Subgroups — #16927603

Order 4352.

346 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#11762562	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	#11762562	—	2	1	no	—
11	#11762562	—	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	#8689	—	17	1	no	maximal, Sylow
82	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
83	Q₈×C₁₇	—	4	2	no	—
84	Q₈×C₁₇	—	4	1	no	—
85	Q₈×C₁₇	—	4	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	4	1	no	—
91	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₇	#135	1	2	yes	—
98	C₂×C₄×C₁₇	#137	1	2	yes	—
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	—	2	1	no	—
101	C₈×C₁₇	—	2	1	no	—
102	C₈×C₁₇	—	2	1	no	—
103	C₈×C₁₇	—	4	2	no	—
104	#3622	—	8	2	no	—
105	#3622	—	8	2	no	—
106	#2960	D₁₇	1	1	yes	characteristic
107	#1837	—	17	1	no	—
108	C₂×#335	—	17	1	no	—
109	#1550	—	17	2	no	—
110	#1292	—	17	2	no	—
111	C₂²×C₁₇	—	2	1	no	—
112	C₂²×C₁₇	—	2	1	no	—
113	C₂²×C₁₇	#380	1	1	yes	characteristic
114	C₂²×C₁₇	#479	1	2	yes	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	2	1	no	—
118	C₄×C₁₇	—	4	2	no	—
119	C₄×C₁₇	—	4	2	no	—
120	C₄×C₁₇	#416	1	1	yes	characteristic
121	C₄×C₁₇	#419	1	1	yes	characteristic
122	C₄×C₁₇	#464	1	2	yes	—
123	C₄×C₁₇	#472	1	1	yes	characteristic
124	C₄×C₁₇	#473	1	1	yes	characteristic
125	#533	C₂×D₁₇	1	1	yes	characteristic
126	#446	—	2	1	no	—
127	#446	Dic₁₇	1	2	yes	—
128	C₈×Q₈	—	2	1	no	—
129	C₈×Q₈	Dic₁₇	1	2	yes	—
130	#424	—	2	1	no	—
131	#423	—	17	2	no	—
132	#423	C₂×D₁₇	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	17	2	no	—
134	#405	—	2	1	no	—
135	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
136	#341	—	34	1	no	—
137	#341	—	34	1	no	—
138	#335	—	34	1	no	—
139	#335	—	34	1	no	—
140	#333	—	34	2	no	—
141	#332	—	34	2	no	—
142	C₂×C₁₇	—	2	2	no	—
143	C₂×C₁₇	#1805	1	1	yes	characteristic
144	C₂×C₁₇	#1813	1	1	yes	characteristic
145	C₂×C₁₇	#1816	1	1	yes	characteristic
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	—	34	2	no	—
148	C₂²×C₈	—	34	2	no	—
149	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
151	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
152	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
153	#125	—	2	1	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	Dic₃₄	1	2	yes	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
158	#117	—	2	1	no	—
159	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	34	2	no	—
164	C₄⋊C₈	—	34	2	no	—
165	C₄⋊C₈	—	34	2	no	—
166	C₄⋊C₈	—	34	2	no	—
167	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
168	#97	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
171	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
174	C₁₇	#8689	1	1	yes	characteristic, complemented, Sylow
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₄⋊Dic₁₇	1	2	yes	—
177	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
178	C₂²×C₄	#63252	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	68	2	no	—
182	M₁₆	—	68	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	68	2	no	—
187	C₂×C₈	—	68	2	no	—
188	C₂×C₈	—	136	2	no	—
189	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂×D₆₈	1	1	yes	characteristic
202	C₄²	#63252	1	1	yes	characteristic
203	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
205	C₂³	#10587610	1	1	yes	characteristic
206	Q₈	—	4	2	no	—
207	Q₈	—	4	1	no	—
208	Q₈	—	4	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	#10587610	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
216	C₂×C₄	#10587612	1	1	yes	characteristic
217	C₂×C₄	#10587616	1	1	yes	characteristic
218	C₂×C₄	#10587616	1	1	yes	characteristic
219	C₂×C₄	#10587628	1	2	yes	—
220	C₂×C₄	#10587636	1	2	yes	—
221	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	136	2	no	—
228	C₈	—	136	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#11762669	1	1	yes	characteristic
232	C₂²	#11763116	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	#11762805	1	1	yes	characteristic
239	C₄	#11762818	1	1	yes	characteristic
240	C₄	#11763031	1	2	yes	—
241	C₄	#11763079	1	1	yes	characteristic
242	C₄	#11763084	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#14528799	1	1	yes	characteristic
245	C₂	#14528834	1	1	yes	characteristic
246	C₂	#14528848	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	#8689
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
174. C₁₇	#8689	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×#11762562	2
#14527474	2
#14526658	2
C₁₇×#2960	2
#8689	17