Subgroups — #16950839

Order 4352.

404 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587490	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762638	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#63214	C₂	1	1	yes	maximal, characteristic
4	#14526877	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63223	—	2	1	no	—
7	C₂²×#63214	C₂²	1	2	yes	—
8	C₂×#10587490	—	2	4	no	—
9	C₂×#10587490	—	2	1	no	—
10	C₂×#10587490	C₂²	1	2	yes	—
11	#11762638	—	2	4	no	—
12	C₂²⋊#63214	C₄	1	2	yes	—
13	C₂²⋊#63214	C₄	1	2	yes	—
14	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
17	C₁₇×#407	—	2	1	no	—
18	C₁₇×#341	—	2	1	no	—
19	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	—	2	1	no	—
21	C₂×M₁₆×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	—	2	2	no	—
23	C₂²×C₈×C₁₇	—	2	2	no	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×#63223	—	2	4	no	—
31	C₂×#63223	—	2	2	no	—
32	C₂×#63223	—	4	2	no	—
33	C₂×#63223	—	4	1	no	—
34	C₂²×#3622	—	4	1	no	—
35	C₁₇×#108	—	2	2	no	—
36	C₁₇×#108	—	2	2	no	—
37	C₁₇×C₂²⋊C₈	—	4	1	no	—
38	C₁₇×C₂²⋊C₈	—	4	1	no	—
39	#10587490	—	2	4	no	—
40	#10587490	—	2	2	no	—
41	#10587490	D₄	1	4	yes	—
42	C₂×#63214	—	2	2	no	—
43	C₂×#63214	—	2	2	no	—
44	C₂×#63214	C₂×C₄	1	4	yes	—
45	C₂⁴×C₁₇	Q₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	—	2	1	no	—
48	C₂²×C₄×C₁₇	—	2	1	no	—
49	C₂²×C₄×C₁₇	—	2	1	no	—
50	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	Q₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₁₇	—	4	1	no	—
57	M₁₆×C₁₇	—	4	1	no	—
58	M₁₆×C₁₇	—	4	1	no	—
59	M₁₆×C₁₇	—	4	1	no	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	—	2	2	no	—
64	C₂×C₈×C₁₇	—	2	2	no	—
65	C₂×C₈×C₁₇	—	2	2	no	—
66	C₂×C₈×C₁₇	—	4	1	no	—
67	C₂×C₈×C₁₇	—	4	1	no	—
68	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₁₇	#41	1	2	yes	—
74	#63223	—	2	4	no	—
75	#63223	—	4	4	no	—
76	#63223	—	4	2	no	—
77	C₂×#3622	—	4	4	no	—
78	C₂×#3622	—	4	2	no	—
79	#63214	—	2	4	no	—
80	#12922	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	—	2	2	no	—
82	C₂³×C₁₇	—	2	1	no	—
83	C₂³×C₁₇	—	2	1	no	—
84	C₂³×C₁₇	—	4	1	no	—
85	C₂³×C₁₇	#107	1	1	yes	characteristic
86	C₂³×C₁₇	#107	1	1	yes	characteristic
87	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
99	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
106	C₈×C₁₇	—	2	1	no	—
107	C₈×C₁₇	—	2	1	no	—
108	C₈×C₁₇	—	8	1	no	—
109	C₈×C₁₇	—	8	1	no	—
110	C₈×C₁₇	C₄×D₄	1	2	yes	—
111	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
112	#3622	—	4	4	no	—
113	C₂²×#108	—	17	1	no	—
114	C₂×#368	—	17	2	no	—
115	C₂×C₂²⋊C₁₆	—	17	1	no	—
116	#1810	D₁₇	1	1	yes	characteristic
117	#1381	—	17	2	no	—
118	C₂²×C₁₇	—	2	1	no	—
119	C₂²×C₁₇	—	2	1	no	—
120	C₂²×C₁₇	—	2	1	no	—
121	C₂²×C₁₇	—	2	1	no	—
122	C₂²×C₁₇	—	2	1	no	—
123	C₂²×C₁₇	—	2	1	no	—
124	C₂²×C₁₇	—	2	1	no	—
125	C₂²×C₁₇	—	4	2	no	—
126	C₂²×C₁₇	—	4	2	no	—
127	C₂²×C₁₇	#368	1	2	yes	complemented
128	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
129	C₂²×C₁₇	#451	1	1	yes	characteristic
130	C₂²×C₁₇	#484	1	1	yes	characteristic
131	C₄×C₁₇	—	2	1	no	—
132	C₄×C₁₇	—	2	1	no	—
133	C₄×C₁₇	#385	1	1	yes	characteristic
134	C₄×C₁₇	#428	1	1	yes	characteristic
135	C₄×C₁₇	#447	1	1	yes	characteristic
136	C₄×C₁₇	#480	1	1	yes	characteristic
137	C₂²×M₁₆	—	34	1	no	—
138	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
139	C₂²×C₁₆	—	17	2	no	—
140	C₂×#108	—	17	2	no	—
141	C₂×#108	—	34	4	no	—
142	C₂×#108	—	34	1	no	—
143	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
144	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
145	#407	—	2	1	no	—
