Subgroups — #17898003

Order 4352.

2666 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541987	C₂	1	2	yes	maximal
2	#14537384	C₂	1	8	yes	maximal, complemented
3	#14537379	C₂	1	8	yes	maximal, complemented
4	#14537374	C₂	1	6	yes	maximal, complemented
5	C₂×#11762847	C₂	1	6	yes	maximal, complemented
6	C₂²×C₁₇×#131	C₂	1	1	yes	maximal, characteristic
7	#11763292	C₂²	1	16	yes	—
8	#11763287	—	2	2	no	—
9	#11763284	—	2	6	no	—
10	#41×Dic₁₇	—	2	8	no	—
11	C₂×#10587612	C₂²	1	12	yes	—
12	#11762851	C₂²	1	48	yes	complemented
13	#11762847	C₂²	1	4	yes	—
14	#11762847	C₂²	1	12	yes	—
15	#11762847	C₂²	1	12	yes	complemented
16	#11762847	C₂²	1	12	yes	complemented
17	C₂×#10587489	C₂²	1	12	yes	complemented
18	C₂×#10587488	C₂²	1	12	yes	complemented
19	C₂²×#41×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₇×#131	C₄	1	16	yes	complemented
21	C₂×C₁₇×#131	C₂²	1	4	yes	—
22	C₂×C₁₇×#131	C₂²	1	4	yes	—
23	C₂²×M₁₆×C₁₇	C₂²	1	3	yes	—
24	C₂³×C₈×C₁₇	C₂²	1	3	yes	—
25	C₂×#41×C₁₇	C₂×C₄	1	8	yes	—
26	C₂×#41×C₁₇	D₄	1	4	yes	—
27	C₂×#41×C₁₇	Q₈	1	8	yes	—
28	C₂×#41×C₁₇	C₂³	1	4	yes	—
29	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₁₇	D₄	1	3	yes	—
31	C₂³×C₄×C₁₇	C₂³	1	3	yes	—
32	C₁₇×#131	C₂×C₄	1	32	yes	complemented
33	C₁₇×#131	C₂×C₄	1	32	yes	complemented
34	C₂×M₁₆×C₁₇	C₂×C₄	1	24	yes	complemented
35	C₂×M₁₆×C₁₇	C₂³	1	12	yes	—
36	C₂²×C₈×C₁₇	C₂×C₄	1	24	yes	complemented
37	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
39	C₂²×C₈×C₁₇	C₂³	1	6	yes	—
40	C₂²×C₈×C₁₇	C₂³	1	6	yes	—
41	C₂×C₂²⋊Dic₁₇	—	2	12	no	—
42	Q₈×Dic₁₇	—	2	16	no	—
43	D₄×Dic₁₇	—	2	48	no	—
44	#10587612	—	2	12	no	—
45	#10587612	—	2	12	no	—
46	#10587612	C₂³	1	8	yes	—
47	#10587612	C₂³	1	24	yes	—
48	C₂×C₄⋊Dic₁₇	C₂³	1	24	yes	—
49	C₂×C₄×Dic₁₇	—	2	12	no	—
50	#10587489	C₂³	1	8	yes	—
51	#10587489	C₂³	1	24	yes	—
52	#10587488	C₂³	1	8	yes	—
53	#10587488	C₂³	1	24	yes	—
54	C₂⁴×C₁₇	C₂×D₄	1	3	yes	—
55	#41×C₁₇	C₄⋊C₄	1	32	yes	—
56	#41×C₁₇	C₂²×C₄	1	16	yes	—
57	#41×C₁₇	C₂×Q₈	1	16	yes	—
58	C₂×Q₈×C₁₇	C₄⋊C₄	1	8	yes	—
59	C₂×Q₈×C₁₇	C₂×D₄	1	4	yes	—
60	C₂×D₄×C₁₇	C₄⋊C₄	1	24	yes	—
61	C₂×D₄×C₁₇	C₂×D₄	1	12	yes	—
62	C₂²×C₄×C₁₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₁₇	C₂×Q₈	1	12	yes	—
66	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
68	C₂²×Dic₁₇	—	2	24	no	—
69	M₁₆×C₁₇	C₂²×C₄	1	48	yes	—
70	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₁₇	C₂²×C₄	1	12	yes	—
72	C₂×C₈×C₁₇	C₂²×C₄	1	24	yes	—
73	C₂×C₈×C₁₇	C₂²×C₄	1	24	yes	—
74	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂⁴	1	6	yes	—
77	C₂²⋊Dic₁₇	—	2	8	no	—
78	C₂²⋊Dic₁₇	—	2	24	no	—
79	C₄⋊Dic₁₇	—	2	24	no	—
80	C₄⋊Dic₁₇	C₂⁴	1	16	yes	—
81	C₄×Dic₁₇	—	2	8	no	—
82	C₄×Dic₁₇	—	2	24	no	—
83	#60108	—	17	1	no	maximal, Sylow
84	C₂³×C₁₇	—	4	6	no	—
85	C₂³×C₁₇	C₂×C₄⋊C₄	1	12	yes	—
86	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
87	C₂³×C₁₇	C₂²×D₄	1	6	yes	—
88	Q₈×C₁₇	C₂×C₄⋊C₄	1	16	yes	—
