Subgroups — #14542941

Order 2176.

1336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763548	C₂	1	3	yes	maximal, complemented
2	#11763489	C₂	1	3	yes	maximal, complemented
3	#11763452	C₂	1	6	yes	maximal, complemented
4	#11763441	C₂	1	3	yes	maximal, complemented
5	#11763393	C₂	1	6	yes	maximal, complemented
6	#11763377	C₂	1	3	yes	maximal, complemented
7	#11763369	C₂	1	3	yes	maximal, complemented
8	#120×D₁₇	C₂	1	3	yes	maximal, complemented
9	C₁₇×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₇	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₂²⋊D₁₇	C₂²	1	3	yes	complemented
12	C₂×#63256	C₂²	1	3	yes	complemented
13	C₂×D₄×D₁₇	C₂²	1	3	yes	complemented
14	C₂×D₄×D₁₇	C₂²	1	6	yes	complemented
15	C₁₇×#124	C₂²	1	3	yes	complemented
16	C₁₇×#124	C₂²	1	6	yes	complemented
17	C₁₇×#120	C₂²	1	3	yes	complemented
18	C₁₇×#120	C₂²	1	3	yes	complemented
19	#10587639	C₂²	1	3	yes	complemented
20	#10587639	C₂²	1	1	yes	characteristic, complemented
21	#10587633	C₂²	1	3	yes	complemented
22	#10587632	C₂²	1	6	yes	complemented
23	#10587626	C₂²	1	3	yes	complemented
24	#10587625	C₂²	1	6	yes	complemented
25	#10587624	C₂²	1	6	yes	complemented
26	#10587623	C₂²	1	3	yes	—
27	#10587623	C₂²	1	6	yes	complemented
28	#10587621	C₂²	1	3	yes	complemented
29	#10587572	C₂²	1	3	yes	complemented
30	#10587570	C₂²	1	6	yes	complemented
31	#10587570	C₂²	1	6	yes	complemented
32	#10587570	C₂²	1	6	yes	complemented
33	#10587569	C₂²	1	6	yes	complemented
34	#10587569	C₂²	1	6	yes	complemented
35	#10587568	C₂²	1	6	yes	complemented
36	#10587568	C₂²	1	3	yes	complemented
37	#10587568	C₂²	1	6	yes	complemented
38	#10587567	C₂²	1	6	yes	complemented
39	C₂²⋊C₄×D₁₇	C₂²	1	6	yes	complemented
40	C₂²⋊C₄×D₁₇	C₂²	1	3	yes	complemented
41	C₂²⋊C₄×D₁₇	C₂²	1	3	yes	complemented
42	#10587565	C₂²	1	6	yes	complemented
43	#10587562	C₂²	1	2	yes	complemented
44	#10587560	C₂²	1	1	yes	characteristic, complemented
45	C₄×D₆₈	C₂²	1	6	yes	complemented
46	#10587558	C₂²	1	3	yes	complemented
47	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic
48	C₂⁴×C₁₇	C₂³	1	1	yes	characteristic, complemented
49	C₂³×D₁₇	—	4	6	no	—
50	C₂³×D₁₇	C₂³	1	3	yes	—
51	C₂³×D₁₇	C₂³	1	3	yes	—
