Subgroups — #16944413

Order 4352.

378 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×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527543	C₂	1	1	yes	maximal, characteristic, complemented
3	#14526745	C₂	1	4	yes	maximal, complemented
4	C₁₇×#3169	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	—	2	1	no	—
6	#11762833	—	2	1	no	—
7	C₄×#63214	—	2	1	no	—
8	C₄×#63214	C₂²	1	2	yes	—
9	#11762562	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	C₁₇×#532	—	2	1	no	—
12	C₁₇×#500	C₄	1	1	yes	characteristic
13	C₁₇×#493	C₄	1	1	yes	characteristic
14	C₁₇×#486	—	2	1	no	—
15	C₁₇×#466	—	2	1	no	—
16	C₁₇×#466	C₄	1	2	yes	—
17	C₄×D₈×C₁₇	—	2	1	no	—
18	C₄×D₈×C₁₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×#41×C₁₇	—	4	1	no	—
21	C₂×D₈×C₁₇	—	2	1	no	—
22	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₁₇×C₄⋊Q₈	—	2	1	no	—
26	C₁₇×#127	—	2	1	no	—
27	C₁₇×#124	—	4	1	no	—
28	C₁₇×#121	—	2	2	no	—
29	C₁₇×#121	—	4	1	no	—
30	C₄×D₄×C₁₇	—	2	2	no	—
31	C₄×D₄×C₁₇	—	4	1	no	—
32	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
33	C₂×#63223	—	4	1	no	—
34	C₁₇×#107	—	2	1	no	—
35	C₁₇×#107	C₂×C₄	1	2	yes	—
36	C₁₇×#102	—	2	2	no	—
37	C₁₇×#102	—	2	2	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×#63214	—	2	4	no	—
43	C₂×#63214	—	2	2	no	—
44	C₄⋊#3622	—	2	2	no	—
45	C₄⋊#3622	—	4	1	no	—
46	#41×C₁₇	—	8	2	no	—
47	#41×C₁₇	—	8	2	no	—
48	C₂×Q₈×C₁₇	—	4	1	no	—
49	C₂×D₄×C₁₇	—	2	2	no	—
50	C₂×D₄×C₁₇	—	4	2	no	—
51	C₂×D₄×C₁₇	—	4	1	no	—
52	C₂×D₄×C₁₇	—	4	1	no	—
53	C₂²×C₄×C₁₇	—	4	2	no	—
54	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
57	D₈×C₁₇	—	2	4	no	—
58	D₈×C₁₇	—	4	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₇	—	2	2	no	—
66	C₄⋊C₄×C₁₇	—	4	1	no	—
67	C₂²⋊C₄×C₁₇	—	4	2	no	—
68	C₂²⋊C₄×C₁₇	—	4	2	no	—
69	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₁₇	D₈	1	1	yes	characteristic
71	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
72	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
73	#63223	—	8	2	no	—
74	C₂×#3622	—	4	2	no	—
75	#63214	—	2	4	no	—
76	#11671	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	—	4	2	no	—
78	C₂³×C₁₇	#102	1	1	yes	characteristic
79	Q₈×C₁₇	—	8	2	no	—
80	D₄×C₁₇	—	4	2	no	—
81	D₄×C₁₇	—	4	2	no	—
82	D₄×C₁₇	—	8	4	no	—
83	D₄×C₁₇	—	8	2	no	—
84	D₄×C₁₇	—	8	2	no	—
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	—	4	2	no	—
89	C₂×C₄×C₁₇	—	8	2	no	—
90	C₂×C₄×C₁₇	—	8	2	no	—
91	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
94	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
95	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
96	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	—	2	1	no	—
101	C₈×C₁₇	#104	1	2	yes	—
102	C₈×C₁₇	#131	1	2	yes	—
103	#3622	—	8	2	no	—
104	#3169	D₁₇	1	1	yes	characteristic
105	C₂×C₄×C₁₆	—	17	1	no	—
106	#1567	—	17	1	no	—
107	#1331	—	17	4	no	—
108	C₂²×C₁₇	—	2	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	—	8	2	no	—
111	C₂²×C₁₇	—	8	2	no	—
112	C₂²×C₁₇	—	8	2	no	—
113	C₂²×C₁₇	#357	1	2	yes	—
114	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	8	2	no	—
118	C₄×C₁₇	#357	1	2	yes	—
119	C₄×C₁₇	#359	1	2	yes	—
120	C₄×C₁₇	#408	1	1	yes	characteristic
121	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
122	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
123	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
124	#532	—	2	1	no	—
125	C₂²×C₁₆	—	34	1	no	—
