Subgroups — #102×Dic₁₇ #14528601

Order 2176.

396 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587503	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762572	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762571	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#102	C₂	1	1	yes	maximal, characteristic
8	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₇	—	2	1	no	—
10	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₁₇	—	2	1	no	—
13	D₄×Dic₁₇	—	2	2	no	—
14	D₄×Dic₁₇	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	—	2	1	no	—
18	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂²×#3622	C₂²	1	1	yes	characteristic
20	C₁₇×#102	C₄	1	4	yes	complemented
21	#10587503	C₄	1	4	yes	complemented
22	#10587502	—	2	1	no	—
23	C₈×Dic₁₇	—	2	2	no	—
24	C₂⁴×C₁₇	—	2	1	no	—
25	C₂×D₄×C₁₇	—	2	1	no	—
26	C₂×D₄×C₁₇	—	2	1	no	—
27	C₂×D₄×C₁₇	—	2	1	no	—
28	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₇	—	2	1	no	—
32	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₇	—	2	1	no	—
34	C₂²×Dic₁₇	—	2	1	no	—
35	C₂²×Dic₁₇	—	2	4	no	—
36	C₂²×Dic₁₇	—	4	2	no	—
37	C₂²×Dic₁₇	—	4	2	no	—
38	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	complemented
43	C₄⋊C₄×C₁₇	—	2	1	no	—
44	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	complemented
45	C₂²⋊Dic₁₇	—	2	4	no	—
46	C₄⋊Dic₁₇	—	2	1	no	—
47	C₄⋊Dic₁₇	—	2	1	no	—
48	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	complemented
50	C₄×Dic₁₇	—	4	2	no	—
51	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₇	D₄	1	2	yes	—
53	C₂×#3622	—	2	1	no	—
54	C₂×#3622	—	2	1	no	—
55	C₂×#3622	C₂×C₄	1	2	yes	complemented
56	C₂³×C₁₇	—	2	2	no	—
57	C₂³×C₁₇	—	2	2	no	—
58	C₂³×C₁₇	—	2	2	no	—
59	C₂³×C₁₇	—	4	1	no	—
60	C₂³×C₁₇	—	4	1	no	—
61	C₂³×C₁₇	—	4	1	no	—
62	C₂³×C₁₇	—	4	1	no	—
63	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
64	D₄×C₁₇	—	2	2	no	—
65	D₄×C₁₇	—	2	2	no	—
66	D₄×C₁₇	—	2	2	no	—
67	D₄×C₁₇	C₄²	1	4	yes	complemented
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	4	1	no	—
70	C₂×C₄×C₁₇	—	4	1	no	—
71	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	2	2	no	—
80	C₂×Dic₁₇	—	2	2	no	—
81	C₂×Dic₁₇	—	4	1	no	—
82	C₂×Dic₁₇	—	4	1	no	—
83	C₂×Dic₁₇	—	4	1	no	—
84	C₂×Dic₁₇	—	4	1	no	—
85	C₂×Dic₁₇	—	4	8	no	—
86	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
87	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
88	C₂×Dic₁₇	D₈	1	2	yes	—
89	C₂×Dic₁₇	SD₁₆	1	2	yes	—
90	C₈×C₁₇	—	2	2	no	—
91	#3622	—	2	2	no	—
92	C₄×#102	—	17	1	no	maximal, Sylow
93	C₂²×C₁₇	—	2	4	no	—
94	C₂²×C₁₇	—	4	2	no	—
95	C₂²×C₁₇	—	4	2	no	—
96	C₂²×C₁₇	—	4	2	no	—
97	C₂²×C₁₇	—	4	2	no	—
98	C₂²×C₁₇	—	4	2	no	—
99	C₂²×C₁₇	—	4	2	no	—
100	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
102	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
104	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂²×C₁₇	#135	1	1	yes	characteristic
106	C₂²×C₁₇	#135	1	1	yes	characteristic
107	C₄×C₁₇	—	4	2	no	—
108	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
109	C₄×C₁₇	#117	1	1	yes	characteristic
110	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
111	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
112	Dic₁₇	—	2	2	no	—
113	Dic₁₇	—	4	2	no	—
114	Dic₁₇	—	4	2	no	—
115	Dic₁₇	#102	1	4	yes	direct product factor
116	C₂×C₄×D₄	—	17	1	no	—
117	C₂×#102	—	17	1	no	—
118	C₂×#102	D₁₇	1	1	yes	characteristic
119	C₂×C₄×C₈	—	17	1	no	—
120	C₄×C₄⋊C₄	—	17	1	no	—
121	#340	—	17	1	no	—
122	#340	—	17	1	no	—
123	C₂×C₁₇	—	4	4	no	—
124	C₂×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₁₇	C₂×#102	1	1	yes	characteristic
126	C₂×C₁₇	#416	1	1	yes	characteristic
127	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
128	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
129	C₂×C₁₇	C₄×SD₁₆	1	1	yes	characteristic
130	C₂×C₁₇	C₄×SD₁₆	1	1	yes	characteristic
131	C₂²×D₄	C₂×D₁₇	1	1	yes	characteristic
132	C₂³×C₄	—	34	1	no	—
133	C₂²×C₈	—	17	1	no	—
134	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
135	C₄×D₄	—	17	4	no	—
136	C₄×D₄	—	34	2	no	—
137	C₂×C₄⋊C₄	—	17	1	no	—
138	C₂×C₄⋊C₄	—	17	1	no	—
139	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	34	1	no	—
141	C₂×C₄²	—	17	1	no	—
142	C₂×C₄²	—	34	1	no	—
143	#102	—	17	4	no	—
144	#102	Dic₁₇	1	4	yes	direct product factor
145	C₄×C₈	—	34	2	no	—
146	#95	—	34	1	no	—
147	C₁₇	C₄×#102	1	1	yes	characteristic, complemented, Sylow
148	C₂⁴	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₄×D₁₇	1	2	yes	—
153	C₂×D₄	C₄×D₁₇	1	2	yes	—
154	C₂×D₄	C₂×Dic₁₇	1	2	yes	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	34	4	no	—
159	C₂²×C₄	—	34	1	no	—
160	C₂²×C₄	—	34	1	no	—
161	C₂²×C₄	—	68	2	no	—
162	C₂²×C₄	—	68	2	no	—
163	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	34	1	no	—
168	C₂×C₈	—	34	1	no	—
169	C₂×C₈	C₂×Dic₁₇	1	2	yes	complemented
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	17	2	no	—
172	C₄⋊C₄	—	17	2	no	—
173	C₄⋊C₄	—	34	1	no	—
174	C₄⋊C₄	—	34	1	no	—
175	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	complemented
176	C₂²⋊C₄	—	34	4	no	—
177	C₄²	—	17	2	no	—
178	C₄²	—	17	2	no	—
179	C₄²	—	68	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	1	no	—
184	C₂³	—	4	1	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	1	no	—
187	C₂³	D₄×D₁₇	1	1	yes	characteristic
188	D₄	—	2	2	no	—
189	D₄	—	2	2	no	—
190	D₄	—	2	2	no	—
191	D₄	C₄×Dic₁₇	1	4	yes	complemented
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	34	1	no	—
200	C₂×C₄	—	34	1	no	—
201	C₂×C₄	—	34	2	no	—
202	C₂×C₄	—	34	2	no	—
203	C₂×C₄	—	68	8	no	—
204	C₂×C₄	—	68	1	no	—
205	C₂×C₄	—	68	1	no	—
206	C₂×C₄	—	68	1	no	—
207	C₂×C₄	—	68	1	no	—
208	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
210	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
211	C₂×C₄	#63254	1	1	yes	characteristic
212	C₂×C₄	#63254	1	1	yes	characteristic
213	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	34	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
224	C₂²	#10587567	1	1	yes	characteristic
225	C₂²	D₈×D₁₇	1	1	yes	characteristic
226	C₂²	#10587598	1	1	yes	characteristic
227	C₂²	SD₁₆×D₁₇	1	1	yes	characteristic
228	C₂²	#10587602	1	1	yes	characteristic
229	C₂²	D₄×Dic₁₇	1	1	yes	characteristic
230	C₄	—	4	2	no	—
231	C₄	—	17	4	no	—
232	C₄	—	34	2	no	—
233	C₄	—	68	2	no	—
234	C₄	—	68	2	no	—
235	C₄	#10587563	1	1	yes	characteristic
236	C₄	#10587567	1	1	yes	characteristic
237	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
238	C₄	D₄×Dic₁₇	1	1	yes	characteristic
239	C₂	—	4	4	no	—
240	C₂	C₂²⋊C₄×Dic₁₇	1	1	yes	characteristic
241	C₂	#102×D₁₇	1	1	yes	characteristic
242	C₂	#11762802	1	1	yes	characteristic
243	C₂	#11762889	1	1	yes	characteristic
244	C₂	D₈×Dic₁₇	1	1	yes	characteristic
245	C₂	SD₁₆×Dic₁₇	1	1	yes	characteristic
246	C₂	#11762900	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#102	C₂
socle	C₂³×C₁₇	C₂×D₄
prime radical	C₂²×D₄×C₁₇	C₂²
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#102	D₁₇
hypocenter	C₁₇	C₄×#102
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#102

