Subgroups — #16945552

Order 4352.

418 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11762560	C₂	1	1	yes	maximal, characteristic, complemented
3	#14526827	C₂	1	4	yes	maximal, complemented
4	C₁₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	—	2	1	no	—
6	C₂×C₄⋊#3622	—	2	1	no	—
7	C₄⋊#63214	—	2	1	no	—
8	C₄⋊#63214	C₂²	1	2	yes	—
9	#11762560	—	2	1	no	—
10	#11762560	C₂²	1	2	yes	—
11	C₂×C₁₇×#135	—	2	1	no	—
12	C₂×C₁₇×#135	C₄	1	2	yes	—
13	C₁₇×#532	—	2	1	no	—
14	C₁₇×#495	—	2	1	no	—
15	C₁₇×#495	C₂²	1	2	yes	—
16	C₁₇×#494	C₄	1	1	yes	characteristic
17	C₁₇×#493	C₄	1	1	yes	characteristic
18	C₁₇×#492	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×#41×C₁₇	—	2	2	no	—
21	C₁₇×#135	—	4	4	no	—
22	C₁₇×#135	—	4	2	no	—
23	C₁₇×#135	—	4	2	no	—
24	C₂×Q₁₆×C₁₇	—	2	1	no	—
25	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₁₇	—	2	2	no	—
27	C₂×SD₁₆×C₁₇	—	2	2	no	—
28	C₂×D₈×C₁₇	—	2	1	no	—
29	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₁₇×C₄⋊Q₈	—	2	1	no	—
33	C₁₇×#127	—	2	1	no	—
34	C₁₇×#124	—	2	2	no	—
35	C₁₇×#121	—	4	1	no	—
36	C₁₇×#121	—	4	1	no	—
37	C₄×D₄×C₁₇	—	4	1	no	—
38	C₄×D₄×C₁₇	—	4	1	no	—
39	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
40	C₂²×#3622	—	4	1	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	#10587483	—	4	2	no	—
46	C₂×#63214	—	2	2	no	—
47	C₄⋊#3622	—	2	2	no	—
48	C₄⋊#3622	—	4	1	no	—
49	#41×C₁₇	—	4	4	no	—
50	#41×C₁₇	—	4	4	no	—
51	C₂×Q₈×C₁₇	—	2	2	no	—
52	C₂×D₄×C₁₇	—	2	2	no	—
53	C₂×D₄×C₁₇	—	4	2	no	—
54	C₂×D₄×C₁₇	—	4	1	no	—
55	C₂×D₄×C₁₇	—	4	1	no	—
56	C₂²×C₄×C₁₇	—	4	2	no	—
57	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
60	Q₁₆×C₁₇	—	4	2	no	—
61	Q₁₆×C₁₇	—	4	2	no	—
62	SD₁₆×C₁₇	—	4	4	no	—
63	SD₁₆×C₁₇	—	4	2	no	—
64	SD₁₆×C₁₇	—	4	2	no	—
65	D₈×C₁₇	—	4	2	no	—
66	D₈×C₁₇	—	4	2	no	—
67	C₂×C₈×C₁₇	—	2	2	no	—
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	D₈	1	2	yes	—
71	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
72	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	—	4	1	no	—
75	C₄⋊C₄×C₁₇	—	4	1	no	—
76	C₂²⋊C₄×C₁₇	—	4	2	no	—
77	C₂²⋊C₄×C₁₇	—	4	2	no	—
78	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₁₇	D₈	1	1	yes	characteristic
80	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
81	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×#3622	—	4	2	no	—
83	C₂×#3622	—	8	2	no	—
84	#63214	—	4	2	no	—
85	#11925	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	—	4	2	no	—
87	C₂³×C₁₇	#102	1	1	yes	characteristic
88	Q₈×C₁₇	—	4	2	no	—
89	Q₈×C₁₇	—	4	2	no	—
90	D₄×C₁₇	—	4	2	no	—
91	D₄×C₁₇	—	4	2	no	—
92	D₄×C₁₇	—	8	2	no	—
93	D₄×C₁₇	—	8	4	no	—
94	D₄×C₁₇	—	8	2	no	—
95	C₂×C₄×C₁₇	—	2	2	no	—
96	C₂×C₄×C₁₇	—	4	1	no	—
97	C₂×C₄×C₁₇	—	4	2	no	—
98	C₂×C₄×C₁₇	—	8	4	no	—
99	C₂×C₄×C₁₇	#100	1	2	yes	—
100	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#102	1	2	yes	—
103	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₈×C₁₇	—	2	1	no	—
108	C₈×C₁₇	—	2	1	no	—
109	C₈×C₁₇	#102	1	2	yes	—
110	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
111	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
112	#3622	—	8	2	no	—
113	#3147	D₁₇	1	1	yes	characteristic
114	#2150	—	17	1	no	—
115	C₂×#334	—	17	1	no	—
116	#1360	—	17	4	no	—
117	C₂²×C₁₇	—	2	1	no	—
118	C₂²×C₁₇	—	2	1	no	—
119	C₂²×C₁₇	—	8	2	no	—
120	C₂²×C₁₇	—	8	2	no	—
121	C₂²×C₁₇	—	8	2	no	—
122	C₂²×C₁₇	#331	1	2	yes	—
123	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	8	2	no	—
128	C₄×C₁₇	#331	1	2	yes	—
129	C₄×C₁₇	#360	1	2	yes	—
130	C₄×C₁₇	C₂×#100	1	1	yes	characteristic
