Subgroups — #16952699

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762635	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762614	C₂	1	2	yes	maximal
4	#14529645	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₈	C₂²	1	2	yes	—
7	C₂×Dic₁₃₆	—	2	2	no	—
8	C₂×Dic₁₃₆	—	2	2	no	—
9	C₂×Dic₁₃₆	C₄	1	4	yes	complemented
10	C₂×#10587489	C₂²	1	2	yes	—
11	C₂×#10587487	—	2	2	no	—
12	#11762715	—	2	2	no	—
13	#11762635	—	2	2	no	—
14	#11762614	—	2	4	no	—
15	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₇	—	2	2	no	—
19	C₂²×Dic₃₄	—	2	2	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×Dic₆₈	—	2	2	no	—
25	C₂×Dic₆₈	—	2	2	no	—
26	C₂×Dic₆₈	—	2	2	no	—
27	C₂×Dic₆₈	—	2	2	no	—
28	C₂×Dic₆₈	—	2	2	no	—
29	C₂×Dic₆₈	C₂×C₄	1	4	yes	—
30	C₂×C₁₆×C₁₇	—	2	1	no	—
31	C₂×C₁₆×C₁₇	—	2	1	no	—
32	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
34	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
35	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
36	C₄×C₈×C₁₇	—	2	1	no	—
37	C₄×C₈×C₁₇	—	2	1	no	—
38	#10587502	—	4	4	no	—
39	#10587489	—	2	2	no	—
40	#10587489	—	2	2	no	—
41	#10587487	—	4	4	no	—
42	Dic₁₃₆	—	2	8	no	—
43	C₂²×C₄×C₁₇	D₈	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	—	4	2	no	—
46	C₂²×Dic₁₇	—	4	2	no	—
47	C₂×Dic₃₄	—	2	4	no	—
48	C₂×Dic₃₄	—	4	2	no	—
49	C₂×Dic₃₄	—	4	2	no	—
50	C₂×Dic₃₄	—	4	2	no	—
51	C₂×Dic₃₄	—	4	2	no	—
52	C₂×Dic₃₄	—	4	2	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	D₈	1	2	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
62	C₄²×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	—	2	1	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	C₄⋊Dic₁₇	—	4	2	no	—
66	Dic₆₈	—	2	4	no	—
67	Dic₆₈	—	2	4	no	—
68	Dic₆₈	—	4	2	no	—
69	Dic₆₈	—	4	2	no	—
70	C₁₆×C₁₇	—	2	2	no	—
71	C₁₆×C₁₇	C₂²⋊C₄	1	4	yes	—
72	#13236	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	—	2	2	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	Q₃₂	1	4	yes	complemented
77	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
83	C₂×Dic₁₇	—	4	4	no	—
84	C₂×Dic₁₇	—	8	2	no	—
85	C₂×Dic₁₇	—	8	2	no	—
86	C₂×Dic₁₇	—	8	2	no	—
87	C₂×Dic₁₇	—	8	4	no	—
88	Dic₃₄	—	4	4	no	—
89	Dic₃₄	—	4	4	no	—
90	Dic₃₄	—	8	2	no	—
91	Dic₃₄	—	8	2	no	—
92	C₈×C₁₇	—	2	2	no	—
93	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
95	C₈×C₁₇	#121	1	1	yes	characteristic
96	C₈×C₁₇	#124	1	1	yes	characteristic
97	C₂²×Q₃₂	—	17	1	no	—
98	C₂×#366	—	17	1	no	—
99	C₂×#358	—	17	2	no	—
100	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
101	#1969	—	17	2	no	—
102	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
103	C₂²×C₁₇	#466	1	1	yes	characteristic
104	C₂²×C₁₇	#493	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
106	C₂²×C₁₇	#508	1	2	yes	—
107	C₄×C₁₇	—	2	4	no	—
108	C₄×C₁₇	#390	1	1	yes	characteristic
109	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
110	C₄×C₁₇	#466	1	1	yes	characteristic
111	C₄×C₁₇	#495	1	1	yes	characteristic
112	Dic₁₇	—	8	4	no	—
113	Dic₁₇	—	16	2	no	—
114	C₂²×Q₁₆	—	17	2	no	—
115	C₂×Q₃₂	—	17	4	no	—
116	C₂×Q₃₂	—	34	2	no	—
117	C₂×Q₃₂	—	34	2	no	—
118	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
119	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
120	C₂×#107	—	17	2	no	—
121	C₂×#103	—	34	2	no	—
122	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
123	#391	—	34	2	no	—
124	#366	—	34	2	no	—
125	#358	—	34	4	no	—
126	C₄×C₁₆	—	2	2	no	—
