Subgroups — #14545561

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763888	C₂	1	2	yes	maximal, complemented
2	#11763854	C₂	1	4	yes	maximal, complemented
3	#11763853	C₂	1	2	yes	maximal, complemented
4	#11763851	C₂	1	2	yes	maximal, complemented
5	C₂×#10587536	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763806	C₂	1	2	yes	maximal, complemented
7	#11763772	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₇	C₂	1	1	yes	maximal, characteristic
9	#41×D₁₇	C₂²	1	2	yes	—
10	#10587680	C₂²	1	4	yes	complemented
11	#10587678	—	2	4	no	—
12	#10587678	C₂²	1	4	yes	complemented
13	#10587677	C₂²	1	1	yes	characteristic
14	D₁₇⋊M₁₆	—	2	2	no	—
15	D₁₇⋊M₁₆	C₂²	1	1	yes	characteristic
16	C₂×#63243	—	2	2	no	—
17	C₂×#63243	C₂²	1	2	yes	—
18	C₁₇×#135	C₄	1	1	yes	characteristic, complemented
19	#10587636	C₄	1	2	yes	complemented
20	#10587605	C₂²	1	1	yes	characteristic
21	Q₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
22	#10587602	C₂²	1	2	yes	—
23	SD₁₆×D₁₇	C₄	1	2	yes	complemented
24	#10587598	C₂²	1	1	yes	characteristic
25	D₈×D₁₇	C₄	1	1	yes	characteristic, complemented
26	#10587590	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	#10587550	C₂²	1	4	yes	complemented
29	#10587548	C₂²	1	4	yes	complemented
30	#10587536	C₂²	1	2	yes	complemented
31	#10587536	C₂²	1	2	yes	complemented
32	#10587532	C₂²	1	2	yes	complemented
33	C₈×#595	C₂²	1	2	yes	complemented
34	#41×C₁₇	C₂×C₄	1	2	yes	—
35	#63256	C₂³	1	2	yes	—
36	Q₈×D₁₇	C₂×C₄	1	2	yes	—
37	#63254	—	2	2	no	—
38	#63254	C₂³	1	2	yes	—
39	D₄×D₁₇	—	2	2	no	—
40	D₄×D₁₇	C₂×C₄	1	2	yes	—
41	#63252	—	2	2	no	—
42	#63252	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₇	—	2	2	no	—
44	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
45	C₂²⋊#595	—	2	1	no	—
46	#63247	—	2	4	no	—
47	#63247	D₄	1	2	yes	—
48	C₂×#3624	—	2	2	no	—
49	C₂×#3624	—	2	4	no	—
50	C₄⋊#595	—	2	1	no	—
51	C₄×#595	C₂³	1	2	yes	—
52	#63243	—	2	2	no	—
53	#63243	—	2	4	no	—
54	#63243	D₄	1	1	yes	characteristic
55	#63243	D₄	1	1	yes	characteristic
56	#63243	C₂³	1	4	yes	—
57	#63242	—	2	4	no	—
58	#63242	C₂³	1	2	yes	—
59	Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
60	SD₁₆×C₁₇	C₂×C₄	1	2	yes	—
61	D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
63	#63231	C₂×C₄	1	2	yes	—
64	#63230	C₂×C₄	1	2	yes	—
65	#63229	C₂×C₄	1	2	yes	—
66	#63228	C₂×C₄	1	2	yes	—
67	C₂×#3622	C₂×C₄	1	1	yes	characteristic
68	Dic₆₈	C₂×C₄	1	1	yes	characteristic
69	D₁₃₆	C₂×C₄	1	1	yes	characteristic
70	#63219	C₂×C₄	1	2	yes	—
71	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
72	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
73	C₈×D₁₇	C₂³	1	1	yes	characteristic
74	C₈×D₁₇	C₂³	1	1	yes	characteristic
75	C₂²×D₁₇	—	2	2	no	—
76	C₂²×D₁₇	#41	1	1	yes	characteristic
77	C₂×#595	—	2	2	no	—
78	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
