Subgroups — #14537822

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#11762888	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762887	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762886	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762885	C₂	1	2	yes	maximal, complemented
6	#11762884	C₂	1	2	yes	maximal, complemented
7	#11762861	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762860	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762772	C₂	1	2	yes	maximal, complemented
10	#11762771	C₂	1	2	yes	maximal, complemented
11	C₁₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₇	C₂²	1	2	yes	—
14	C₂×M₁₆×C₁₇	C₂²	1	2	yes	complemented
15	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
16	D₄×Dic₁₇	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₇	C₂²	1	2	yes	—
18	C₂×#63223	C₂²	1	2	yes	complemented
19	C₂×#63218	C₂²	1	2	yes	complemented
20	C₂×#63218	C₂²	1	2	yes	complemented
21	#10587579	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₇	C₄	1	1	yes	characteristic
23	#10587567	C₄	1	2	yes	—
24	C₂²⋊C₄×D₁₇	C₄	1	2	yes	—
25	C₄×D₆₈	C₂²	1	1	yes	characteristic
26	C₄²×D₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₁₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊#3622	C₂²	1	2	yes	complemented
31	#10587491	C₂²	1	1	yes	characteristic
32	#10587491	C₂²	1	2	yes	—
33	#10587491	C₂²	1	1	yes	characteristic
34	#10587491	C₂²	1	2	yes	complemented
35	#10587491	C₂²	1	2	yes	complemented
36	#10587485	C₂²	1	1	yes	characteristic, complemented
37	#10587485	C₂²	1	1	yes	characteristic, complemented
38	C₈×Dic₁₇	C₂²	1	2	yes	complemented
39	C₄⋊#3622	C₂²	1	1	yes	characteristic, complemented
40	C₄×#3622	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₇	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₁₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
46	D₄×D₁₇	—	2	4	no	—
47	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₇	—	4	2	no	—
49	C₂×C₄×D₁₇	—	4	2	no	—
50	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₁₇	C₂³	1	2	yes	—
56	M₁₆×C₁₇	—	4	2	no	—
57	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂³	1	2	yes	—
59	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
62	C₄²×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₇	C₂×C₄	1	2	yes	—
64	#63227	C₂×C₄	1	2	yes	—
65	#63227	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
67	#63225	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
71	#63223	—	4	2	no	—
72	C₂×#3622	C₂³	1	1	yes	characteristic
73	C₂×#3622	C₂³	1	1	yes	characteristic
74	C₂×#3622	C₂³	1	2	yes	—
75	#63218	—	4	2	no	—
76	#63218	—	4	2	no	—
77	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₇	—	2	4	no	—
79	C₂²×D₁₇	—	4	2	no	—
80	C₂²×D₁₇	—	4	2	no	—
81	C₂²×D₁₇	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₁₇	—	2	4	no	—
91	C₂×Dic₁₇	—	4	2	no	—
92	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
