Subgroups — #14537697

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587636	C₂	1	1	yes	maximal, characteristic
2	#11763576	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763575	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762928	C₂	1	4	yes	maximal, complemented
5	#11762923	C₂	1	4	yes	maximal, complemented
6	C₂×#10587493	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762827	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762765	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#421	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×#63252	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#117	C₂²	1	1	yes	characteristic
14	#10587636	C₄	1	2	yes	complemented
15	#10587636	C₄	1	2	yes	complemented
16	#10587636	C₂²	1	2	yes	—
17	#10587636	C₂²	1	2	yes	—
18	#10587634	—	2	2	no	—
19	#10587634	C₂²	1	4	yes	complemented
20	C₂×#63231	C₄	1	1	yes	characteristic, complemented
21	C₂×#63230	C₄	1	1	yes	characteristic, complemented
22	C₂×#63229	C₄	1	1	yes	characteristic, complemented
23	C₂×#63228	C₄	1	1	yes	characteristic, complemented
24	C₂×#63223	—	2	2	no	—
25	C₂×#63223	C₂²	1	1	yes	characteristic
26	C₂×#63223	C₂²	1	1	yes	characteristic
27	C₂²×#3622	C₂²	1	1	yes	characteristic
28	#10587593	C₂²	1	4	yes	complemented
29	M₁₆×D₁₇	—	2	2	no	—
30	#10587587	—	2	2	no	—
31	C₁₇×#104	C₂²	1	4	yes	complemented
32	#10587493	C₂²	1	2	yes	complemented
33	#10587493	C₂²	1	2	yes	complemented
34	#10587476	C₂²	1	4	yes	complemented
35	#10587474	C₂²	1	2	yes	complemented
36	C₄×#3622	C₂²	1	2	yes	complemented
37	#41×C₁₇	—	2	1	no	—
38	#41×C₁₇	—	2	1	no	—
39	#41×C₁₇	C₂×C₄	1	2	yes	—
40	#41×C₁₇	C₂³	1	2	yes	—
41	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₇	—	2	1	no	—
43	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₇	—	2	1	no	—
47	#63252	—	2	1	no	—
48	#63252	—	2	1	no	—
49	#63252	C₂×C₄	1	2	yes	—
50	#63252	C₂³	1	2	yes	—
51	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₇	—	2	1	no	—
53	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₁₇	—	2	1	no	—
55	M₁₆×C₁₇	C₂³	1	2	yes	—
56	C₂×C₈×C₁₇	—	2	1	no	—
57	C₄⋊C₄×C₁₇	—	2	1	no	—
58	C₂²⋊C₄×C₁₇	—	2	1	no	—
59	C₄²×C₁₇	C₂³	1	2	yes	—
60	#63231	C₂×C₄	1	2	yes	—
61	#63231	C₂×C₄	1	2	yes	—
62	#63230	C₂×C₄	1	2	yes	—
63	#63230	C₂×C₄	1	2	yes	—
64	#63229	C₂×C₄	1	2	yes	—
65	#63229	C₂×C₄	1	2	yes	—
66	#63228	C₂×C₄	1	2	yes	—
67	#63228	C₂×C₄	1	2	yes	—
68	#63223	—	2	1	no	—
69	#63223	—	2	4	no	—
70	#63223	D₄	1	2	yes	—
71	#63223	D₄	1	2	yes	—
72	#63223	C₂³	1	2	yes	—
73	C₂×#3622	—	2	1	no	—
74	C₂×#3622	—	2	4	no	—
75	C₂×#3622	C₂×C₄	1	1	yes	characteristic
76	C₂×#3622	C₂×C₄	1	1	yes	characteristic
77	C₂×#3622	C₂×C₄	1	1	yes	characteristic
78	C₂×#3622	C₂×C₄	1	1	yes	characteristic
79	C₂×#3622	C₂³	1	1	yes	characteristic
80	C₂×#3622	C₂³	1	1	yes	characteristic
81	C₂×#3622	C₂³	1	2	yes	—
82	#63218	—	2	4	no	—
83	C₈×D₁₇	—	2	4	no	—
84	C₂³×C₁₇	—	2	1	no	—
85	C₂³×C₁₇	#41	1	1	yes	characteristic