Complements

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

N	H ≅ G/N	type
3. C₂×C₈×Dic₁₇	C₂	semidirect
4. C₄⋊C₄×Dic₁₇	C₂	semidirect
5. #11762572	C₂	semidirect
6. #11762571	C₂	semidirect
14. D₄×Dic₁₇	C₄	semidirect
20. C₁₇×#102	C₄	semidirect
21. #10587503	C₄	semidirect
42. C₂×C₈×C₁₇	C₂×C₄	semidirect
44. C₄⋊C₄×C₁₇	C₂×C₄	semidirect
49. C₄⋊Dic₁₇	C₂×C₄	semidirect
55. C₂×#3622	C₂×C₄	semidirect
67. D₄×C₁₇	C₄²	semidirect
115. Dic₁₇	#102	direct (N × H)
144. #102	Dic₁₇	direct (N × H)
147. C₁₇	C₄×#102	semidirect
169. C₂×C₈	C₂×Dic₁₇	semidirect
175. C₄⋊C₄	C₂×Dic₁₇	semidirect
191. D₄	C₄×Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₇	2
C₂×#10587503	2
C₂×C₈×Dic₁₇	2
C₄⋊C₄×Dic₁₇	2
#11762572	2
#11762571	2
C₂×C₁₇×#102	2
C₄×#102	17

Subgroups — #102×Dic17 #14528601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #102×Dic₁₇ #14528601