89	D₄×C₁₇	C₂×C₄⋊C₄	1	48	yes	—
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	12	yes	—
91	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	24	yes	—
92	C₂×C₄×C₁₇	C₂³×C₄	1	2	yes	—
93	C₂×C₄×C₁₇	C₂³×C₄	1	12	yes	—
94	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
95	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₁₇	C₂²×Q₈	1	12	yes	—
98	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
99	C₂×Dic₁₇	—	2	16	no	—
100	C₂×Dic₁₇	—	8	24	no	—
101	C₈×C₁₇	C₂³×C₄	1	2	yes	—
102	C₈×C₁₇	C₂³×C₄	1	2	yes	—
103	C₈×C₁₇	C₂³×C₄	1	12	yes	—
104	C₂²×#131	D₁₇	1	1	yes	characteristic
105	#3428	—	17	2	no	—
106	#2913	—	17	8	no	—
107	#2912	—	17	8	no	—
108	#2911	—	17	6	no	—
109	C₂×#427	—	17	6	no	—
110	C₂²×C₁₇	—	2	12	no	—
111	C₂²×C₁₇	—	2	2	no	—
112	C₂²×C₁₇	—	4	12	no	—
113	C₂²×C₁₇	C₂²×C₄⋊C₄	1	2	yes	—
114	C₂²×C₁₇	C₂²×C₄⋊C₄	1	12	yes	—
115	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
116	C₄×C₁₇	C₂²×C₄⋊C₄	1	2	yes	—
117	C₄×C₁₇	C₂²×C₄⋊C₄	1	12	yes	—
118	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₁₇	C₂³×Q₈	1	1	yes	characteristic
120	Dic₁₇	—	4	16	no	—
121	C₂²×#41	C₂×D₁₇	1	1	yes	characteristic
122	C₂×#131	Dic₁₇	1	16	yes	complemented
123	C₂×#131	C₂×D₁₇	1	4	yes	—
124	C₂×#131	C₂×D₁₇	1	4	yes	—
125	C₂²×M₁₆	C₂×D₁₇	1	3	yes	—
126	C₂³×C₈	C₂×D₁₇	1	3	yes	—
127	#520	—	17	16	no	—
128	#519	—	34	2	no	—
129	#518	—	34	6	no	—
130	C₄×#41	—	34	8	no	—
131	C₂×#117	—	17	12	no	—
132	#428	—	17	48	no	—
133	#427	—	17	4	no	—
134	#427	—	17	12	no	—
135	#427	—	17	12	no	—
136	#427	—	17	12	no	—
137	C₂×#107	—	17	12	no	—
138	C₂×#106	—	17	12	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	—	2	12	no	—
141	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₁₇	#3584	1	2	yes	—
143	C₂×#41	Dic₃₄	1	8	yes	—
144	C₂×#41	D₆₈	1	4	yes	—
145	C₂×#41	C₂×Dic₁₇	1	8	yes	—
146	C₂×#41	C₂²×D₁₇	1	4	yes	—
147	C₂²×Q₈	D₆₈	1	1	yes	characteristic
148	C₂²×D₄	D₆₈	1	3	yes	—
149	C₂³×C₄	C₂²×D₁₇	1	3	yes	—
150	#131	C₂×Dic₁₇	1	32	yes	complemented
151	#131	C₂×Dic₁₇	1	32	yes	complemented
152	C₂×M₁₆	C₂×Dic₁₇	1	24	yes	complemented
153	C₂×M₁₆	C₂²×D₁₇	1	12	yes	—
154	C₂²×C₈	C₂×Dic₁₇	1	24	yes	complemented
155	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
157	C₂²×C₈	C₂²×D₁₇	1	6	yes	—
158	C₂²×C₈	C₂²×D₁₇	1	6	yes	—
159	C₄×Q₈	—	34	16	no	—
160	C₄×D₄	—	34	48	no	—
161	#117	—	17	8	no	—
162	#117	—	17	24	no	—
163	#117	—	34	12	no	—
164	#117	—	34	12	no	—
165	C₂×C₄⋊C₄	—	17	24	no	—
166	C₂×C₂²⋊C₄	—	34	12	no	—
167	C₂×C₄²	—	34	12	no	—
168	#107	—	17	8	no	—
169	#107	—	17	24	no	—
170	#106	—	17	8	no	—
171	#106	—	17	24	no	—
172	C₁₇	#60108	1	1	yes	characteristic, complemented, Sylow
173	C₂⁴	C₂×D₆₈	1	3	yes	—
174	#41	C₄⋊Dic₁₇	1	32	yes	—
175	#41	C₂×Dic₃₄	1	16	yes	—
176	#41	C₂²×Dic₁₇	1	16	yes	—
177	C₂×Q₈	C₄⋊Dic₁₇	1	8	yes	—
178	C₂×Q₈	C₂×D₆₈	1	4	yes	—
179	C₂×D₄	C₄⋊Dic₁₇	1	24	yes	—
180	C₂×D₄	C₂×D₆₈	1	12	yes	—
181	C₂²×C₄	—	34	24	no	—
182	C₂²×C₄	C₂×Dic₃₄	1	12	yes	—