86	C₂²×D₁₇	—	2	1	no	—
87	Q₈×C₁₇	—	2	1	no	—
88	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
89	D₄×C₁₇	—	2	2	no	—
90	D₄×C₁₇	—	2	2	no	—
91	D₄×C₁₇	—	2	1	no	—
92	D₄×C₁₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	2	no	—
95	C₂×C₄×C₁₇	—	2	2	no	—
96	C₂×C₄×C₁₇	—	2	2	no	—
97	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₁₇	—	2	2	no	—
104	C₂²⋊D₁₇	—	2	2	no	—
105	C₂×Dic₁₇	—	2	1	no	—
106	D₆₈	—	2	1	no	—
107	D₆₈	C₂²×C₄	1	2	yes	—
108	C₄×D₁₇	—	2	2	no	—
109	C₄×D₁₇	—	2	2	no	—
110	Dic₃₄	—	2	1	no	—
111	Dic₃₄	C₂²×C₄	1	2	yes	—
112	C₈×C₁₇	—	2	2	no	—
113	#3622	—	2	2	no	—
114	#3622	C₂²×C₄	1	1	yes	characteristic
115	#3622	C₂²×C₄	1	1	yes	characteristic
116	#3622	C₂²×C₄	1	1	yes	characteristic
117	#3622	C₂²×C₄	1	1	yes	characteristic
118	#3622	C₂×D₄	1	4	yes	—
119	#2957	—	17	1	no	maximal, Sylow
120	C₂²×C₁₇	—	2	2	no	—
121	C₂²×C₁₇	—	4	1	no	—
122	C₂²×C₁₇	—	8	1	no	—
123	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
126	C₂×D₁₇	—	2	2	no	—
127	C₂×D₁₇	—	8	1	no	—
128	C₄×C₁₇	—	2	2	no	—
129	C₄×C₁₇	—	4	2	no	—
130	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
131	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
134	Dic₁₇	—	2	2	no	—
135	C₂×#135	—	17	1	no	—
136	#568	—	17	1	no	—
137	#568	—	17	1	no	—
138	#444	—	17	4	no	—
139	#443	—	17	4	no	—
140	C₂×#108	—	17	1	no	—
141	#421	—	17	1	no	—
142	#421	D₁₇	1	1	yes	characteristic, complemented
143	#405	—	17	1	no	—
144	C₂×C₁₇	—	2	1	no	—
145	C₂×C₁₇	—	2	1	no	—
146	C₂×C₁₇	—	2	1	no	—
147	C₂×C₁₇	—	4	2	no	—
148	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
149	D₁₇	—	4	2	no	—
150	C₂×#41	—	17	1	no	—
151	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
152	#135	—	17	2	no	—
153	#135	—	17	2	no	—
154	#135	—	17	2	no	—
155	#135	—	17	2	no	—
156	C₂×Q₁₆	—	17	1	no	—
157	C₂×SD₁₆	—	17	1	no	—
158	C₂×SD₁₆	—	17	1	no	—
159	C₂×D₈	—	17	1	no	—
160	#131	—	17	4	no	—
161	#131	—	17	4	no	—
162	#131	—	34	2	no	—
163	#131	—	34	2	no	—
164	C₂×M₁₆	—	17	1	no	—
165	C₂×M₁₆	—	17	1	no	—
166	C₂×M₁₆	—	34	2	no	—
167	C₂×M₁₆	—	34	2	no	—
168	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
169	C₂²×C₈	—	17	1	no	—
170	#117	C₂×D₁₇	1	1	yes	characteristic
171	#108	—	17	2	no	—
172	#108	—	17	2	no	—
173	#104	—	17	4	no	—
174	#104	C₂×D₁₇	1	4	yes	complemented
175	#97	—	17	2	no	—
176	C₄×C₈	—	17	2	no	—
177	C₁₇	#2957	1	1	yes	characteristic, complemented, Sylow
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	17	2	no	—
181	#41	—	17	2	no	—
182	#41	—	34	1	no	—
183	#41	—	34	1	no	—
184	#41	C₄×D₁₇	1	2	yes	—
185	#41	C₂²×D₁₇	1	2	yes	—
186	C₂×Q₈	—	17	1	no	—
187	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	17	1	no	—
190	C₂×D₄	—	34	1	no	—
191	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	34	1	no	—
194	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
195	Q₁₆	—	17	2	no	—
196	Q₁₆	—	17	2	no	—
197	SD₁₆	—	17	2	no	—
198	SD₁₆	—	17	2	no	—
199	SD₁₆	—	17	2	no	—
200	SD₁₆	—	17	2	no	—
201	D₈	—	17	2	no	—