96	D₆₈	—	2	2	no	—
97	C₄×D₁₇	—	2	1	no	—
98	C₄×D₁₇	—	2	1	no	—
99	C₄×D₁₇	—	2	2	no	—
100	C₄×D₁₇	—	2	2	no	—
101	C₄×D₁₇	—	4	2	no	—
102	C₄×D₁₇	—	4	2	no	—
103	C₄×D₁₇	—	4	1	no	—
104	C₄×D₁₇	M₁₆	1	4	yes	complemented
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	—	2	2	no	—
107	C₈×C₁₇	—	2	1	no	—
108	#3622	—	2	1	no	—
109	#3622	—	2	2	no	—
110	#3622	—	2	1	no	—
111	#2975	—	17	1	no	maximal, Sylow
112	C₂²×C₁₇	—	2	2	no	—
113	C₂²×C₁₇	—	2	2	no	—
114	C₂²×C₁₇	—	2	2	no	—
115	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₇	—	2	1	no	—
117	C₂×D₁₇	—	2	1	no	—
118	C₂×D₁₇	—	2	2	no	—
119	C₂×D₁₇	—	2	2	no	—
120	C₂×D₁₇	—	2	2	no	—
121	C₂×D₁₇	—	4	8	no	—
122	C₂×D₁₇	C₂×M₁₆	1	2	yes	—
123	C₄×C₁₇	—	2	2	no	—
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
126	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
128	Dic₁₇	—	2	1	no	—
129	Dic₁₇	—	2	2	no	—
130	Dic₁₇	—	2	1	no	—
131	Dic₁₇	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×D₄	—	17	1	no	—
133	#436	—	17	2	no	—
134	#436	—	17	1	no	—
135	#436	—	17	1	no	—
136	#436	—	17	2	no	—
137	#436	—	17	1	no	—
138	#436	D₁₇	1	1	yes	characteristic, complemented
139	#431	—	17	1	no	—
140	#431	—	17	1	no	—
141	#407	—	17	2	no	—
142	#407	—	17	2	no	—
143	C₂×C₁₇	—	4	2	no	—
144	C₂×C₁₇	#518	1	1	yes	characteristic
145	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
146	C₂×C₁₇	#568	1	1	yes	characteristic
147	D₁₇	—	2	2	no	—
148	D₁₇	—	4	2	no	—
149	C₂²×D₄	—	17	1	no	—
150	C₂³×C₄	—	17	2	no	—
151	C₂×M₁₆	—	17	2	no	—
152	C₂×M₁₆	—	17	2	no	—
153	C₂×M₁₆	—	17	2	no	—
154	C₂×M₁₆	C₂×D₁₇	1	2	yes	complemented
155	C₄×D₄	—	17	1	no	—
156	C₄×D₄	—	17	2	no	—
157	C₄×D₄	—	17	1	no	—
158	C₄×D₄	—	17	2	no	—
159	C₄×D₄	—	17	1	no	—
160	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	17	1	no	—
162	C₂×C₂²⋊C₄	—	17	2	no	—
163	C₂×C₄²	—	17	1	no	—
164	C₄⋊C₈	—	17	1	no	—
165	C₄⋊C₈	—	17	1	no	—
166	C₄⋊C₈	—	17	1	no	—
167	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
168	C₂²⋊C₈	—	17	2	no	—
169	C₂²⋊C₈	—	17	2	no	—
170	C₂²⋊C₈	—	17	1	no	—
171	C₂²⋊C₈	—	17	1	no	—
172	C₂²⋊C₈	—	17	2	no	—
173	C₂²⋊C₈	—	17	2	no	—
174	C₂²⋊C₈	C₂×D₁₇	1	2	yes	complemented
175	C₄×C₈	—	17	2	no	—
176	C₄×C₈	—	17	1	no	—
177	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
178	C₁₇	#2975	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	17	2	no	—
180	C₂×D₄	—	17	2	no	—
181	C₂×D₄	—	17	1	no	—
182	C₂×D₄	—	34	4	no	—
183	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
184	C₂²×C₄	—	17	1	no	—
185	C₂²×C₄	—	17	2	no	—
186	C₂²×C₄	—	17	1	no	—
187	C₂²×C₄	—	17	2	no	—
188	C₂²×C₄	—	17	1	no	—
189	C₂²×C₄	—	17	2	no	—
190	C₂²×C₄	—	17	2	no	—
191	C₂²×C₄	—	68	2	no	—
192	C₂²×C₄	—	68	2	no	—
193	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	68	2	no	—
196	M₁₆	—	68	2	no	—
197	M₁₆	—	68	2	no	—
198	C₂×C₈	—	17	1	no	—
199	C₂×C₈	—	17	2	no	—
200	C₂×C₈	—	17	1	no	—
201	C₂×C₈	C₂²×D₁₇	1	2	yes	—