131	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
132	C₂×#135	—	2	1	no	—
133	C₂×#135	Dic₁₇	1	2	yes	—
134	#532	—	2	1	no	—
135	C₂×M₃₂	—	34	1	no	—
136	#495	—	2	1	no	—
137	#495	C₂×D₁₇	1	2	yes	—
138	#494	Dic₁₇	1	1	yes	characteristic
139	#493	Dic₁₇	1	1	yes	characteristic
140	#492	—	2	1	no	—
141	C₂×C₄⋊C₈	—	34	1	no	—
142	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
143	C₄⋊C₁₆	—	17	2	no	—
144	C₄⋊C₁₆	—	34	1	no	—
145	#334	—	17	2	no	—
146	#334	—	34	1	no	—
147	C₂×C₁₇	—	2	2	no	—
148	C₂×C₁₇	—	16	2	no	—
149	C₂×C₁₇	C₂×#331	1	1	yes	characteristic
150	C₂×C₁₇	#2143	1	1	yes	characteristic
151	C₂×C₁₇	C₂×#360	1	1	yes	characteristic
152	C₂×#41	—	2	2	no	—
153	#135	—	4	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
162	C₂²×C₈	—	68	1	no	—
163	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	1	no	—
166	#127	—	2	1	no	—
167	#124	—	2	2	no	—
168	#121	—	4	1	no	—
169	#121	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
173	M₃₂	—	68	2	no	—
174	C₂×C₁₆	—	34	2	no	—
175	C₄⋊C₈	—	34	2	no	—
176	C₄⋊C₈	—	68	1	no	—
177	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
181	C₁₇	#11925	1	1	yes	characteristic, complemented, Sylow
182	#41	—	4	4	no	—
183	#41	—	4	4	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	#63228	1	1	yes	characteristic
191	C₂²×C₄	#63229	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
193	Q₁₆	—	4	2	no	—
194	Q₁₆	—	4	2	no	—
195	SD₁₆	—	4	4	no	—
196	SD₁₆	—	4	2	no	—
197	SD₁₆	—	4	2	no	—
198	D₈	—	4	2	no	—
199	D₈	—	4	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	68	2	no	—
202	C₂×C₈	—	136	2	no	—
203	C₂×C₈	#63228	1	2	yes	—
204	C₂×C₈	#63229	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
207	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₂²⋊C₄	—	4	2	no	—
213	C₄²	#63228	1	1	yes	characteristic
214	C₄²	#63229	1	1	yes	characteristic
215	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
217	C₁₆	—	68	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	#10587503	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	4	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	#10587503	1	1	yes	characteristic
232	C₂×C₄	#10587503	1	2	yes	—
233	C₂×C₄	#10587503	1	1	yes	characteristic
234	C₂×C₄	#10587503	1	1	yes	characteristic
235	C₂×C₄	#10587504	1	2	yes	—
236	C₂×C₄	C₂×#63228	1	1	yes	characteristic
237	C₂×C₄	C₂×#63229	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	136	2	no	—
242	C₈	#10587503	1	2	yes	—
243	C₈	C₂×#63228	1	1	yes	characteristic
244	C₈	C₂×#63229	1	1	yes	characteristic
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	#11762553	1	2	yes	—
251	C₂²	C₂×#10587503	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	2	no	—
256	C₄	#11762553	1	2	yes	—
257	C₄	#11762619	1	2	yes	—
258	C₄	C₂×#10587504	1	1	yes	characteristic
259	C₄	C₂×#10587503	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	16	2	no	—
262	C₂	C₂×#11762553	1	1	yes	characteristic
263	C₂	#14530512	1	1	yes	characteristic
264	C₂	C₂×#11762619	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587503
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting, prime radical	C₁₇×#3147	C₂
socle	C₂²×C₁₇	C₂×#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#3147	D₁₇
hypocenter	C₁₇	#11925
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #14530540	C₂	semidirect
2. C₂×#11762560	C₂	semidirect
3. #14526827	C₂	semidirect
181. C₁₇	#11925	semidirect

Maximal subgroups

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

subgroup	index
#14530540	2
C₂×#11762560	2
#14526827	2
C₁₇×#3147	2
#11925	17