183	C₂²×C₄	C₂×D₆₈	1	3	yes	—
184	C₂²×C₄	C₂×D₆₈	1	6	yes	—
185	C₂²×C₄	C₂²×Dic₁₇	1	12	yes	—
186	C₂²×C₄	C₂³×D₁₇	1	6	yes	—
187	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
188	M₁₆	C₂²×Dic₁₇	1	48	yes	—
189	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
190	C₂×C₈	C₂²×Dic₁₇	1	12	yes	—
191	C₂×C₈	C₂²×Dic₁₇	1	24	yes	—
192	C₂×C₈	C₂²×Dic₁₇	1	24	yes	—
193	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
194	C₂×C₈	C₂³×D₁₇	1	6	yes	—
195	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
196	C₄⋊C₄	—	17	16	no	—
197	C₄⋊C₄	—	34	24	no	—
198	C₂²⋊C₄	—	34	8	no	—
199	C₂²⋊C₄	—	34	24	no	—
200	C₄²	—	34	24	no	—
201	C₄²	—	34	8	no	—
202	C₂³	—	4	6	no	—
203	C₂³	C₂×C₄⋊Dic₁₇	1	12	yes	—
204	C₂³	C₂²×D₆₈	1	1	yes	characteristic
205	C₂³	C₂²×D₆₈	1	6	yes	—
206	Q₈	C₂×C₄⋊Dic₁₇	1	16	yes	—
207	D₄	C₂×C₄⋊Dic₁₇	1	48	yes	—
208	C₂×C₄	—	34	16	no	—
209	C₂×C₄	—	136	24	no	—
210	C₂×C₄	C₂×C₄⋊Dic₁₇	1	12	yes	—
211	C₂×C₄	C₂×C₄⋊Dic₁₇	1	24	yes	—
212	C₂×C₄	C₂²×Dic₃₄	1	2	yes	—
213	C₂×C₄	C₂²×Dic₃₄	1	12	yes	—
214	C₂×C₄	C₂²×D₆₈	1	6	yes	—
215	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
216	C₂×C₄	C₂³×Dic₁₇	1	2	yes	—
217	C₂×C₄	C₂³×Dic₁₇	1	12	yes	—
218	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
219	C₈	C₂³×Dic₁₇	1	2	yes	—
220	C₈	C₂³×Dic₁₇	1	2	yes	—
221	C₈	C₂³×Dic₁₇	1	12	yes	—
222	C₂²	—	2	12	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	12	no	—
225	C₂²	C₂²×C₄⋊Dic₁₇	1	2	yes	—
226	C₂²	C₂²×C₄⋊Dic₁₇	1	12	yes	—
227	C₂²	C₂³×D₆₈	1	1	yes	characteristic
228	C₄	—	68	16	no	—
229	C₄	C₂²×C₄⋊Dic₁₇	1	12	yes	—
230	C₄	C₂²×C₄⋊Dic₁₇	1	2	yes	—
231	C₄	C₂⁴×Dic₁₇	1	1	yes	characteristic
232	C₄	C₂³×Dic₃₄	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	2	12	no	—
235	C₂	C₂³×C₄⋊Dic₁₇	1	1	yes	characteristic
236	C₂	#14543558	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₄×C₁₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂²×C₁₇×#131	C₂
socle	C₂²×C₁₇	C₂³×D₄
prime radical	C₂²×#41×C₁₇	C₂²
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂²×#131	D₁₇
hypocenter	C₁₇	#60108
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #14537384	C₂	semidirect
3. #14537379	C₂	semidirect
4. #14537374	C₂	semidirect
5. C₂×#11762847	C₂	semidirect
12. #11762851	C₂²	semidirect
15. #11762847	C₂²	semidirect
16. #11762847	C₂²	semidirect
17. C₂×#10587489	C₂²	semidirect
18. C₂×#10587488	C₂²	semidirect
20. C₂×C₁₇×#131	C₄	semidirect
32. C₁₇×#131	C₂×C₄	semidirect
33. C₁₇×#131	C₂×C₄	semidirect
34. C₂×M₁₆×C₁₇	C₂×C₄	semidirect
36. C₂²×C₈×C₁₇	C₂×C₄	semidirect
122. C₂×#131	Dic₁₇	semidirect
150. #131	C₂×Dic₁₇	semidirect
151. #131	C₂×Dic₁₇	semidirect
152. C₂×M₁₆	C₂×Dic₁₇	semidirect
154. C₂²×C₈	C₂×Dic₁₇	semidirect
172. C₁₇	#60108	semidirect

Maximal subgroups

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

subgroup	index
#14541987	2
#14537384	2
#14537379	2
#14537374	2
C₂×#11762847	2
C₂²×C₁₇×#131	2
#60108	17