Subgroups — #135×#595 #14545534

Order 2176.

500 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#595	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#595	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#595	C₂	1	4	yes	maximal, complemented
4	D₈×#595	C₂	1	2	yes	maximal, complemented
5	#11763819	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763799	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#595	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₇	C₂	1	1	yes	maximal, characteristic
9	#41×D₁₇	C₂²	1	2	yes	—
10	Q₈×#595	C₂²	1	4	yes	—
11	D₄×#595	—	2	4	no	—
12	D₄×#595	C₂²	1	4	yes	—
13	#10587677	—	2	2	no	—
14	#10587677	C₂²	1	2	yes	—
15	C₂×C₄×#595	—	2	2	no	—
16	C₂×C₄×#595	C₂²	1	1	yes	characteristic
17	C₂×#63242	C₂²	1	1	yes	characteristic
18	C₁₇×#135	C₄	1	1	yes	characteristic, complemented
19	#10587636	C₄	1	2	yes	complemented
20	#10587605	C₄	1	1	yes	characteristic, complemented
21	Q₁₆×D₁₇	C₂²	1	1	yes	characteristic
22	#10587602	C₄	1	2	yes	complemented
23	SD₁₆×D₁₇	C₂²	1	2	yes	—
24	#10587598	C₄	1	1	yes	characteristic, complemented
25	D₈×D₁₇	C₂²	1	1	yes	characteristic
26	#10587590	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	#10587549	C₂²	1	4	yes	—
29	#10587547	C₂²	1	4	yes	—
30	#10587534	C₂²	1	2	yes	—
31	#10587533	C₂²	1	2	yes	—
32	C₈×#595	C₂²	1	2	yes	—
33	C₈×#595	C₂²	1	2	yes	—
34	C₂²×#595	—	2	4	no	—
35	C₂²×#595	D₄	1	2	yes	—
36	#41×C₁₇	C₂×C₄	1	2	yes	complemented
37	#63256	C₂×C₄	1	2	yes	complemented
38	Q₈×D₁₇	C₂³	1	2	yes	—
39	#63254	—	2	2	no	—
40	#63254	C₂×C₄	1	2	yes	complemented
41	D₄×D₁₇	—	2	2	no	—
42	D₄×D₁₇	C₂³	1	2	yes	—
43	#63252	—	2	2	no	—
44	#63252	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₁₇	—	2	2	no	—
46	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
47	C₂²⋊#595	—	2	2	no	—
48	C₂²⋊#595	—	2	4	no	—
49	C₂×#3624	—	2	1	no	—
50	C₄⋊#595	—	2	4	no	—
51	C₄⋊#595	C₂³	1	4	yes	—
52	C₄×#595	—	2	4	no	—
53	C₄×#595	—	2	2	no	—
54	C₄×#595	D₄	1	1	yes	characteristic
55	C₄×#595	D₄	1	1	yes	characteristic
56	C₄×#595	C₂³	1	2	yes	—
57	#63242	—	2	1	no	—
58	#63242	C₂³	1	2	yes	—
59	Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
60	SD₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
61	D₈×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
63	#63231	C₂×C₄	1	2	yes	complemented
64	#63230	C₂×C₄	1	2	yes	complemented
65	#63229	C₂×C₄	1	2	yes	complemented
66	#63228	C₂×C₄	1	2	yes	complemented
67	C₂×#3622	C₂×C₄	1	1	yes	characteristic, complemented
68	Dic₆₈	C₂×C₄	1	1	yes	characteristic, complemented
69	D₁₃₆	C₂×C₄	1	1	yes	characteristic, complemented
70	#63219	C₂×C₄	1	2	yes	complemented
71	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic, complemented
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	C₂×D₄	1	1	yes	characteristic
77	C₂×#595	—	2	4	no	—
78	C₂×#595	—	2	4	no	—
79	C₂×#595	—	4	8	no	—
80	C₂×#595	C₂×D₄	1	2	yes	—
81	C₂×#595	C₂×D₄	1	2	yes	—
82	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
83	D₄×C₁₇	—	2	2	no	—
