Subgroups — #14545563

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#11763856	C₂	1	2	yes	maximal, complemented
3	#11763855	C₂	1	2	yes	maximal, complemented
4	#11763852	C₂	1	4	yes	maximal, complemented
5	C₂×#10587535	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	#10587535	C₂²	1	2	yes	complemented
31	#10587535	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	2	yes	—
55	#63243	C₂³	1	4	yes	—
56	#63242	—	2	4	no	—
57	#63242	C₂³	1	2	yes	—
58	Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
59	SD₁₆×C₁₇	C₂×C₄	1	2	yes	—
60	D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
62	#63231	C₂×C₄	1	2	yes	—
63	#63230	C₂×C₄	1	2	yes	—
64	#63229	C₂×C₄	1	2	yes	—
65	#63228	C₂×C₄	1	2	yes	—
66	C₂×#3622	C₂×C₄	1	1	yes	characteristic
67	Dic₆₈	C₂×C₄	1	1	yes	characteristic
68	D₁₃₆	C₂×C₄	1	1	yes	characteristic
69	#63219	C₂×C₄	1	2	yes	—
70	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
71	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
72	C₈×D₁₇	C₂³	1	1	yes	characteristic
73	C₈×D₁₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₇	—	2	2	no	—
75	C₂²×D₁₇	#41	1	1	yes	characteristic
76	C₂×#595	—	2	2	no	—
77	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
78	D₄×C₁₇	—	2	2	no	—
79	D₄×C₁₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₇	—	2	2	no	—
81	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₁₇	—	2	2	no	—
83	C₂²⋊D₁₇	—	2	2	no	—
84	C₂²⋊D₁₇	—	2	2	no	—
85	C₂×Dic₁₇	—	2	2	no	—
86	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
87	D₆₈	—	2	2	no	—
88	D₆₈	C₂²×C₄	1	2	yes	—
89	C₄×D₁₇	—	2	2	no	—
90	C₄×D₁₇	—	2	2	no	—
91	C₄×D₁₇	—	2	2	no	—
92	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
94	C₄×D₁₇	#41	1	1	yes	characteristic
95	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
96	Dic₃₄	—	2	2	no	—
97	Dic₃₄	C₂²×C₄	1	2	yes	—
98	#3624	—	2	4	no	—
99	#3624	C₂×D₄	1	4	yes	—
100	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	#3622	C₂²×C₄	1	1	yes	characteristic
103	#3622	C₂²×C₄	1	1	yes	characteristic
104	#2957	—	17	1	no	maximal, Sylow
105	C₂²×C₁₇	—	2	2	no	—
106	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
107	C₂×D₁₇	—	2	2	no	—
108	C₂×D₁₇	—	4	1	no	—
109	C₂×D₁₇	—	8	2	no	—
110	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
111	C₂×D₁₇	C₂×#41	1	1	yes	characteristic
112	#595	—	4	2	no	—
113	C₄×C₁₇	—	2	2	no	—
114	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
115	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
116	Dic₁₇	—	2	2	no	—
117	Dic₁₇	C₄×D₄	1	1	yes	characteristic
118	Dic₁₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×#135	—	17	1	no	—
120	#568	—	17	2	no	—
121	#444	—	17	2	no	—
122	#444	—	17	2	no	—
123	#443	—	17	4	no	—
124	C₂×#108	—	17	1	no	—
125	#421	—	17	2	no	—
126	#405	—	17	1	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	—	4	2	no	—
129	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
130	D₁₇	—	2	1	no	—
131	D₁₇	—	2	1	no	—
132	D₁₇	—	4	2	no	—
133	C₂×#41	—	17	2	no	—
134	#135	—	17	2	no	—
135	#135	—	17	1	no	—
136	#135	—	17	2	no	—
137	#135	—	17	1	no	—
138	#135	—	17	1	no	—