146	#368	—	34	4	no	—
147	C₂²⋊C₁₆	—	17	2	no	—
148	C₂²⋊C₁₆	—	17	2	no	—
149	#341	—	2	1	no	—
150	C₂×C₁₇	—	2	2	no	—
151	C₂×C₁₇	—	2	1	no	—
152	C₂×C₁₇	—	2	1	no	—
153	C₂×C₁₇	—	4	1	no	—
154	C₂×C₁₇	#1852	1	1	yes	characteristic
155	C₂×C₁₇	C₂×#368	1	1	yes	characteristic
156	C₂×C₁₇	#2161	1	1	yes	characteristic
157	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	34	4	no	—
161	C₂×M₁₆	—	34	2	no	—
162	C₂×M₁₆	—	68	2	no	—
163	C₂×M₁₆	—	68	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	68	1	no	—
167	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
168	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
169	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
173	C₂×C₁₆	—	17	4	no	—
174	C₂×C₁₆	—	34	2	no	—
175	C₂×C₁₆	—	34	2	no	—
176	#108	—	2	2	no	—
177	#108	—	2	2	no	—
178	#108	—	17	4	no	—
179	#108	—	34	4	no	—
180	#108	—	34	2	no	—
181	C₂²⋊C₈	—	4	1	no	—
182	C₂²⋊C₈	—	4	1	no	—
183	C₁₇	#12922	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	#63231	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	#63225	1	1	yes	characteristic
190	C₂²×C₄	#63225	1	1	yes	characteristic
191	C₂²×C₄	#63228	1	1	yes	characteristic
192	C₂²×C₄	#63228	1	1	yes	characteristic
193	C₂²×C₄	#63231	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
195	M₁₆	—	4	1	no	—
196	M₁₆	—	4	1	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	34	4	no	—
200	M₁₆	—	68	4	no	—
201	M₁₆	—	68	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	68	4	no	—
211	C₂×C₈	—	68	2	no	—
212	C₂×C₈	#63225	1	1	yes	characteristic
213	C₂×C₈	#63225	1	1	yes	characteristic
214	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
216	C₂×C₈	D₄×D₁₇	1	2	yes	—
217	C₂×C₈	#63254	1	2	yes	—
218	C₁₆	—	34	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	1	no	—
223	C₂³	#10587478	1	1	yes	characteristic
224	C₂³	#10587478	1	1	yes	characteristic
225	C₂³	C₂×#63231	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	#10587478	1	1	yes	characteristic
237	C₂×C₄	#10587478	1	1	yes	characteristic
238	C₂×C₄	C₂×#63225	1	1	yes	characteristic
239	C₂×C₄	C₂×#63228	1	1	yes	characteristic
240	C₂×C₄	#10587621	1	1	yes	characteristic
241	C₂×C₄	#10587623	1	1	yes	characteristic
242	C₂×C₄	#10587635	1	1	yes	characteristic
243	C₂×C₄	#10587637	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	8	1	no	—
247	C₈	—	8	1	no	—
248	C₈	—	68	4	no	—
249	C₈	#10587564	1	2	yes	—
250	C₈	#10587567	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#11762638	1	2	yes	complemented
261	C₂²	C₂×#10587478	1	1	yes	characteristic
262	C₂²	#11762964	1	1	yes	characteristic
263	C₂²	#11763142	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	#11762681	1	1	yes	characteristic
267	C₄	#11762852	1	1	yes	characteristic
268	C₄	#11762942	1	1	yes	characteristic
269	C₄	#11763120	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	1	no	—
274	C₂	#14528971	1	1	yes	characteristic
275	C₂	C₂×#11762638	1	1	yes	characteristic
276	C₂	#14530578	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1810	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂⁴×C₁₇	Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1810	D₁₇
hypocenter	C₁₇	#12922
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

Complements

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

N	H ≅ G/N	type
2. C₂×#11762638	C₂	semidirect
4. #14526877	C₂	semidirect
127. C₂²×C₁₇	#368	semidirect
183. C₁₇	#12922	semidirect
260. C₂²	#11762638	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587490	2
C₂×#11762638	2
C₂×C₂²⋊#63214	2
#14526877	2
C₁₇×#1810	2
#12922	17