Subgroups — #16952909

Order 4352.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14530540	C₂	1	1	yes	maximal, characteristic
3	#14526891	C₂	1	4	yes	maximal
4	C₄×C₁₇×#106	C₂	1	1	yes	maximal, characteristic
5	#11763366	—	2	1	no	—
6	C₂×#10587483	C₂²	1	2	yes	—
7	#11763214	C₄	1	1	yes	characteristic, complemented
8	#11763214	C₄	1	1	yes	characteristic, complemented
9	#11763211	—	2	1	no	—
10	#11763211	C₄	1	2	yes	complemented
11	#11763207	—	2	1	no	—
12	#11762847	—	2	1	no	—
13	#11762847	C₂²	1	2	yes	—
14	C₄⋊#63214	—	2	1	no	—
15	C₄⋊#63214	—	2	1	no	—
16	C₂×C₁₇×#106	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
19	C₁₇×#340	—	2	1	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄²×C₁₇	—	4	1	no	—
24	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
25	#10587612	—	2	2	no	—
26	#10587610	—	4	1	no	—
27	#10587610	—	4	1	no	—
28	#10587556	—	2	2	no	—
29	#10587555	—	2	1	no	—
30	#10587555	—	2	1	no	—
31	C₄×Dic₃₄	—	4	1	no	—
32	C₄×Dic₃₄	—	4	1	no	—
33	C₁₇×#106	—	2	2	no	—
34	C₁₇×#106	Q₈	1	4	yes	complemented
35	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
39	C₁₇×#95	—	4	1	no	—
40	#10587489	—	2	1	no	—
41	#10587489	—	2	1	no	—
42	#10587489	C₂×C₄	1	2	yes	—
43	#10587489	C₂×C₄	1	2	yes	—
44	#10587488	—	2	2	no	—
45	#10587488	—	2	2	no	—
46	#10587483	—	2	4	no	—
47	C₂×#63214	—	2	2	no	—
48	C₂²×C₄×C₁₇	—	4	2	no	—
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₃₄	—	4	1	no	—
53	C₂×Dic₃₄	—	4	1	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₇	#41	1	2	yes	—
61	C₄⋊C₄×C₁₇	—	2	4	no	—
62	C₄⋊C₄×C₁₇	—	4	2	no	—
63	C₄²×C₁₇	—	8	2	no	—
64	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
67	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
68	C₂²⋊Dic₁₇	—	4	2	no	—
69	C₄⋊Dic₁₇	—	2	2	no	—
70	C₄⋊Dic₁₇	—	2	2	no	—
71	C₄⋊Dic₁₇	—	4	1	no	—
72	C₄⋊Dic₁₇	—	4	1	no	—
73	#63225	—	4	2	no	—
74	#63225	—	4	2	no	—
75	C₄×Dic₁₇	—	4	2	no	—
76	#63214	—	4	2	no	—
77	#13278	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	4	1	no	—
81	C₂×C₄×C₁₇	—	4	4	no	—
82	C₂×C₄×C₁₇	—	8	2	no	—
83	C₂×C₄×C₁₇	—	8	2	no	—
84	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#137	1	2	yes	—
92	C₂×Dic₁₇	—	4	2	no	—
93	C₂×Dic₁₇	—	4	2	no	—
94	Dic₃₄	—	8	2	no	—
95	Dic₃₄	—	8	2	no	—
96	C₈×C₁₇	—	2	1	no	—
97	C₈×C₁₇	—	2	1	no	—
98	C₈×C₁₇	C₄×Q₈	1	2	yes	—
99	C₈×C₁₇	#124	1	1	yes	characteristic
100	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
101	#3160	—	17	1	no	—
102	#2150	—	17	1	no	—
103	C₄×#106	D₁₇	1	1	yes	characteristic
104	#1386	—	17	4	no	—
105	C₂²×C₁₇	—	2	1	no	—
106	C₂²×C₁₇	—	2	1	no	—
107	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
108	C₂²×C₁₇	#474	1	2	yes	—
109	C₄×C₁₇	—	2	1	no	—
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	—	8	4	no	—
113	C₄×C₁₇	#360	1	2	yes	—
114	C₄×C₁₇	#391	1	1	yes	characteristic
115	C₄×C₁₇	#419	1	1	yes	characteristic
116	C₄×C₁₇	#475	1	2	yes	—
117	Dic₁₇	—	8	2	no	—
118	Dic₁₇	—	8	2	no	—
119	#533	—	34	1	no	—
120	C₂×M₃₂	—	17	2	no	—
121	#500	—	17	1	no	—
122	#500	—	17	1	no	—
123	#499	—	17	2	no	—
124	#499	—	34	1	no	—
125	#498	—	34	1	no	—
126	#427	—	17	2	no	—
127	#427	—	34	1	no	—
128	C₂×#106	C₂×D₁₇	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₁₆	—	34	1	no	—
132	C₄⋊C₁₆	—	34	1	no	—
133	#340	—	2	1	no	—
134	C₂×C₁₇	—	2	2	no	—
135	C₂×C₁₇	#1988	1	1	yes	characteristic