139	#135	#595	1	1	yes	characteristic, complemented
140	C₂×Q₁₆	—	17	1	no	—
141	C₂×SD₁₆	—	17	2	no	—
142	C₂×D₈	—	17	1	no	—
143	#131	—	17	4	no	—
144	#131	—	17	4	no	—
145	#131	—	34	4	no	—
146	C₂×M₁₆	—	17	1	no	—
147	C₂×M₁₆	—	17	2	no	—
148	C₂×M₁₆	—	34	2	no	—
149	C₂×M₁₆	—	34	2	no	—
150	C₂²×C₈	—	17	1	no	—
151	#117	—	17	1	no	—
152	#108	—	17	2	no	—
153	#108	—	17	2	no	—
154	#104	—	17	4	no	—
155	#104	—	17	4	no	—
156	#97	—	17	2	no	—
157	C₄×C₈	—	17	2	no	—
158	C₁₇	#2957	1	1	yes	characteristic, complemented, Sylow
159	#41	—	17	2	no	—
160	#41	—	17	2	no	—
161	#41	—	17	2	no	—
162	#41	—	34	2	no	—
163	#41	—	34	2	no	—
164	#41	C₂×#595	1	2	yes	—
165	C₂×Q₈	—	17	2	no	—
166	C₂×D₄	—	17	2	no	—
167	C₂×D₄	—	34	2	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	—	34	2	no	—
170	Q₁₆	—	17	2	no	—
171	Q₁₆	—	17	1	no	—
172	Q₁₆	C₂×#595	1	1	yes	characteristic
173	SD₁₆	—	17	2	no	—
174	SD₁₆	—	17	2	no	—
175	SD₁₆	—	17	2	no	—
176	SD₁₆	C₂×#595	1	2	yes	—
177	D₈	—	17	2	no	—
178	D₈	—	17	1	no	—
179	D₈	C₂×#595	1	1	yes	characteristic
180	M₁₆	—	17	2	no	—
181	M₁₆	—	17	2	no	—
182	M₁₆	—	17	4	no	—
183	M₁₆	—	34	4	no	—
184	M₁₆	—	34	4	no	—
185	M₁₆	—	34	2	no	—
186	C₂×C₈	—	17	2	no	—
187	C₂×C₈	—	17	1	no	—
188	C₂×C₈	—	17	1	no	—
189	C₂×C₈	—	17	1	no	—
190	C₂×C₈	—	17	1	no	—
191	C₂×C₈	—	17	1	no	—
192	C₂×C₈	—	34	4	no	—
193	C₂×C₈	—	34	4	no	—
194	C₂×C₈	—	34	2	no	—
195	C₂×C₈	C₂×#595	1	1	yes	characteristic
196	C₄⋊C₄	—	34	1	no	—
197	C₂²⋊C₄	—	34	1	no	—
198	C₄²	—	17	2	no	—
199	C₂³	—	17	1	no	—
200	C₂³	—	34	2	no	—
201	Q₈	—	17	2	no	—
202	Q₈	—	34	2	no	—
203	Q₈	C₂²×#595	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	17	2	no	—
206	D₄	—	34	2	no	—
207	D₄	—	34	2	no	—
208	D₄	—	34	2	no	—
209	D₄	—	34	2	no	—
210	D₄	C₂²×#595	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	17	1	no	—
213	C₂×C₄	—	17	1	no	—
214	C₂×C₄	—	17	1	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	34	2	no	—
218	C₂×C₄	—	34	2	no	—
219	C₂×C₄	—	34	2	no	—
220	C₂×C₄	—	34	2	no	—
221	C₂×C₄	—	34	2	no	—
222	C₂×C₄	C₂²×#595	1	1	yes	characteristic
223	C₈	—	17	4	no	—
224	C₈	—	17	1	no	—
225	C₈	—	17	1	no	—
226	C₈	—	34	4	no	—
227	C₈	C₂²×#595	1	1	yes	characteristic
228	C₈	C₂²×#595	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	17	1	no	—
231	C₂²	—	17	1	no	—
232	C₂²	—	34	2	no	—
233	C₂²	—	68	1	no	—
234	C₂²	—	136	2	no	—
235	C₂²	D₄×#595	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	17	1	no	—
238	C₄	—	17	1	no	—
239	C₄	—	34	2	no	—
240	C₄	—	68	2	no	—
241	C₄	D₄×#595	1	1	yes	characteristic
242	C₄	C₂³×#595	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	34	1	no	—
246	C₂	—	34	1	no	—
247	C₂	—	68	2	no	—
248	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. #11763856	C₂	semidirect
3. #11763855	C₂	semidirect
4. #11763852	C₂	semidirect
5. C₂×#10587535	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. #10587535	C₂²	semidirect
31. #10587535	C₂²	semidirect
32. #10587532	C₂²	semidirect
33. C₈×#595	C₂²	semidirect
139. #135	#595	semidirect
158. 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
#11763856	2
#11763855	2
#11763852	2
C₂×#10587535	2
#11763806	2
#11763772	2
#135×D₁₇	2
#2957	17