂³	C₂×#63223	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₄⋊#3622	1	2	yes	—
218	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
219	C₂×C₄	C₂×#63223	1	1	yes	characteristic
220	C₂×C₄	C₂×#63223	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
222	C₂×C₄	#10587612	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	C₂²⋊#3622	1	4	yes	—
227	C₈	#10587610	1	2	yes	—
228	C₈	#10587616	1	2	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	#11762773	1	1	yes	characteristic
235	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
236	C₂²	#11762866	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	#11762669	1	1	yes	characteristic
241	C₄	C₂×C₂²⋊#3622	1	1	yes	characteristic
242	C₄	#11762833	1	1	yes	characteristic
243	C₄	#11762862	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	4	2	no	—
247	C₂	#14528764	1	1	yes	characteristic
248	C₂	#14530596	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10587612
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	#432
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₇
hypocenter	C₁₇	#12791
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11762630	C₂	semidirect
3. #14526864	C₂	semidirect
165. C₁₇	#12791	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762630	2
C₂×#11762591	2
#14526864	2
C₂×C₄×M₁₆×C₁₇	2
#12791	17