Subgroups — #16933603

Order 4352.

442 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	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14529732	C₂	1	2	yes	maximal
3	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762557	C₂	1	2	yes	maximal
5	C₁₇×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	2	yes	—
7	#11763214	—	2	2	no	—
8	#11763214	C₄	1	4	yes	complemented
9	C₂×#10587489	—	2	1	no	—
10	C₂×#10587489	—	2	1	no	—
11	C₂×C₄⋊#3622	C₂²	1	2	yes	—
12	#11762571	—	2	4	no	—
13	#11762560	—	2	2	no	—
14	#11762557	—	2	4	no	—
15	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#97	—	2	1	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#336	—	2	1	no	—
20	C₂²×Dic₃₄	—	2	2	no	—
21	C₂²×C₈×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
25	C₂×C₄²×C₁₇	—	2	1	no	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂²×#3622	—	2	2	no	—
30	#10587555	—	2	2	no	—
31	#10587555	—	2	2	no	—
32	#10587555	—	2	2	no	—
33	#10587555	C₂×C₄	1	4	yes	—
34	C₁₇×C₄⋊C₈	—	2	1	no	—
35	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
36	C₁₇×#97	—	4	2	no	—
37	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₇	D₄	1	2	yes	—
39	C₁₇×#95	—	2	1	no	—
40	#10587489	—	2	4	no	—
41	#10587489	—	2	4	no	—
42	C₄⋊#3622	—	2	2	no	—
43	C₄⋊#3622	—	2	2	no	—
44	C₂²×C₄×C₁₇	—	2	2	no	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
48	C₂²×Dic₁₇	—	4	2	no	—
49	C₂×Dic₃₄	—	2	4	no	—
50	C₂×Dic₃₄	—	4	2	no	—
51	C₂×Dic₃₄	—	8	4	no	—
52	C₂×C₈×C₁₇	—	2	1	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	4	1	no	—
56	C₂×C₈×C₁₇	—	4	1	no	—
57	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
60	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
61	C₄⋊C₄×C₁₇	—	2	4	no	—
62	C₄²×C₁₇	—	4	1	no	—
63	C₄²×C₁₇	—	4	1	no	—
64	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₇	#41	1	1	yes	characteristic
68	C₄⋊Dic₁₇	—	2	4	no	—
69	C₄⋊Dic₁₇	—	2	4	no	—
70	C₄⋊Dic₁₇	—	4	2	no	—
71	C₄⋊Dic₁₇	—	4	2	no	—
72	C₂×#3622	—	4	2	no	—
73	C₂×#3622	—	4	4	no	—
74	#9739	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	4	2	no	—
78	C₂×C₄×C₁₇	—	4	2	no	—
79	C₂×C₄×C₁₇	—	4	2	no	—
80	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#137	1	4	yes	—
89	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
91	C₂×Dic₁₇	—	4	4	no	—
92	C₂×Dic₁₇	—	8	4	no	—
93	Dic₃₄	—	8	4	no	—
94	Dic₃₄	—	8	4	no	—
95	C₈×C₁₇	—	2	2	no	—
96	C₈×C₁₇	—	4	2	no	—
97	C₈×C₁₇	#103	1	4	yes	—
98	#3622	—	8	2	no	—
99	C₂×#500	—	17	1	no	—
100	#1987	—	17	2	no	—
101	#1770	D₁₇	1	1	yes	characteristic
102	C₂×#334	—	17	1	no	—
103	C₂×#333	—	17	2	no	—
104	C₂²×C₁₇	#390	1	1	yes	characteristic
105	C₂²×C₁₇	#461	1	1	yes	characteristic
106	C₂²×C₁₇	#462	1	1	yes	characteristic
107	C₂²×C₁₇	#470	1	2	yes	—
108	C₂²×C₁₇	#472	1	2	yes	—
109	C₄×C₁₇	—	2	1	no	—
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	8	2	no	—
112	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
113	C₄×C₁₇	#419	1	1	yes	characteristic
114	C₄×C₁₇	#441	1	2	yes	—
115	C₄×C₁₇	#461	1	1	yes	characteristic
116	C₄×C₁₇	#462	1	1	yes	characteristic
117	C₄×C₁₇	#497	1	2	yes	—
118	Dic₁₇	—	8	4	no	—
119	C₂×C₄⋊Q₈	—	17	2	no	—
120	#500	—	17	4	no	—
121	#500	—	34	2	no	—
122	C₂×#107	—	34	1	no	—
123	C₂×#107	—	34	1	no	—
124	C₂×C₄⋊C₈	—	17	2	no	—
125	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
126	C₂×#97	—	2	1	no	—
127	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
129	#340	—	34	4	no	—