84	D₄×C₁₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊D₁₇	—	2	2	no	—
88	C₂²⋊D₁₇	—	2	2	no	—
89	C₂²⋊D₁₇	—	2	2	no	—
90	C₂×Dic₁₇	—	2	2	no	—
91	C₂×Dic₁₇	#41	1	1	yes	characteristic
92	D₆₈	—	2	2	no	—
93	D₆₈	C₂²×C₄	1	2	yes	—
94	C₄×D₁₇	—	2	2	no	—
95	C₄×D₁₇	—	2	2	no	—
96	C₄×D₁₇	—	2	2	no	—
97	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
99	C₄×D₁₇	#41	1	1	yes	characteristic
100	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
101	Dic₃₄	—	2	2	no	—
102	Dic₃₄	C₂²×C₄	1	2	yes	—
103	#3624	—	2	2	no	—
104	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
106	#3622	C₂²×C₄	1	1	yes	characteristic
107	#3622	C₂²×C₄	1	1	yes	characteristic
108	C₄×#135	—	17	1	no	maximal, Sylow
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
111	C₂×D₁₇	—	2	2	no	—
112	C₂×D₁₇	—	2	2	no	—
113	C₂×D₁₇	—	4	4	no	—
114	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
115	C₂×D₁₇	C₂²×D₄	1	1	yes	characteristic
116	#595	—	2	2	no	—
117	#595	—	4	4	no	—
118	#595	#135	1	4	yes	direct product factor
119	C₄×C₁₇	—	2	2	no	—
120	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
121	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
122	Dic₁₇	—	2	2	no	—
123	Dic₁₇	C₄×D₄	1	1	yes	characteristic
124	Dic₁₇	C₂×#41	1	1	yes	characteristic
125	C₂×#135	—	17	1	no	—
126	C₄×#41	—	17	2	no	—
127	C₄×Q₁₆	—	17	2	no	—
128	C₄×SD₁₆	—	17	4	no	—
129	C₄×D₈	—	17	2	no	—
130	#427	—	17	1	no	—
131	#416	—	17	2	no	—
132	C₂×C₄×C₈	—	17	1	no	—
133	C₂×C₁₇	—	2	1	no	—
134	C₂×C₁₇	—	4	2	no	—
135	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
136	D₁₇	—	2	1	no	—
137	D₁₇	—	4	2	no	—
138	D₁₇	C₂×#135	1	2	yes	—
139	C₂×#41	—	17	2	no	—
140	#135	—	17	2	no	—
141	#135	—	17	1	no	—
142	#135	—	17	2	no	—
143	#135	—	17	1	no	—
144	#135	—	17	1	no	—
145	#135	#595	1	1	yes	characteristic, direct product factor
146	C₂×Q₁₆	—	17	1	no	—
147	C₂×SD₁₆	—	17	2	no	—
148	C₂×D₈	—	17	1	no	—
149	C₂²×C₈	—	17	1	no	—
150	C₂²×C₈	—	17	1	no	—
151	C₄×Q₈	—	17	4	no	—
152	C₄×D₄	—	17	4	no	—
153	C₄×D₄	—	34	4	no	—
154	#117	—	17	2	no	—
155	#117	—	34	2	no	—
156	C₂×C₄²	—	17	1	no	—
157	C₂×C₄²	—	34	2	no	—
158	#107	—	17	2	no	—
159	#106	—	17	2	no	—
160	#103	—	17	4	no	—
161	#102	—	17	4	no	—
162	C₄×C₈	—	17	2	no	—
163	C₄×C₈	—	17	2	no	—
164	C₁₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
165	#41	—	17	2	no	—
166	#41	—	17	2	no	—
167	#41	—	17	2	no	—
168	#41	—	34	2	no	—
169	#41	—	34	2	no	—
170	#41	C₂×#595	1	2	yes	complemented
171	C₂×Q₈	—	17	2	no	—
172	C₂×D₄	—	17	2	no	—
173	C₂×D₄	—	34	2	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂²×C₄	—	17	2	no	—
176	C₂²×C₄	—	34	2	no	—
177	C₂²×C₄	—	34	4	no	—
178	Q₁₆	—	17	2	no	—
179	Q₁₆	—	17	1	no	—
180	Q₁₆	C₂×#595	1	1	yes	characteristic, complemented
181	SD₁₆	—	17	2	no	—
182	SD₁₆	—	17	2	no	—
183	SD₁₆	—	17	2	no	—
184	SD₁₆	C₂×#595	1	2	yes	complemented
185	D₈	—	17	2	no	—
186	D₈	—	17	1	no	—
187	D₈	C₂×#595	1	1	yes	characteristic, complemented
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	—	17	1	no	—
193	C₂×C₈	—	17	2	no	—
194	C₂×C₈	—	34	1	no	—
195	C₂×C₈	—	34	1	no	—