52	C₂³×D₁₇	C₂³	1	3	yes	—
53	C₂×Q₈×C₁₇	C₂³	1	3	yes	complemented
54	C₂×D₄×C₁₇	C₂³	1	3	yes	—
55	C₂×D₄×C₁₇	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₇	—	4	6	no	—
57	C₂×C₂²⋊D₁₇	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₁₇	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₁₇	C₂³	1	3	yes	—
60	C₂×C₂²⋊D₁₇	C₂³	1	3	yes	—
61	C₂×C₂²⋊D₁₇	C₂³	1	6	yes	complemented
62	C₂²×Dic₁₇	C₂³	1	3	yes	complemented
63	#63256	—	4	6	no	—
64	D₄×D₁₇	—	4	6	no	—
65	D₄×D₁₇	—	4	12	no	—
66	C₂×D₆₈	C₂³	1	6	yes	—
67	C₂×D₆₈	C₂³	1	3	yes	—
68	C₂×D₆₈	C₂³	1	3	yes	complemented
69	C₂×C₄×D₁₇	C₂³	1	3	yes	—
70	C₂×C₄×D₁₇	C₂³	1	6	yes	—
71	C₂×C₄×D₁₇	C₂³	1	6	yes	complemented
72	C₂²⋊C₄×C₁₇	C₂³	1	6	yes	—
73	C₂²⋊C₄×C₁₇	C₂³	1	3	yes	—
74	C₂²⋊C₄×C₁₇	C₂³	1	3	yes	complemented
75	C₂²⋊C₄×C₁₇	C₂³	1	6	yes	complemented
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic, complemented
77	C₄²×C₁₇	C₂³	1	2	yes	complemented
78	C₂²⋊Dic₁₇	C₂³	1	3	yes	—
79	C₂²⋊Dic₁₇	C₂³	1	6	yes	—
80	C₂²⋊Dic₁₇	C₂³	1	3	yes	complemented
81	#63227	C₂³	1	6	yes	—
82	#63227	C₂³	1	6	yes	—
83	#63227	C₂³	1	6	yes	—
84	#63227	C₂³	1	6	yes	complemented
85	C₄⋊Dic₁₇	C₂³	1	6	yes	complemented
86	#63225	C₂³	1	6	yes	complemented
87	#63225	C₂³	1	6	yes	complemented
88	C₄×Dic₁₇	C₂³	1	3	yes	complemented
89	C₂³×C₁₇	—	4	3	no	—
90	C₂³×C₁₇	—	4	3	no	—
91	C₂³×C₁₇	C₂⁴	1	3	yes	—
92	C₂³×C₁₇	C₂⁴	1	3	yes	—
93	C₂²×D₁₇	—	2	12	no	—
94	C₂²×D₁₇	—	2	6	no	—
95	C₂²×D₁₇	—	2	6	no	—
96	C₂²×D₁₇	—	4	3	no	—
97	C₂²×D₁₇	—	4	6	no	—
98	C₂²×D₁₇	—	4	6	no	—
99	C₂²×D₁₇	—	4	16	no	—
100	C₂²×D₁₇	C₂⁴	1	3	yes	—
101	C₂²×D₁₇	C₂⁴	1	1	yes	characteristic
102	C₂²×D₁₇	C₂⁴	1	3	yes	—
103	Q₈×C₁₇	—	4	3	no	—
104	D₄×C₁₇	—	4	3	no	—
105	D₄×C₁₇	—	4	6	no	—
106	C₂×C₄×C₁₇	C₂⁴	1	6	yes	—
107	C₂×C₄×C₁₇	C₂⁴	1	3	yes	—
108	C₂²⋊D₁₇	—	2	12	no	—
109	C₂²⋊D₁₇	—	4	3	no	—
110	C₂²⋊D₁₇	—	4	3	no	—
111	C₂²⋊D₁₇	—	4	6	no	—
112	C₂²⋊D₁₇	—	4	6	no	—
113	C₂×Dic₁₇	—	4	3	no	—
114	C₂×Dic₁₇	C₂⁴	1	3	yes	—
115	C₂×Dic₁₇	C₂⁴	1	6	yes	—
116	D₆₈	—	2	12	no	—
117	D₆₈	—	4	3	no	—
118	D₆₈	—	4	3	no	—
119	C₄×D₁₇	—	2	6	no	—
120	C₄×D₁₇	—	4	6	no	—
121	C₄×D₁₇	—	4	6	no	—
122	#3532	—	17	1	no	maximal, Sylow
123	C₂²×C₁₇	—	2	3	no	—
124	C₂²×C₁₇	—	2	6	no	—
125	C₂²×C₁₇	—	2	3	no	—