130	#336	—	2	1	no	—
131	#334	—	34	2	no	—
132	#333	—	34	4	no	—
133	C₂×C₁₇	#1695	1	2	yes	—
134	C₂×C₁₇	#1696	1	2	yes	—
135	C₂×C₁₇	#1971	1	1	yes	characteristic
136	C₂×C₁₇	#1978	1	1	yes	characteristic
137	C₂×C₁₇	#1981	1	1	yes	characteristic
138	C₂²×Q₈	—	34	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	—	34	2	no	—
141	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	17	4	no	—
144	C₄⋊Q₈	—	34	2	no	—
145	C₄⋊Q₈	—	34	2	no	—
146	C₄⋊Q₈	—	34	2	no	—
147	C₂×C₄⋊C₄	—	34	2	no	—
148	C₂×C₄⋊C₄	—	34	2	no	—
149	C₂×C₄⋊C₄	D₆₈	1	2	yes	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
152	#107	—	34	4	no	—
153	#107	—	34	4	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	34	2	no	—
156	C₄⋊C₈	—	34	2	no	—
157	C₄⋊C₈	D₆₈	1	2	yes	—
158	#97	—	4	2	no	—
159	C₄×C₈	C₄×D₁₇	1	2	yes	—
160	C₄×C₈	D₆₈	1	2	yes	—
161	#95	—	2	1	no	—
162	C₁₇	#9739	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	—	34	4	no	—
164	C₂×Q₈	—	68	2	no	—
165	C₂×Q₈	—	136	4	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	68	2	no	—
168	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
169	C₂²×C₄	#63252	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	68	2	no	—
177	C₂×C₈	—	68	4	no	—
178	C₂×C₈	#63227	1	2	yes	—
179	C₂×C₈	#63227	1	2	yes	—
180	C₂×C₈	#63229	1	2	yes	—
181	C₂×C₈	#63231	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	34	4	no	—
184	C₄⋊C₄	—	34	4	no	—
185	C₄⋊C₄	—	68	2	no	—
186	C₄⋊C₄	—	68	2	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
190	C₄²	C₂×D₆₈	1	1	yes	characteristic
191	C₄²	C₂×D₆₈	1	1	yes	characteristic
192	C₄²	#63252	1	1	yes	characteristic
193	C₂³	#10587616	1	1	yes	characteristic
194	Q₈	—	136	4	no	—
195	Q₈	—	136	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	68	4	no	—
201	C₂×C₄	—	136	4	no	—
202	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
203	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
204	C₂×C₄	#10587560	1	1	yes	characteristic
205	C₂×C₄	#10587561	1	1	yes	characteristic
206	C₂×C₄	#10587595	1	4	yes	—
207	C₂×C₄	C₂×#63227	1	1	yes	characteristic
208	C₂×C₄	#10587616	1	1	yes	characteristic
209	C₂×C₄	C₂×#63229	1	1	yes	characteristic
210	C₂×C₄	C₂×#63231	1	1	yes	characteristic
211	C₂×C₄	#10587637	1	1	yes	characteristic
212	C₂×C₄	#10587637	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	4	2	no	—
215	C₈	—	136	2	no	—
216	C₈	#10587481	1	4	yes	—
217	C₂²	#11762711	1	1	yes	characteristic
218	C₂²	#11763014	1	1	yes	characteristic
219	C₂²	#11763019	1	1	yes	characteristic
220	C₂²	#11763067	1	2	yes	—
221	C₂²	#11763079	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	2	no	—
225	C₄	—	136	4	no	—
226	C₄	C₂×#10587481	1	1	yes	characteristic
227	C₄	#11762817	1	1	yes	characteristic
228	C₄	#11762917	1	2	yes	—
229	C₄	#11763014	1	1	yes	characteristic
230	C₄	#11763019	1	1	yes	characteristic
231	C₄	#11763203	1	2	yes	—
232	C₂	#14528307	1	2	yes	—
233	C₂	#14528312	1	2	yes	—
234	C₂	#14529655	1	1	yes	characteristic
235	C₂	#14529694	1	1	yes	characteristic
236	C₂	#14529707	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1770	C₂
socle, prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1770	D₁₇
hypocenter	C₁₇	#9739
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
8. #11763214	C₄	semidirect
162. C₁₇	#9739	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
#14529732	2
C₂×#11762560	2
C₂×#11762557	2
C₁₇×#1770	2
#9739	17