202	D₈	—	17	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	17	2	no	—
205	M₁₆	—	17	2	no	—
206	M₁₆	—	17	2	no	—
207	M₁₆	—	34	4	no	—
208	M₁₆	—	34	4	no	—
209	M₁₆	—	34	1	no	—
210	M₁₆	C₂²×D₁₇	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	17	1	no	—
213	C₂×C₈	—	17	1	no	—
214	C₂×C₈	—	17	1	no	—
215	C₂×C₈	—	17	1	no	—
216	C₂×C₈	—	17	1	no	—
217	C₂×C₈	—	17	2	no	—
218	C₂×C₈	—	17	1	no	—
219	C₂×C₈	—	34	4	no	—
220	C₂×C₈	—	34	4	no	—
221	C₂×C₈	—	34	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	C₂²×D₁₇	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	34	1	no	—
227	C₂³	#63254	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	17	2	no	—
230	Q₈	—	34	1	no	—
231	Q₈	C₂×C₄×D₁₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	17	2	no	—
236	D₄	—	34	2	no	—
237	D₄	—	34	2	no	—
238	D₄	—	34	1	no	—
239	D₄	C₂×C₄×D₁₇	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	34	1	no	—
245	C₂×C₄	—	34	2	no	—
246	C₂×C₄	—	34	2	no	—
247	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
249	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
250	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
251	C₂×C₄	#63254	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	17	1	no	—
255	C₈	—	17	1	no	—
256	C₈	—	17	1	no	—
257	C₈	—	17	1	no	—
258	C₈	—	17	4	no	—
259	C₈	—	34	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	34	2	no	—
264	C₂²	—	136	1	no	—
265	C₂²	#10587567	1	1	yes	characteristic
266	C₂²	#10587567	1	1	yes	characteristic
267	C₂²	C₂×#63254	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	34	2	no	—
271	C₄	#10587567	1	1	yes	characteristic
272	C₄	#10587567	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
274	C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	68	2	no	—
280	C₂	C₂×#10587567	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#421	C₂
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	C₂×#10587636	C₂
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#421	D₁₇
hypocenter	C₁₇	#2957
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #11763576	C₂	semidirect
3. #11763575	C₂	semidirect
4. #11762928	C₂	semidirect
5. #11762923	C₂	semidirect
6. C₂×#10587493	C₂	semidirect
7. #11762827	C₂	semidirect
8. #11762765	C₂	semidirect
9. C₁₇×#421	C₂	semidirect
14. #10587636	C₄	semidirect
15. #10587636	C₄	semidirect
19. #10587634	C₂²	semidirect
20. C₂×#63231	C₄	semidirect
21. C₂×#63230	C₄	semidirect
22. C₂×#63229	C₄	semidirect
23. C₂×#63228	C₄	semidirect
28. #10587593	C₂²	semidirect
31. C₁₇×#104	C₂²	semidirect
32. #10587493	C₂²	semidirect
33. #10587493	C₂²	semidirect
34. #10587476	C₂²	semidirect
35. #10587474	C₂²	semidirect
36. C₄×#3622	C₂²	semidirect
142. #421	D₁₇	semidirect
174. #104	C₂×D₁₇	semidirect
177. C₁₇	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587636	2
#11763576	2
#11763575	2
#11762928	2
#11762923	2
C₂×#10587493	2
#11762827	2
#11762765	2
C₁₇×#421	2
#2957	17