Subgroups — #17187417

Order 4352.

598 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	#14530591	C₂	1	4	yes	maximal, complemented
2	#14530583	C₂	1	4	yes	maximal, complemented
3	#14530543	C₂	1	2	yes	maximal, complemented
4	#14530398	C₂	1	2	yes	maximal
5	C₄×#10587483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₇×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587483	C₂²	1	2	yes	—
9	C₂²×#63214	C₂²	1	2	yes	—
10	C₄⋊#63214	C₂²	1	8	yes	—
11	C₂²⋊#63214	C₂²	1	8	yes	—
12	#11762585	C₂²	1	2	yes	—
13	C₄×#63214	C₂²	1	2	yes	—
14	C₄×#63214	C₂²	1	2	yes	—
15	C₄×#63214	C₂²	1	2	yes	—
16	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
17	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
18	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
19	C₁₇×#405	C₄	1	2	yes	—
20	C₁₇×#405	C₄	1	1	yes	characteristic
21	C₁₇×#405	C₂²	1	2	yes	—
22	C₁₇×#405	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
24	C₄×M₁₆×C₁₇	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₁₇×#131	—	2	4	no	—
29	C₁₇×#131	—	2	4	no	—
30	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₁₇	C₈	1	2	yes	—
39	C₄×D₄×C₁₇	C₈	1	4	yes	—
40	C₄×D₄×C₁₇	C₈	1	2	yes	—
41	C₁₇×#117	C₂×C₄	1	2	yes	—
42	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₁₇×#97	C₂×C₄	1	2	yes	—
46	C₁₇×#97	C₂×C₄	1	2	yes	—
47	C₁₇×#97	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₇	C₂³	1	2	yes	—
54	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
55	#10587483	—	2	4	no	—
56	C₂×#63214	—	2	4	no	—
57	C₂×#63214	C₂³	1	4	yes	—
58	C₂×#63214	C₂³	1	4	yes	—
59	#41×C₁₇	—	2	2	no	—
60	#41×C₁₇	—	2	2	no	—
61	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₁₇	—	2	2	no	—
69	M₁₆×C₁₇	—	2	2	no	—
70	M₁₆×C₁₇	—	2	4	no	—
71	M₁₆×C₁₇	—	2	4	no	—
72	C₂×C₈×C₁₇	—	2	4	no	—
73	C₂×C₈×C₁₇	—	2	4	no	—
74	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	#41	1	4	yes	—
81	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
83	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
85	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
86	C₄²×C₁₇	C₂×C₈	1	4	yes	—
87	C₄²×C₁₇	C₂×C₈	1	2	yes	—
88	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
91	#63214	—	2	4	no	—
92	#63214	—	2	4	no	—
93	#33420	—	17	1	no	maximal, Sylow
94	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
95	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
96	Q₈×C₁₇	—	2	2	no	—
97	D₄×C₁₇	—	2	4	no	—
98	D₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	4	no	—
102	C₂×C₄×C₁₇	—	2	8	no	—
103	C₂×C₄×C₁₇	#117	1	2	yes	—
104	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₁₇	—	2	2	no	—
112	C₈×C₁₇	—	2	2	no	—
113	C₈×C₁₇	#117	1	2	yes	—
114	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
115	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
116	C₈×C₁₇	C₂×#41	1	2	yes	—
117	C₄×#131	D₁₇	1	1	yes	characteristic
118	#2164	—	17	4	no	—
119	#2162	—	17	4	no	—
120	#2151	—	17	2	no	—
121	#2113	—	17	2	no	—
122	C₄×M₃₂	—	17	1	no	—
123	C₂×C₄×C₁₆	—	17	1	no	—
124	C₂²×C₁₇	—	2	2	no	—
125	C₂²×C₁₇	—	2	4	no	—
126	C₂²×C₁₇	—	2	2	no	—
127	C₂²×C₁₇	#431	1	2	yes	—
128	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₁₇	—	2	2	no	—
130	C₄×C₁₇	—	2	4	no	—
131	C₄×C₁₇	—	2	2	no	—
132	C₄×C₁₇	#431	1	2	yes	—
133	C₄×C₁₇	#504	1	4	yes	—
134	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
135	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
136	C₂×#131	Dic₁₇	1	1	yes	characteristic
137	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
138	C₄×#41	Dic₁₇	1	1	yes	characteristic
139	C₂×M₃₂	—	17	2	no	—
140	C₂²×C₁₆	—	17	2	no	—
141	#405	Dic₁₇	1	2	yes	—
142	#405	Dic₁₇	1	1	yes	characteristic
143	#405	C₂×D₁₇	1	1	yes	characteristic
144	#405	C₂×D₁₇	1	2	yes	—
145	C₄×M₁₆	Dic₁₇	1	1	yes	characteristic
146	C₄×M₁₆	Dic₁₇	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
149	C₄⋊C₁₆	—	17	8	no	—