136	C₂×C₁₇	C₂×#360	1	1	yes	characteristic
137	C₂×C₁₇	#2145	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	34	1	no	—
141	C₄⋊Q₈	—	34	1	no	—
142	#125	—	34	2	no	—
143	C₂²⋊Q₈	—	68	1	no	—
144	C₂²⋊Q₈	—	68	1	no	—
145	C₄×Q₈	—	68	1	no	—
146	C₄×Q₈	—	68	1	no	—
147	#117	—	34	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
151	M₃₂	—	34	4	no	—
152	C₂×C₁₆	—	34	2	no	—
153	#107	—	17	2	no	—
154	#107	—	17	2	no	—
155	#107	—	34	1	no	—
156	#107	—	34	1	no	—
157	#106	—	2	2	no	—
158	#106	—	34	2	no	—
159	#106	—	34	2	no	—
160	#106	Dic₃₄	1	4	yes	complemented
161	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
162	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
163	C₄×C₈	D₆₈	1	1	yes	characteristic
164	C₄×C₈	D₆₈	1	1	yes	characteristic
165	#95	—	4	1	no	—
166	C₁₇	#13278	1	1	yes	characteristic, complemented, Sylow
167	C₂×Q₈	—	68	1	no	—
168	C₂×Q₈	—	68	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	#63228	1	1	yes	characteristic
171	C₂²×C₄	#63228	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	#63227	1	1	yes	characteristic
175	C₂×C₈	#63227	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
178	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
179	C₂×C₈	#63252	1	2	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	34	2	no	—
183	C₄⋊C₄	—	34	2	no	—
184	C₄⋊C₄	—	68	2	no	—
185	C₄⋊C₄	—	68	2	no	—
186	C₄⋊C₄	—	68	1	no	—
187	C₄⋊C₄	—	68	1	no	—
188	C₂²⋊C₄	—	68	2	no	—
189	C₄²	—	8	2	no	—
190	C₄²	—	68	2	no	—
191	C₄²	#63227	1	1	yes	characteristic
192	C₄²	#63227	1	1	yes	characteristic
193	C₄²	#63230	1	1	yes	characteristic
194	C₄²	#63230	1	1	yes	characteristic
195	C₁₆	—	68	2	no	—
196	C₂³	C₂×#63228	1	1	yes	characteristic
197	Q₈	—	136	2	no	—
198	Q₈	—	136	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	68	2	no	—
205	C₂×C₄	—	68	2	no	—
206	C₂×C₄	#10587480	1	1	yes	characteristic
207	C₂×C₄	#10587480	1	1	yes	characteristic
208	C₂×C₄	#10587480	1	1	yes	characteristic
209	C₂×C₄	#10587480	1	1	yes	characteristic
210	C₂×C₄	#10587610	1	2	yes	—
211	C₂×C₄	C₂×#63227	1	1	yes	characteristic
212	C₂×C₄	C₂×#63230	1	1	yes	characteristic
213	C₂×C₄	#10587637	1	2	yes	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	C₄×Dic₃₄	1	2	yes	—
217	C₈	#10587555	1	1	yes	characteristic
218	C₈	#10587561	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#10587480	1	1	yes	characteristic
222	C₂²	#11763089	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	4	no	—
227	C₄	—	136	2	no	—
228	C₄	—	136	2	no	—
229	C₄	#11762618	1	2	yes	—
230	C₄	#11762715	1	1	yes	characteristic
231	C₄	#11762817	1	1	yes	characteristic
232	C₄	#11763094	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	#14529736	1	1	yes	characteristic
235	C₂	C₂×#11762618	1	1	yes	characteristic
236	C₂	#14530519	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587480
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×#106	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×#106	D₁₇
hypocenter	C₁₇	#13278
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
7. #11763214	C₄	semidirect
8. #11763214	C₄	semidirect
10. #11763211	C₄	semidirect
34. C₁₇×#106	Q₈	semidirect
160. #106	Dic₃₄	semidirect
166. C₁₇	#13278	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14530540	2
#14526891	2
C₄×C₁₇×#106	2
#13278	17