79	D₄×C₁₇	—	2	2	no	—
80	D₄×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊D₁₇	—	2	2	no	—
84	C₂²⋊D₁₇	—	2	2	no	—
85	C₂²⋊D₁₇	—	2	2	no	—
86	C₂×Dic₁₇	—	2	2	no	—
87	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
88	D₆₈	—	2	2	no	—
89	D₆₈	C₂²×C₄	1	2	yes	—
90	C₄×D₁₇	—	2	2	no	—
91	C₄×D₁₇	—	2	2	no	—
92	C₄×D₁₇	—	2	2	no	—
93	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
95	C₄×D₁₇	#41	1	1	yes	characteristic
96	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
97	Dic₃₄	—	2	2	no	—
98	Dic₃₄	C₂²×C₄	1	2	yes	—
99	#3624	—	2	4	no	—
100	#3624	C₂×D₄	1	2	yes	—
101	#3624	C₂×D₄	1	2	yes	—
102	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
104	#3622	C₂²×C₄	1	1	yes	characteristic
105	#3622	C₂²×C₄	1	1	yes	characteristic
106	#2957	—	17	1	no	maximal, Sylow
107	C₂²×C₁₇	—	2	2	no	—
108	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
109	C₂×D₁₇	—	2	2	no	—
110	C₂×D₁₇	—	4	1	no	—
111	C₂×D₁₇	—	8	2	no	—
112	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
113	C₂×D₁₇	C₂×#41	1	1	yes	characteristic
114	#595	—	4	2	no	—
115	C₄×C₁₇	—	2	2	no	—
116	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
117	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
118	Dic₁₇	—	2	2	no	—
119	Dic₁₇	C₄×D₄	1	1	yes	characteristic
120	Dic₁₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×#135	—	17	1	no	—
122	#568	—	17	2	no	—
123	#444	—	17	4	no	—
124	#443	—	17	2	no	—
125	#443	—	17	2	no	—
126	C₂×#108	—	17	1	no	—
127	#421	—	17	2	no	—
128	#405	—	17	1	no	—
129	C₂×C₁₇	—	2	1	no	—
130	C₂×C₁₇	—	4	2	no	—
131	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
132	D₁₇	—	2	1	no	—
133	D₁₇	—	2	1	no	—
134	D₁₇	—	4	2	no	—
135	C₂×#41	—	17	2	no	—
136	#135	—	17	2	no	—
137	#135	—	17	1	no	—
138	#135	—	17	2	no	—
139	#135	—	17	1	no	—
140	#135	—	17	1	no	—
141	#135	#595	1	1	yes	characteristic, complemented
142	C₂×Q₁₆	—	17	1	no	—
143	C₂×SD₁₆	—	17	2	no	—
144	C₂×D₈	—	17	1	no	—
145	#131	—	17	4	no	—
146	#131	—	17	4	no	—
147	#131	—	34	4	no	—
148	C₂×M₁₆	—	17	1	no	—
149	C₂×M₁₆	—	17	2	no	—
150	C₂×M₁₆	—	34	2	no	—
151	C₂×M₁₆	—	34	2	no	—
152	C₂²×C₈	—	17	1	no	—
153	#117	—	17	1	no	—
154	#108	—	17	2	no	—
155	#108	—	17	2	no	—
156	#104	—	17	4	no	—
157	#104	—	17	4	no	—
158	#97	—	17	2	no	—
159	C₄×C₈	—	17	2	no	—
160	C₁₇	#2957	1	1	yes	characteristic, complemented, Sylow
161	#41	—	17	2	no	—
162	#41	—	17	2	no	—
163	#41	—	17	2	no	—
164	#41	—	34	2	no	—
165	#41	—	34	2	no	—
166	#41	C₂×#595	1	2	yes	—
167	C₂×Q₈	—	17	2	no	—
168	C₂×D₄	—	17	2	no	—
169	C₂×D₄	—	34	2	no	—
170	C₂²×C₄	—	17	1	no	—
171	C₂²×C₄	—	34	2	no	—
172	Q₁₆	—	17	2	no	—
173	Q₁₆	—	17	1	no	—
174	Q₁₆	C₂×#595	1	1	yes	characteristic
175	SD₁₆	—	17	2	no	—
176	SD₁₆	—	17	2	no	—
177	SD₁₆	—	17	2	no	—
178	SD₁₆	C₂×#595	1	2	yes	—
179	D₈	—	17	2	no	—
180	D₈	—	17	1	no	—