150	C₂²⋊C₁₆	—	17	8	no	—
151	#346	—	17	2	no	—
152	C₄×C₁₆	—	17	2	no	—
153	C₄×C₁₆	—	17	2	no	—
154	C₄×C₁₆	—	17	2	no	—
155	C₂×C₁₇	—	2	2	no	—
156	C₂×C₁₇	—	4	2	no	—
157	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
158	C₂×C₁₇	C₂×#504	1	2	yes	—
159	C₂×#41	C₂×Dic₁₇	1	1	yes	characteristic
160	#131	—	2	4	no	—
161	#131	—	2	4	no	—
162	C₂×M₁₆	C₂×Dic₁₇	1	1	yes	characteristic
163	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
164	C₂×M₁₆	C₂×Dic₁₇	1	1	yes	characteristic
165	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
166	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
167	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
169	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
170	C₄×Q₈	#3622	1	2	yes	—
171	C₄×D₄	#3622	1	4	yes	—
172	C₄×D₄	#3622	1	2	yes	—
173	#117	C₂×Dic₁₇	1	2	yes	—
174	#117	C₂×Dic₁₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
176	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
177	M₃₂	—	34	4	no	—
178	C₂×C₁₆	—	17	4	no	—
179	C₂×C₁₆	—	17	4	no	—
180	C₂×C₁₆	—	34	4	no	—
181	#97	C₂×Dic₁₇	1	2	yes	—
182	#97	C₂×Dic₁₇	1	2	yes	—
183	#97	C₂×Dic₁₇	1	2	yes	—
184	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
185	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
186	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
187	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₇	1	2	yes	—
190	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
191	C₁₇	#33420	1	1	yes	characteristic, complemented, Sylow
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₂×#3622	1	1	yes	characteristic
195	C₂×D₄	C₂×#3622	1	2	yes	—
196	C₂×D₄	C₂×#3622	1	1	yes	characteristic
197	C₂²×C₄	C₂×#3622	1	4	yes	—
198	C₂²×C₄	C₂×#3622	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	#63223	1	4	yes	—
208	C₂×C₈	#63252	1	4	yes	—
209	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
210	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
211	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
214	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂×#3622	1	4	yes	—
216	C₄⋊C₄	C₂×#3622	1	2	yes	—
217	C₂²⋊C₄	C₂×#3622	1	4	yes	—
218	C₂²⋊C₄	C₂×#3622	1	2	yes	—
219	C₄²	C₂×#3622	1	4	yes	—
220	C₄²	C₂×#3622	1	2	yes	—
221	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
222	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
223	C₄²	C₂²×Dic₁₇	1	2	yes	—
224	C₁₆	—	34	4	no	—
225	C₁₆	—	34	4	no	—
226	C₂³	C₂²×#3622	1	2	yes	—
227	C₂³	C₂²×#3622	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	C₂²×#3622	1	2	yes	—
236	C₂×C₄	C₂²×#3622	1	4	yes	—
237	C₂×C₄	C₂²×#3622	1	2	yes	—
238	C₂×C₄	C₂²×#3622	1	2	yes	—
239	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
240	C₂×C₄	C₂×#63223	1	2	yes	—
241	C₂×C₄	#10587612	1	2	yes	—
242	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	C₂×#63223	1	2	yes	—
246	C₈	C₂×#63223	1	2	yes	—
247	C₈	#10587612	1	2	yes	—
248	C₈	C₂×#63252	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#11762862	1	2	yes	—
253	C₂²	C₂³×#3622	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	#11762862	1	2	yes	—
258	C₄	#11763226	1	4	yes	—
259	C₄	C₂×#10587612	1	1	yes	characteristic
260	C₄	C₂²×#63223	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#11762862	1	1	yes	characteristic
264	C₂	C₂×#11763226	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₄×C₁₇×#131	C₂
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₄×#131	D₁₇
hypocenter	C₁₇	#33420
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #14530591	C₂	semidirect
2. #14530583	C₂	semidirect
3. #14530543	C₂	semidirect
5. C₄×#10587483	C₂	semidirect
6. C₂×C₄×#63214	C₂	semidirect
191. C₁₇	#33420	semidirect

Maximal subgroups

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

subgroup	index
#14530591	2
#14530583	2
#14530543	2
#14530398	2
C₄×#10587483	2
C₂×C₄×#63214	2
C₄×C₁₇×#131	2
#33420	17