127	C₂×C₁₇	#1970	1	1	yes	characteristic
128	C₂×C₁₇	C₄×Q₃₂	1	2	yes	—
129	C₂×C₁₇	#2216	1	2	yes	—
130	C₂×C₁₇	#2247	1	1	yes	characteristic
131	C₂×C₁₇	#2249	1	1	yes	characteristic
132	C₂²×Q₈	—	34	2	no	—
133	C₂×Q₁₆	—	17	4	no	—
134	C₂×Q₁₆	—	34	2	no	—
135	C₂×Q₁₆	—	34	2	no	—
136	C₂×Q₁₆	—	34	2	no	—
137	C₂×Q₁₆	—	34	2	no	—
138	C₂×Q₁₆	—	34	2	no	—
139	C₂²×C₈	D₆₈	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	34	2	no	—
142	C₂×C₄²	D₆₈	1	1	yes	characteristic
143	Q₃₂	—	34	8	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
147	C₂×C₁₆	D₆₈	1	2	yes	—
148	C₂×C₁₆	D₆₈	1	2	yes	—
149	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
150	#107	—	34	2	no	—
151	#107	—	34	2	no	—
152	#103	—	68	4	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	#95	—	68	4	no	—
156	C₁₇	#13236	1	1	yes	characteristic, complemented, Sylow
157	C₂×Q₈	—	34	4	no	—
158	C₂×Q₈	—	68	2	no	—
159	C₂×Q₈	—	68	2	no	—
160	C₂×Q₈	—	68	2	no	—
161	C₂×Q₈	—	68	2	no	—
162	C₂×Q₈	—	68	2	no	—
163	C₂²×C₄	—	68	2	no	—
164	C₂²×C₄	—	68	2	no	—
165	C₂²×C₄	D₁₃₆	1	2	yes	—
166	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
167	Q₁₆	—	34	4	no	—
168	Q₁₆	—	34	4	no	—
169	Q₁₆	—	68	2	no	—
170	Q₁₆	—	68	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	D₁₃₆	1	2	yes	—
174	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
175	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
176	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
177	C₂×C₈	#63252	1	1	yes	characteristic
178	C₂×C₈	#63252	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
180	C₄⋊C₄	—	68	2	no	—
181	C₄⋊C₄	—	68	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	#63227	1	4	yes	—
186	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
187	Q₈	—	68	4	no	—
188	Q₈	—	68	4	no	—
189	Q₈	—	136	2	no	—
190	Q₈	—	136	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	68	4	no	—
194	C₂×C₄	—	136	4	no	—
195	C₂×C₄	—	136	2	no	—
196	C₂×C₄	—	136	2	no	—
197	C₂×C₄	—	136	2	no	—
198	C₂×C₄	Dic₁₃₆	1	4	yes	complemented
199	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
200	C₂×C₄	#10587560	1	1	yes	characteristic
201	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
202	C₂×C₄	#10587590	1	1	yes	characteristic
203	C₂×C₄	#10587590	1	1	yes	characteristic
204	C₂×C₄	#10587616	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₄×D₆₈	1	1	yes	characteristic
207	C₈	#10587561	1	1	yes	characteristic
208	C₈	C₂×#63227	1	1	yes	characteristic
209	C₈	#10587616	1	1	yes	characteristic
210	C₂²	C₄×D₁₃₆	1	1	yes	characteristic
211	C₂²	#11763045	1	1	yes	characteristic
212	C₂²	#11763186	1	1	yes	characteristic
213	C₂²	C₂×Dic₁₃₆	1	2	yes	—
214	C₂²	#11763241	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	272	2	no	—
217	C₄	—	136	4	no	—
218	C₄	#11762711	1	1	yes	characteristic
219	C₄	C₄×D₁₃₆	1	1	yes	characteristic
220	C₄	#11763045	1	1	yes	characteristic
221	C₄	#11763193	1	1	yes	characteristic
222	C₂	#14529650	1	1	yes	characteristic
223	C₂	C₄×Dic₁₃₆	1	2	yes	—
224	C₂	#14530875	1	2	yes	—
225	C₂	#14531037	1	1	yes	characteristic
226	C₂	#14531045	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#13236
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₂×Dic₁₃₆	C₄	semidirect
76. C₂×C₄×C₁₇	Q₃₂	semidirect
156. C₁₇	#13236	semidirect
198. C₂×C₄	Dic₁₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₃₆	2
C₂×#11762635	2
C₂×#11762614	2
#14529645	2
C₂×C₄×C₁₆×C₁₇	2
#13236	17