196	C₂×C₈	C₂×#595	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	17	4	no	—
198	C₄⋊C₄	—	34	4	no	—
199	C₂²⋊C₄	—	34	4	no	—
200	C₂²⋊C₄	—	34	2	no	—
201	C₄²	—	17	1	no	—
202	C₄²	—	17	1	no	—
203	C₄²	—	17	2	no	—
204	C₄²	—	34	2	no	—
205	C₄²	—	34	4	no	—
206	C₂³	—	17	1	no	—
207	C₂³	—	34	2	no	—
208	Q₈	—	17	2	no	—
209	Q₈	—	34	2	no	—
210	Q₈	C₂²×#595	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	17	2	no	—
213	D₄	—	34	2	no	—
214	D₄	—	34	2	no	—
215	D₄	—	34	2	no	—
216	D₄	—	34	2	no	—
217	D₄	C₂²×#595	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	17	2	no	—
220	C₂×C₄	—	17	1	no	—
221	C₂×C₄	—	17	1	no	—
222	C₂×C₄	—	17	1	no	—
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	2	no	—
226	C₂×C₄	—	34	2	no	—
227	C₂×C₄	—	34	2	no	—
228	C₂×C₄	—	34	2	no	—
229	C₂×C₄	—	34	4	no	—
230	C₂×C₄	—	34	2	no	—
231	C₂×C₄	—	34	4	no	—
232	C₂×C₄	—	68	8	no	—
233	C₂×C₄	C₂²×#595	1	1	yes	characteristic
234	C₈	—	17	1	no	—
235	C₈	—	17	1	no	—
236	C₈	—	34	2	no	—
237	C₈	C₂²×#595	1	1	yes	characteristic
238	C₈	C₂²×#595	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	17	1	no	—
241	C₂²	—	17	1	no	—
242	C₂²	—	34	2	no	—
243	C₂²	—	34	2	no	—
244	C₂²	—	68	4	no	—
245	C₂²	D₄×#595	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	17	4	no	—
248	C₄	—	17	1	no	—
249	C₄	—	17	1	no	—
250	C₄	—	34	2	no	—
251	C₄	—	34	2	no	—
252	C₄	—	68	4	no	—
253	C₄	D₄×#595	1	1	yes	characteristic
254	C₄	C₂³×#595	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	17	2	no	—
258	C₂	—	34	1	no	—
259	C₂	—	68	2	no	—
260	C₂	C₂×D₄×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₄	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	C₄×#135
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#595	C₂	semidirect
2. Q₁₆×#595	C₂	semidirect
3. SD₁₆×#595	C₂	semidirect
4. D₈×#595	C₂	semidirect
5. #11763819	C₂	semidirect
6. #11763799	C₂	semidirect
7. C₂×C₈×#595	C₂	semidirect
18. C₁₇×#135	C₄	semidirect
19. #10587636	C₄	semidirect
20. #10587605	C₄	semidirect
22. #10587602	C₄	semidirect
24. #10587598	C₄	semidirect
26. #10587590	C₄	semidirect
36. #41×C₁₇	C₂×C₄	semidirect
37. #63256	C₂×C₄	semidirect
40. #63254	C₂×C₄	semidirect
44. #63252	C₂×C₄	semidirect
59. Q₁₆×C₁₇	C₂×C₄	semidirect
60. SD₁₆×C₁₇	C₂×C₄	semidirect
61. D₈×C₁₇	C₂×C₄	semidirect
62. C₂×C₈×C₁₇	C₂×C₄	semidirect
63. #63231	C₂×C₄	semidirect
64. #63230	C₂×C₄	semidirect
65. #63229	C₂×C₄	semidirect
66. #63228	C₂×C₄	semidirect
67. C₂×#3622	C₂×C₄	semidirect
68. Dic₆₈	C₂×C₄	semidirect
69. D₁₃₆	C₂×C₄	semidirect
70. #63219	C₂×C₄	semidirect
71. C₈×D₁₇	C₂×C₄	semidirect
72. C₈×D₁₇	C₂×C₄	semidirect
118. #595	#135	direct (N × H)
145. #135	#595	direct (N × H)
164. C₁₇	C₄×#135	semidirect
170. #41	C₂×#595	semidirect
180. Q₁₆	C₂×#595	semidirect
184. SD₁₆	C₂×#595	semidirect
187. D₈	C₂×#595	semidirect
196. C₂×C₈	C₂×#595	semidirect

Maximal subgroups

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

subgroup	index
#41×#595	2
Q₁₆×#595	2
SD₁₆×#595	2
D₈×#595	2
#11763819	2
#11763799	2
C₂×C₈×#595	2
#135×D₁₇	2
C₄×#135	17