126	#500	Dic₁₇	1	1	yes	characteristic
127	#493	Dic₁₇	1	1	yes	characteristic
128	#486	—	2	1	no	—
129	#466	—	2	1	no	—
130	#466	Dic₁₇	1	2	yes	—
131	C₄×D₈	—	2	1	no	—
132	C₄×D₈	C₂×D₁₇	1	2	yes	—
133	#423	—	34	1	no	—
134	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
135	C₄×C₁₆	—	17	2	no	—
136	C₄×C₁₆	—	34	1	no	—
137	#335	—	17	2	no	—
138	#335	—	34	1	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	—	16	2	no	—
141	C₂×C₁₇	#1492	1	1	yes	characteristic
142	C₂×C₁₇	C₂×#357	1	1	yes	characteristic
143	C₂×C₁₇	C₂×#359	1	1	yes	characteristic
144	C₂×#41	—	4	1	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
147	C₂×M₁₆	—	68	1	no	—
148	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	1	no	—
151	#127	—	2	1	no	—
152	#124	—	4	1	no	—
153	#121	—	2	2	no	—
154	#121	—	4	1	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	4	1	no	—
157	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₂×C₁₆	—	34	4	no	—
159	C₂×C₁₆	—	34	2	no	—
160	#107	—	2	1	no	—
161	#107	C₂×Dic₁₇	1	2	yes	—
162	C₄⋊C₈	—	34	2	no	—
163	C₄⋊C₈	—	68	1	no	—
164	#102	—	2	2	no	—
165	#102	—	2	2	no	—
166	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
169	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
170	C₁₇	#11671	1	1	yes	characteristic, complemented, Sylow
171	#41	—	8	2	no	—
172	#41	—	8	2	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	#63228	1	1	yes	characteristic
180	C₂²×C₄	#63229	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
182	D₈	—	2	4	no	—
183	D₈	—	4	2	no	—
184	M₁₆	—	136	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	68	2	no	—
188	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	#63228	1	1	yes	characteristic
197	C₄²	#63229	1	1	yes	characteristic
198	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
200	C₁₆	—	34	4	no	—
201	C₂³	—	4	2	no	—
202	C₂³	#10587503	1	1	yes	characteristic
203	Q₈	—	8	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	4	no	—
207	D₄	—	8	2	no	—
208	D₄	—	8	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	#10587497	1	2	yes	—
216	C₂×C₄	#10587498	1	2	yes	—
217	C₂×C₄	#10587503	1	1	yes	characteristic
218	C₂×C₄	#10587503	1	1	yes	characteristic
219	C₂×C₄	#10587503	1	1	yes	characteristic
220	C₂×C₄	C₂×#63228	1	1	yes	characteristic
221	C₂×C₄	C₂×#63229	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	136	2	no	—
226	C₈	#10587508	1	2	yes	—
227	C₈	#10587634	1	2	yes	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	#11762610	1	2	yes	—
234	C₂²	C₂×#10587503	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	8	2	no	—
238	C₄	#11762610	1	2	yes	—
239	C₄	#11762616	1	2	yes	—
240	C₄	#11762776	1	1	yes	characteristic
241	C₄	C₂×#10587508	1	1	yes	characteristic
242	C₄	C₂×#10587497	1	1	yes	characteristic
243	C₄	C₂×#10587498	1	1	yes	characteristic
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	#14527235	1	1	yes	characteristic
247	C₂	C₂×#11762610	1	1	yes	characteristic
248	C₂	C₂×#11762616	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting, prime radical	C₁₇×#3169	C₂
socle	C₂²×C₁₇	C₂×#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#3169	D₁₇
hypocenter	C₁₇	#11671
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×#63214	C₂	semidirect
2. #14527543	C₂	semidirect
3. #14526745	C₂	semidirect
170. C₁₇	#11671	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#63214	2
#14527543	2
#14526745	2
C₁₇×#3169	2
#11671	17