202	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
204	C₄⋊C₄	—	17	2	no	—
205	C₄⋊C₄	—	17	1	no	—
206	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	17	2	no	—
208	C₂²⋊C₄	—	17	2	no	—
209	C₂²⋊C₄	—	17	2	no	—
210	C₂²⋊C₄	C₄×D₁₇	1	2	yes	—
211	C₄²	—	17	1	no	—
212	C₄²	—	17	1	no	—
213	C₄²	—	17	1	no	—
214	C₄²	C₂²×D₁₇	1	1	yes	characteristic
215	C₂³	—	17	1	no	—
216	C₂³	—	17	2	no	—
217	C₂³	—	34	4	no	—
218	C₂³	—	68	2	no	—
219	C₂³	—	68	2	no	—
220	C₂³	C₂×C₄×D₁₇	1	2	yes	—
221	D₄	—	2	2	no	—
222	D₄	—	34	4	no	—
223	D₄	—	34	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	4	no	—
228	C₂×C₄	—	17	1	no	—
229	C₂×C₄	—	17	1	no	—
230	C₂×C₄	—	17	2	no	—
231	C₂×C₄	—	34	2	no	—
232	C₂×C₄	—	34	1	no	—
233	C₂×C₄	—	34	2	no	—
234	C₂×C₄	—	34	1	no	—
235	C₂×C₄	—	68	2	no	—
236	C₂×C₄	—	68	2	no	—
237	C₂×C₄	—	68	1	no	—
238	C₂×C₄	—	68	2	no	—
239	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	34	1	no	—
247	C₈	—	34	2	no	—
248	C₈	—	34	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	17	2	no	—
253	C₂²	—	34	1	no	—
254	C₂²	—	34	1	no	—
255	C₂²	—	34	2	no	—
256	C₂²	—	34	2	no	—
257	C₂²	—	34	2	no	—
258	C₂²	—	68	8	no	—
259	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	17	2	no	—
263	C₄	—	34	1	no	—
264	C₄	—	34	2	no	—
265	C₄	—	34	1	no	—
266	C₄	M₁₆×D₁₇	1	2	yes	—
267	C₄	#10587691	1	1	yes	characteristic
268	C₄	#10587696	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	34	2	no	—
271	C₂	—	68	2	no	—
272	C₂	#11763281	1	1	yes	characteristic
273	C₂	C₂×M₁₆×D₁₇	1	1	yes	characteristic
274	C₂	#11763574	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#436	C₂
prime radical	C₂×D₄×D₁₇	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#436	D₁₇
hypocenter	C₁₇	#2975
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #11762888	C₂	semidirect
3. #11762887	C₂	semidirect
4. #11762886	C₂	semidirect
5. #11762885	C₂	semidirect
6. #11762884	C₂	semidirect
7. #11762861	C₂	semidirect
8. #11762860	C₂	semidirect
9. #11762772	C₂	semidirect
10. #11762771	C₂	semidirect
11. C₁₇×#436	C₂	semidirect
14. C₂×M₁₆×C₁₇	C₂²	semidirect
18. C₂×#63223	C₂²	semidirect
19. C₂×#63218	C₂²	semidirect
20. C₂×#63218	C₂²	semidirect
27. C₁₇×C₄⋊C₈	C₂²	semidirect
28. C₁₇×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₈×C₁₇	C₂²	semidirect
30. C₂²⋊#3622	C₂²	semidirect
34. #10587491	C₂²	semidirect
35. #10587491	C₂²	semidirect
36. #10587485	C₂²	semidirect
37. #10587485	C₂²	semidirect
38. C₈×Dic₁₇	C₂²	semidirect
39. C₄⋊#3622	C₂²	semidirect
40. C₄×#3622	C₂²	semidirect
104. C₄×D₁₇	M₁₆	semidirect
138. #436	D₁₇	semidirect
154. C₂×M₁₆	C₂×D₁₇	semidirect
167. C₄⋊C₈	C₂×D₁₇	semidirect
174. C₂²⋊C₈	C₂×D₁₇	semidirect
177. C₄×C₈	C₂×D₁₇	semidirect
178. C₁₇	#2975	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₁₇	2
#11762888	2
#11762887	2
#11762886	2
#11762885	2
#11762884	2
#11762861	2
#11762860	2
#11762772	2
#11762771	2
C₁₇×#436	2
#2975	17