126	C₂²×C₁₇	—	2	6	no	—
127	C₂²×C₁₇	—	4	4	no	—
128	C₂²×C₁₇	—	8	2	no	—
129	C₂²×C₁₇	C₂⁵	1	1	yes	characteristic
130	C₂×D₁₇	—	2	6	no	—
131	C₂×D₁₇	—	2	6	no	—
132	C₂×D₁₇	—	2	3	no	—
133	C₂×D₁₇	—	2	3	no	—
134	C₂×D₁₇	—	4	12	no	—
135	C₂×D₁₇	—	4	12	no	—
136	C₂×D₁₇	—	8	6	no	—
137	C₂×D₁₇	H(5,𝔽₂)	1	6	yes	—
138	C₄×C₁₇	—	2	3	no	—
139	C₄×C₁₇	—	2	6	no	—
140	Dic₁₇	—	2	3	no	—
141	Dic₁₇	—	2	6	no	—
142	H(3,𝔽₄)	D₁₇	1	1	yes	characteristic, complemented
143	#560	—	17	3	no	—
144	#551	—	17	3	no	—
145	#546	—	17	6	no	—
146	D₄²	—	17	3	no	—
147	#538	—	17	6	no	—
148	#535	—	17	3	no	—
149	#534	—	17	3	no	—
150	C₂×#120	—	17	3	no	—
151	C₂×C₁₇	—	4	3	no	—
152	C₂×C₁₇	—	4	3	no	—
153	C₂×C₁₇	C₂×H(5,𝔽₂)	1	3	yes	—
154	D₁₇	—	2	6	no	—
155	D₁₇	—	4	1	no	—
156	D₁₇	—	4	3	no	—
157	C₂⁵	—	17	1	no	—
158	C₂×#41	—	17	3	no	—
159	C₂²×D₄	—	17	3	no	—
160	C₂²×D₄	—	17	3	no	—
161	C₂²×D₄	—	17	6	no	—
162	#127	—	17	3	no	—
163	#127	—	17	1	no	—
164	#126	—	17	6	no	—
165	#126	—	17	2	no	—
166	#124	—	17	3	no	—
167	#124	C₂×D₁₇	1	3	yes	complemented
168	#124	C₂×D₁₇	1	6	yes	complemented
169	#123	—	17	3	no	—
170	#123	—	17	6	no	—
171	#123	—	17	3	no	—
172	#123	—	17	6	no	—
173	C₂²⋊Q₈	—	17	6	no	—
174	#121	—	17	6	no	—
175	#121	—	17	6	no	—
176	#121	—	17	6	no	—
177	#121	—	17	6	no	—
178	#121	—	17	6	no	—
179	#120	—	17	3	no	—
180	#120	—	17	1	no	—
181	#120	—	17	6	no	—
182	#120	—	17	3	no	—
183	#120	—	17	6	no	—
184	#120	—	17	3	no	—
185	#120	—	17	6	no	—
186	#120	C₂×D₁₇	1	3	yes	complemented
187	#120	C₂×D₁₇	1	3	yes	complemented
188	C₄×D₄	—	17	6	no	—
189	C₄×D₄	—	17	6	no	—
190	#117	—	17	3	no	—
191	C₂×C₂²⋊C₄	—	17	3	no	—
192	C₂×C₂²⋊C₄	—	17	6	no	—
193	C₂×C₂²⋊C₄	—	17	3	no	—
194	C₁₇	#3532	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	—	17	3	no	—
196	C₂⁴	—	17	3	no	—
197	C₂⁴	—	17	3	no	—
198	C₂⁴	—	68	6	no	—
199	C₂⁴	C₂²×D₁₇	1	1	yes	characteristic
200	C₂⁴	C₂²×D₁₇	1	1	yes	characteristic, complemented
201	#41	—	68	6	no	—
202	C₂×Q₈	C₂²×D₁₇	1	3	yes	complemented
203	C₂×D₄	—	17	6	no	—
204	C₂×D₄	—	17	3	no	—
205	C₂×D₄	—	17	3	no	—
206	C₂×D₄	—	17	6	no	—
207	C₂×D₄	—	17	6	no	—
208	C₂×D₄	—	17	6	no	—
209	C₂×D₄	—	17	3	no	—
210	C₂×D₄	—	17	3	no	—
211	C₂×D₄	—	68	6	no	—
212	C₂×D₄	—	68	12	no	—
