Subgroups — #16927593

Order 4352.

346 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528922	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
3	#14527474	C₂	1	2	yes	maximal
4	#14526658	C₂	1	2	yes	maximal
5	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic
6	#11762833	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	2	yes	—
8	#11762574	—	2	1	no	—
9	#11762574	—	2	1	no	—
10	#11762560	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762558	—	2	2	no	—
13	#11762555	—	2	2	no	—
14	C₁₇×#533	C₂²	1	1	yes	characteristic
15	C₁₇×#446	—	2	1	no	—
16	C₁₇×#446	C₄	1	2	yes	—
17	C₈×Q₈×C₁₇	—	2	1	no	—
18	C₈×Q₈×C₁₇	C₄	1	2	yes	—
19	C₁₇×#424	—	2	1	no	—
20	C₁₇×#423	C₂²	1	1	yes	characteristic
21	C₁₇×#405	—	2	1	no	—
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₁₇	—	2	1	no	—
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
27	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
28	C₁₇×#125	—	2	1	no	—
29	C₁₇×C₂²⋊Q₈	—	2	1	no	—
30	C₁₇×C₂²⋊Q₈	Q₈	1	2	yes	—
31	C₄×Q₈×C₁₇	—	2	1	no	—
32	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₁₇×#117	—	2	1	no	—
34	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×#63223	—	2	2	no	—
36	C₂²×#3622	—	2	2	no	—
37	C₁₇×C₄⋊C₈	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	2	no	—
39	C₁₇×C₄⋊C₈	—	2	1	no	—
40	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₁₇×#97	—	2	2	no	—
42	C₄×C₈×C₁₇	—	2	2	no	—
43	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₄⋊#3622	—	2	2	no	—
48	C₄⋊#3622	—	2	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₂×Q₈×C₁₇	—	2	1	no	—
52	C₂×Q₈×C₁₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
56	M₁₆×C₁₇	—	4	2	no	—
57	C₂×C₈×C₁₇	—	2	2	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₇	—	2	2	no	—
65	C₄⋊C₄×C₁₇	—	2	1	no	—
66	C₄⋊C₄×C₁₇	—	2	1	no	—
67	C₄⋊C₄×C₁₇	—	2	1	no	—
68	C₄⋊C₄×C₁₇	—	2	2	no	—
69	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
70	C₂²⋊C₄×C₁₇	—	2	2	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₇	#41	1	1	yes	characteristic
76	#63223	—	4	2	no	—
77	#63223	—	4	2	no	—
78	C₂×#3622	—	4	2	no	—
79	C₂×#3622	—	4	2	no	—
80	C₂×#3622	—	8	2	no	—
81	#8687	—	17	1	no	maximal, Sylow
82	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
83	Q₈×C₁₇	—	4	2	no	—
84	Q₈×C₁₇	—	4	2	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	4	1	no	—
89	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#135	1	2	yes	—
96	C₂×C₄×C₁₇	#137	1	2	yes	—
97	C₈×C₁₇	—	2	1	no	—
98	C₈×C₁₇	—	2	1	no	—
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	—	2	1	no	—
101	C₈×C₁₇	—	4	2	no	—
102	#3622	—	8	2	no	—
103	#3622	—	8	2	no	—
104	#2960	D₁₇	1	1	yes	characteristic
105	#1837	—	17	1	no	—
106	C₂×#334	—	17	1	no	—
107	#1550	—	17	2	no	—
108	#1292	—	17	2	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	—	2	1	no	—
111	C₂²×C₁₇	#380	1	1	yes	characteristic
112	C₂²×C₁₇	#479	1	2	yes	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	4	2	no	—
117	C₄×C₁₇	—	4	2	no	—
118	C₄×C₁₇	#416	1	1	yes	characteristic
119	C₄×C₁₇	#419	1	1	yes	characteristic
120	C₄×C₁₇	#464	1	2	yes	—
121	C₄×C₁₇	#471	1	2	yes	—
122	#533	C₂×D₁₇	1	1	yes	characteristic
123	#446	—	2	1	no	—
124	#446	Dic₁₇	1	2	yes	—
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	Dic₁₇	1	2	yes	—
127	#424	—	2	1	no	—
128	#423	—	17	2	no	—
129	#423	C₂×D₁₇	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	17	2	no	—
131	#405	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
133	#341	—	34	1	no	—
134	#341	—	34	1	no	—
135	#334	—	34	1	no	—
136	#334	—	34	1	no	—
137	#333	—	34	2	no	—
138	#332	—	34	2	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	#1805	1	1	yes	characteristic
141	C₂×C₁₇	#1813	1	1	yes	characteristic
142	C₂×C₁₇	#1815	1	1	yes	characteristic
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	34	2	no	—
145	C₂²×C₈	—	34	2	no	—
146	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
148	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
149	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
150	#125	—	2	1	no	—
151	C₂²⋊Q₈	—	2	1	no	—
152	C₂²⋊Q₈	Dic₃₄	1	2	yes	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
155	#117	—	2	1	no	—
156	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	34	2	no	—
161	C₄⋊C₈	—	34	2	no	—
162	C₄⋊C₈	—	34	2	no	—
163	C₄⋊C₈	—	34	2	no	—
164	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
165	#97	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
168	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
171	C₁₇	#8687	1	1	yes	characteristic, complemented, Sylow
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊Dic₁₇	1	2	yes	—
174	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
175	C₂²×C₄	#63252	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	M₁₆	—	68	2	no	—
179	M₁₆	—	68	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	68	2	no	—
184	C₂×C₈	—	68	2	no	—
185	C₂×C₈	—	136	2	no	—
186	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂×D₆₈	1	1	yes	characteristic
199	C₄²	#63252	1	1	yes	characteristic
200	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
202	C₂³	#10587610	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	#10587610	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
211	C₂×C₄	#10587612	1	1	yes	characteristic
212	C₂×C₄	#10587616	1	1	yes	characteristic
213	C₂×C₄	#10587616	1	1	yes	characteristic
214	C₂×C₄	#10587628	1	2	yes	—
215	C₂×C₄	#10587636	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	136	2	no	—
223	C₈	—	136	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#11762669	1	1	yes	characteristic
227	C₂²	#11763116	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	#11762805	1	1	yes	characteristic
234	C₄	#11762818	1	1	yes	characteristic
235	C₄	#11763031	1	2	yes	—
236	C₄	#11763072	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#14528799	1	1	yes	characteristic
239	C₂	#14528834	1	1	yes	characteristic
240	C₂	#14528843	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#8687
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
171. C₁₇	#8687	semidirect

Maximal subgroups

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

subgroup	index
#14528922	2
C₂×#11762560	2
#14527474	2
#14526658	2
C₁₇×#2960	2
#8687	17