Subgroups — #16950270

Order 4352.

454 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#14530395	C₂	1	2	yes	maximal
3	#14526866	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₂²	1	4	yes	—
6	#11762630	—	2	4	no	—
7	C₂²⋊#63214	—	2	4	no	—
8	C₂²×M₁₆×C₁₇	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₇×#117	C₄	1	1	yes	characteristic
11	C₁₇×#405	—	2	4	no	—
12	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
13	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	—	2	2	no	—
18	C₂×M₁₆×C₁₇	—	2	2	no	—
19	C₂×M₁₆×C₁₇	C₈	1	4	yes	—
20	C₂²×C₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	D₄	1	2	yes	—
26	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#117	—	2	2	no	—
29	C₁₇×#117	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₁₇	—	2	1	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	1	no	—
48	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₁₇	—	2	8	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	—	2	1	no	—
56	C₂×C₈×C₁₇	—	2	1	no	—
57	C₂×C₈×C₁₇	—	2	4	no	—
58	C₂×C₈×C₁₇	—	2	4	no	—
59	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₇	#41	1	2	yes	—
68	C₄⋊C₄×C₁₇	—	2	4	no	—
69	C₂²⋊C₄×C₁₇	—	2	4	no	—
70	C₄²×C₁₇	—	2	1	no	—
71	C₄²×C₁₇	—	2	1	no	—
72	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₇	M₁₆	1	2	yes	—
74	#63214	—	4	4	no	—
75	#12789	—	17	1	no	maximal, Sylow
76	C₂³×C₁₇	—	2	1	no	—
77	C₂³×C₁₇	—	2	1	no	—
78	C₂³×C₁₇	—	4	2	no	—
79	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
80	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	2	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	—	4	2	no	—
90	C₂×C₄×C₁₇	—	4	2	no	—
91	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
98	C₈×C₁₇	—	2	2	no	—
99	C₈×C₁₇	—	2	2	no	—
100	C₈×C₁₇	C₂²⋊C₈	1	4	yes	—
101	C₈×C₁₇	#121	1	2	yes	—
102	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×#405	D₁₇	1	1	yes	characteristic
104	C₂×#364	—	17	2	no	—
105	#2112	—	17	2	no	—
106	#1376	—	17	2	no	—
107	C₂²×C₁₇	—	2	2	no	—
108	C₂²×C₁₇	—	2	2	no	—
109	C₂²×C₁₇	—	2	4	no	—
110	C₂²×C₁₇	—	4	2	no	—
111	C₂²×C₁₇	—	4	2	no	—
112	C₂²×C₁₇	#407	1	1	yes	characteristic
113	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₇	#432	1	1	yes	characteristic
115	C₄×C₁₇	—	2	2	no	—
116	C₄×C₁₇	—	4	2	no	—
117	C₄×C₁₇	—	4	2	no	—
118	C₄×C₁₇	#380	1	1	yes	characteristic
119	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₁₇	#423	1	1	yes	characteristic
121	C₄×C₁₇	#431	1	1	yes	characteristic
122	C₂²×M₁₆	Dic₁₇	1	1	yes	characteristic
123	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
124	C₂×#117	Dic₁₇	1	1	yes	characteristic
125	C₂×M₃₂	—	17	4	no	—
126	#405	—	2	4	no	—
127	C₂×#97	Dic₁₇	1	1	yes	characteristic
128	C₂×#97	Dic₁₇	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
131	#364	—	34	4	no	—
132	C₂²⋊C₁₆	—	34	4	no	—
133	C₂×C₁₇	—	2	1	no	—
134	C₂×C₁₇	—	2	1	no	—
135	C₂×C₁₇	—	4	2	no	—
136	C₂×C₁₇	#1797	1	1	yes	characteristic
137	C₂×C₁₇	#2153	1	2	yes	—
138	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	#3622	1	4	yes	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	Dic₃₄	1	2	yes	—
145	C₂²×C₈	D₆₈	1	2	yes	—
146	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
150	#117	—	2	2	no	—
151	#117	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₂²⋊C₄	—	2	1	no	—
154	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
156	M₃₂	—	34	4	no	—
157	M₃₂	—	34	4	no	—
158	C₂×C₁₆	—	34	4	no	—
159	#97	—	2	2	no	—
160	#97	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	C₂²⋊D₁₇	1	4	yes	—
163	C₁₇	#12789	1	1	yes	characteristic, complemented, Sylow
164	C₂⁴	#63223	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
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	C₂×#3622	1	2	yes	—
172	C₂²×C₄	#63223	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	C₂×#3622	1	2	yes	—
183	C₂×C₈	C₂×#3622	1	2	yes	—
184	C₂×C₈	#63223	1	2	yes	—
185	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
186	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
187	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
188	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
189	C₂×C₈	#63252	1	2	yes	—
190	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	#63223	1	2	yes	—
196	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
197	C₁₆	—	68	4	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	4	2	no	—
201	C₂³	C₄⋊#3622	1	2	yes	—
202	C₂³	C₂×#63223	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₄⋊#3622	1	2	yes	—
214	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
215	C₂×C₄	C₂×#63223	1	1	yes	characteristic
216	C₂×C₄	C₂×#63223	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
218	C₂×C₄	#10587612	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	C₂²⋊#3622	1	4	yes	—
223	C₈	#10587610	1	2	yes	—
224	C₈	#10587616	1	2	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	#11762773	1	1	yes	characteristic
231	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
232	C₂²	#11762866	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	#11762669	1	1	yes	characteristic
237	C₄	C₂×C₂²⋊#3622	1	1	yes	characteristic
238	C₄	#11762833	1	1	yes	characteristic
239	C₄	#11762862	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	4	2	no	—
243	C₂	#14528764	1	1	yes	characteristic
244	C₂	#14530550	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#405	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#405	D₁₇
hypocenter	C₁₇	#12789
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. C₂×#11762630	C₂	semidirect
3. #14526866	C₂	semidirect
163. C₁₇	#12789	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
#14530395	2
#14526866	2
C₂×C₁₇×#405	2
#12789	17