213	C₂×D₄	—	68	6	no	—
214	C₂×D₄	C₂²×D₁₇	1	6	yes	—
215	C₂×D₄	C₂²×D₁₇	1	3	yes	—
216	C₂²×C₄	—	17	3	no	—
217	C₂²×C₄	—	17	3	no	—
218	C₂²×C₄	—	17	6	no	—
219	C₂²×C₄	—	17	6	no	—
220	C₄⋊C₄	—	17	6	no	—
221	C₄⋊C₄	—	17	6	no	—
222	C₄⋊C₄	—	17	6	no	—
223	C₂²⋊C₄	—	17	3	no	—
224	C₂²⋊C₄	—	17	6	no	—
225	C₂²⋊C₄	—	17	3	no	—
226	C₂²⋊C₄	—	17	6	no	—
227	C₂²⋊C₄	—	17	6	no	—
228	C₂²⋊C₄	—	17	6	no	—
229	C₂²⋊C₄	—	17	6	no	—
230	C₂²⋊C₄	C₂²×D₁₇	1	6	yes	—
231	C₂²⋊C₄	C₂²×D₁₇	1	3	yes	—
232	C₂²⋊C₄	C₂²×D₁₇	1	3	yes	complemented
233	C₂²⋊C₄	C₂²×D₁₇	1	6	yes	complemented
234	C₄²	—	17	3	no	—
235	C₄²	C₂²×D₁₇	1	1	yes	characteristic, complemented
236	C₄²	C₂²×D₁₇	1	2	yes	complemented
237	C₂³	—	4	3	no	—
238	C₂³	—	4	3	no	—
239	C₂³	—	17	3	no	—
240	C₂³	—	17	1	no	—
241	C₂³	—	17	3	no	—
242	C₂³	—	34	12	no	—
243	C₂³	—	34	6	no	—
244	C₂³	—	34	6	no	—
245	C₂³	—	68	3	no	—
246	C₂³	—	68	6	no	—
247	C₂³	—	68	6	no	—
248	C₂³	—	68	16	no	—
249	C₂³	C₂³×D₁₇	1	3	yes	—
250	C₂³	C₂³×D₁₇	1	3	yes	—
251	Q₈	—	4	3	no	—
252	D₄	—	4	3	no	—
253	D₄	—	4	6	no	—
254	D₄	—	34	12	no	—
255	D₄	—	34	12	no	—
256	D₄	—	68	3	no	—
257	D₄	—	68	3	no	—
258	D₄	—	68	3	no	—
259	D₄	—	68	3	no	—
260	D₄	—	68	6	no	—
261	D₄	—	68	6	no	—
262	C₂×C₄	—	17	6	no	—
263	C₂×C₄	—	17	3	no	—
264	C₂×C₄	—	34	6	no	—
265	C₂×C₄	—	68	6	no	—
266	C₂×C₄	—	68	6	no	—
267	C₂×C₄	—	68	3	no	—
268	C₂×C₄	C₂³×D₁₇	1	3	yes	—
269	C₂×C₄	C₂³×D₁₇	1	6	yes	—
270	C₂²	—	2	3	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	3	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	17	6	no	—
277	C₂²	—	34	6	no	—
278	C₂²	—	34	6	no	—
279	C₂²	—	34	3	no	—
280	C₂²	—	34	3	no	—
281	C₂²	—	68	12	no	—
282	C₂²	—	68	12	no	—
283	C₂²	—	136	6	no	—
284	C₂²	C₂⁴×D₁₇	1	1	yes	characteristic
285	C₄	—	2	3	no	—
286	C₄	—	2	6	no	—
287	C₄	—	34	3	no	—
288	C₄	—	34	6	no	—
289	C₂	—	4	3	no	—
290	C₂	—	4	3	no	—
291	C₂	—	34	6	no	—
292	C₂	—	68	1	no	—
293	C₂	—	68	3	no	—
294	C₂	H(5,𝔽₂)×D₁₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁵
Fitting	C₁₇×H(3,𝔽₄)	C₂
hypercenter	H(3,𝔽₄)	D₁₇
hypocenter	C₁₇	#3532
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #11763548	C₂	semidirect
2. #11763489	C₂	semidirect
3. #11763452	C₂	semidirect
4. #11763441	C₂	semidirect