181	D₈	C₂×#595	1	1	yes	characteristic
182	M₁₆	—	17	1	no	—
183	M₁₆	—	17	1	no	—
184	M₁₆	—	17	2	no	—
185	M₁₆	—	17	4	no	—
186	M₁₆	—	34	4	no	—
187	M₁₆	—	34	4	no	—
188	M₁₆	—	34	2	no	—
189	C₂×C₈	—	17	2	no	—
190	C₂×C₈	—	17	1	no	—
191	C₂×C₈	—	17	1	no	—
192	C₂×C₈	—	17	1	no	—
193	C₂×C₈	—	17	1	no	—
194	C₂×C₈	—	17	1	no	—
195	C₂×C₈	—	34	4	no	—
196	C₂×C₈	—	34	4	no	—
197	C₂×C₈	—	34	2	no	—
198	C₂×C₈	C₂×#595	1	1	yes	characteristic
199	C₄⋊C₄	—	34	1	no	—
200	C₂²⋊C₄	—	34	1	no	—
201	C₄²	—	17	2	no	—
202	C₂³	—	17	1	no	—
203	C₂³	—	34	2	no	—
204	Q₈	—	17	2	no	—
205	Q₈	—	34	2	no	—
206	Q₈	C₂²×#595	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	17	2	no	—
209	D₄	—	34	2	no	—
210	D₄	—	34	2	no	—
211	D₄	—	34	2	no	—
212	D₄	—	34	2	no	—
213	D₄	C₂²×#595	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	17	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	34	2	no	—
221	C₂×C₄	—	34	2	no	—
222	C₂×C₄	—	34	2	no	—
223	C₂×C₄	—	34	2	no	—
224	C₂×C₄	—	34	2	no	—
225	C₂×C₄	C₂²×#595	1	1	yes	characteristic
226	C₈	—	17	2	no	—
227	C₈	—	17	2	no	—
228	C₈	—	17	1	no	—
229	C₈	—	17	1	no	—
230	C₈	—	34	4	no	—
231	C₈	C₂²×#595	1	1	yes	characteristic
232	C₈	C₂²×#595	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	17	1	no	—
235	C₂²	—	17	1	no	—
236	C₂²	—	34	2	no	—
237	C₂²	—	68	1	no	—
238	C₂²	—	136	2	no	—
239	C₂²	D₄×#595	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	17	1	no	—
242	C₄	—	17	1	no	—
243	C₄	—	34	2	no	—
244	C₄	—	68	2	no	—
245	C₄	D₄×#595	1	1	yes	characteristic
246	C₄	C₂³×#595	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	—	34	1	no	—
250	C₂	—	34	1	no	—
251	C₂	—	68	2	no	—
252	C₂	C₂×D₄×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#595
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×#595
Fitting	C₁₇×#135	C₄
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	#135×D₁₇	C₂
cosocle	C₄×D₁₇	C₂⁴
hypercenter	#135	#595
hypocenter	C₁₇	#2957
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #11763888	C₂	semidirect
2. #11763854	C₂	semidirect
3. #11763853	C₂	semidirect
4. #11763851	C₂	semidirect
5. C₂×#10587536	C₂	semidirect
6. #11763806	C₂	semidirect
7. #11763772	C₂	semidirect
10. #10587680	C₂²	semidirect
12. #10587678	C₂²	semidirect
18. C₁₇×#135	C₄	semidirect
19. #10587636	C₄	semidirect
21. Q₁₆×D₁₇	C₄	semidirect
23. SD₁₆×D₁₇	C₄	semidirect
25. D₈×D₁₇	C₄	semidirect
26. #10587590	C₄	semidirect
28. #10587550	C₂²	semidirect
29. #10587548	C₂²	semidirect
30. #10587536	C₂²	semidirect
31. #10587536	C₂²	semidirect
32. #10587532	C₂²	semidirect
33. C₈×#595	C₂²	semidirect
141. #135	#595	semidirect
160. C₁₇	#2957	semidirect

Maximal subgroups

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

subgroup	index
#11763888	2
#11763854	2
#11763853	2
#11763851	2
C₂×#10587536	2
#11763806	2
#11763772	2
#135×D₁₇	2
#2957	17