5. #11763393	C₂	semidirect
6. #11763377	C₂	semidirect
7. #11763369	C₂	semidirect
8. #120×D₁₇	C₂	semidirect
9. C₁₇×H(3,𝔽₄)	C₂	semidirect
10. C₂⁴×D₁₇	C₂²	semidirect
11. C₂²×C₂²⋊D₁₇	C₂²	semidirect
12. C₂×#63256	C₂²	semidirect
13. C₂×D₄×D₁₇	C₂²	semidirect
14. C₂×D₄×D₁₇	C₂²	semidirect
15. C₁₇×#124	C₂²	semidirect
16. C₁₇×#124	C₂²	semidirect
17. C₁₇×#120	C₂²	semidirect
18. C₁₇×#120	C₂²	semidirect
19. #10587639	C₂²	semidirect
20. #10587639	C₂²	semidirect
21. #10587633	C₂²	semidirect
22. #10587632	C₂²	semidirect
23. #10587626	C₂²	semidirect
24. #10587625	C₂²	semidirect
25. #10587624	C₂²	semidirect
27. #10587623	C₂²	semidirect
28. #10587621	C₂²	semidirect
29. #10587572	C₂²	semidirect
30. #10587570	C₂²	semidirect
31. #10587570	C₂²	semidirect
32. #10587570	C₂²	semidirect
33. #10587569	C₂²	semidirect
34. #10587569	C₂²	semidirect
35. #10587568	C₂²	semidirect
36. #10587568	C₂²	semidirect
37. #10587568	C₂²	semidirect
38. #10587567	C₂²	semidirect
39. C₂²⋊C₄×D₁₇	C₂²	semidirect
40. C₂²⋊C₄×D₁₇	C₂²	semidirect
41. C₂²⋊C₄×D₁₇	C₂²	semidirect
42. #10587565	C₂²	semidirect
43. #10587562	C₂²	semidirect
44. #10587560	C₂²	semidirect
45. C₄×D₆₈	C₂²	semidirect
46. #10587558	C₂²	semidirect
48. C₂⁴×C₁₇	C₂³	semidirect
53. C₂×Q₈×C₁₇	C₂³	semidirect
61. C₂×C₂²⋊D₁₇	C₂³	semidirect
62. C₂²×Dic₁₇	C₂³	semidirect
68. C₂×D₆₈	C₂³	semidirect
71. C₂×C₄×D₁₇	C₂³	semidirect
74. C₂²⋊C₄×C₁₇	C₂³	semidirect
75. C₂²⋊C₄×C₁₇	C₂³	semidirect
76. C₄²×C₁₇	C₂³	semidirect
77. C₄²×C₁₇	C₂³	semidirect
80. C₂²⋊Dic₁₇	C₂³	semidirect
84. #63227	C₂³	semidirect
85. C₄⋊Dic₁₇	C₂³	semidirect
86. #63225	C₂³	semidirect
87. #63225	C₂³	semidirect
88. C₄×Dic₁₇	C₂³	semidirect
142. H(3,𝔽₄)	D₁₇	semidirect
167. #124	C₂×D₁₇	semidirect
168. #124	C₂×D₁₇	semidirect
186. #120	C₂×D₁₇	semidirect
187. #120	C₂×D₁₇	semidirect
194. C₁₇	#3532	semidirect
200. C₂⁴	C₂²×D₁₇	semidirect
202. C₂×Q₈	C₂²×D₁₇	semidirect
232. C₂²⋊C₄	C₂²×D₁₇	semidirect
233. C₂²⋊C₄	C₂²×D₁₇	semidirect
235. C₄²	C₂²×D₁₇	semidirect
236. C₄²	C₂²×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
#11763548	2
#11763489	2
#11763452	2
#11763441	2
#11763393	2
#11763377	2
#11763369	2
#120×D₁₇	2
C₁₇×H(3,𝔽₄)	2
#3532	17