Subgroups — #16950841

Order 4352.

404 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762639	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11762638	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762588	C₂	1	1	yes	maximal, characteristic
5	#14528819	C₂	1	1	yes	maximal, characteristic
6	#14526877	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₁₇	—	2	1	no	—
9	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	1	yes	characteristic
11	C₂×#10587490	C₂²	1	1	yes	characteristic
12	C₂×#10587490	C₂²	1	1	yes	characteristic
13	C₂×#10587493	—	2	1	no	—
14	#108×D₁₇	—	2	2	no	—
15	#108×D₁₇	—	2	2	no	—
16	#11762772	—	2	1	no	—
17	#11762639	—	2	2	no	—
18	#11762638	—	2	2	no	—
19	#11762588	C₄	1	2	yes	—
20	#11762588	C₄	1	2	yes	—
21	#11762573	—	2	1	no	—
22	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
24	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₇	—	2	1	no	—
27	C₂×M₁₆×C₁₇	—	2	1	no	—
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×#63223	—	2	1	no	—
30	C₂×#63223	—	2	1	no	—
31	C₂²×#3622	D₄	1	1	yes	characteristic
32	M₁₆×D₁₇	—	2	4	no	—
33	M₁₆×D₁₇	—	4	2	no	—
34	C₂×#63218	—	4	1	no	—
35	C₂×C₈×D₁₇	—	2	1	no	—
36	C₂×C₈×D₁₇	—	2	1	no	—
37	C₂×C₈×D₁₇	—	2	1	no	—
38	C₂×C₈×D₁₇	—	2	1	no	—
39	C₂×C₈×D₁₇	—	4	1	no	—
40	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
43	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
44	C₂×C₁₆×C₁₇	—	2	1	no	—
45	C₂×C₁₆×C₁₇	—	2	1	no	—
46	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
47	C₁₇×#108	—	2	1	no	—
48	C₁₇×#108	—	2	1	no	—
49	C₁₇×#108	—	2	1	no	—
50	C₁₇×#108	D₄	1	2	yes	—
51	#10587493	—	2	4	no	—
52	#10587491	—	4	1	no	—
53	#10587491	—	4	1	no	—
54	#10587490	—	2	1	no	—
55	#10587490	—	2	1	no	—
56	#10587490	—	2	1	no	—
57	#10587490	D₄	1	2	yes	—
58	C₂×#63214	—	2	1	no	—
59	C₂×#63214	—	2	1	no	—
60	C₂×#63214	C₂×C₄	1	2	yes	—
61	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
71	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
72	M₁₆×C₁₇	—	2	2	no	—
73	M₁₆×C₁₇	—	4	1	no	—
74	M₁₆×C₁₇	—	4	1	no	—
75	M₁₆×C₁₇	—	4	1	no	—
76	C₂×C₈×C₁₇	—	4	1	no	—
77	C₂×C₈×C₁₇	—	4	1	no	—
78	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
84	#63223	—	2	2	no	—
85	#63223	—	4	1	no	—
86	#63223	—	4	1	no	—
87	#63223	—	4	1	no	—
88	C₂×#3622	—	2	1	no	—
89	C₂×#3622	—	2	1	no	—
90	C₂×#3622	—	4	1	no	—
91	C₂×#3622	—	4	1	no	—
92	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
93	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
94	#63218	—	4	4	no	—
95	C₈×D₁₇	—	2	1	no	—
96	C₈×D₁₇	—	2	1	no	—
97	C₈×D₁₇	—	2	1	no	—
98	C₈×D₁₇	—	2	1	no	—
99	C₈×D₁₇	—	2	1	no	—
100	C₈×D₁₇	—	2	1	no	—
101	C₈×D₁₇	—	2	1	no	—
102	C₈×D₁₇	—	2	1	no	—
103	C₈×D₁₇	—	4	4	no	—
104	C₁₆×C₁₇	—	2	2	no	—
105	#63214	—	2	2	no	—
106	#12922	—	17	1	no	maximal, Sylow
107	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₂²×D₁₇	—	2	1	no	—
109	C₂²×D₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	2	no	—
111	C₂²×D₁₇	—	4	1	no	—
112	C₂²×D₁₇	#107	1	1	yes	characteristic
113	C₂²×D₁₇	#107	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	#107	1	1	yes	characteristic
123	C₂×Dic₁₇	#107	1	1	yes	characteristic
124	C₄×D₁₇	—	2	1	no	—
125	C₄×D₁₇	—	2	1	no	—
126	C₄×D₁₇	—	2	1	no	—
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	2	1	no	—
130	C₄×D₁₇	—	2	1	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₈×C₁₇	—	4	2	no	—
133	C₈×C₁₇	—	8	1	no	—
134	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
135	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
136	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
138	#3622	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	—	4	2	no	—
141	#3622	—	8	1	no	—
142	C₂²×#108	—	17	1	no	—
143	C₂×#368	—	17	1	no	—
144	C₂×#368	—	17	1	no	—
145	C₂×C₂²⋊C₁₆	—	17	1	no	—
146	#1810	—	17	1	no	—
147	#1381	—	17	1	no	—
148	#1381	D₁₇	1	1	yes	characteristic, complemented
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
152	C₂²×C₁₇	#451	1	1	yes	characteristic
153	C₂²×C₁₇	#484	1	1	yes	characteristic
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	4	2	no	—
160	C₂×D₁₇	—	4	2	no	—
161	C₂×D₁₇	#368	1	2	yes	complemented
162	C₄×C₁₇	#385	1	1	yes	characteristic
163	C₄×C₁₇	#428	1	1	yes	characteristic
164	C₄×C₁₇	#447	1	1	yes	characteristic
165	C₄×C₁₇	#480	1	1	yes	characteristic
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	C₂²×M₁₆	—	34	1	no	—
169	C₂³×C₈	—	17	1	no	—
170	C₂²×C₁₆	—	17	1	no	—
171	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
172	C₂×#108	—	17	1	no	—
173	C₂×#108	—	17	1	no	—
174	C₂×#108	—	34	2	no	—
175	C₂×#108	—	34	1	no	—
176	C₂×#108	—	34	2	no	—
177	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
178	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
179	#407	—	34	1	no	—
180	#368	—	34	2	no	—
181	#368	—	34	2	no	—
182	C₂²⋊C₁₆	—	17	2	no	—
183	C₂²⋊C₁₆	—	17	2	no	—
184	#341	—	34	1	no	—
185	C₂×C₁₇	—	2	1	no	—
186	C₂×C₁₇	—	2	1	no	—
187	C₂×C₁₇	#1852	1	1	yes	characteristic
188	C₂×C₁₇	C₂×#368	1	1	yes	characteristic
189	C₂×C₁₇	#2161	1	1	yes	characteristic
190	D₁₇	—	2	2	no	—
191	D₁₇	—	4	1	no	—
192	C₂³×C₄	—	17	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	34	4	no	—
196	C₂×M₁₆	—	34	1	no	—
197	C₂×M₁₆	—	34	1	no	—
198	C₂×M₁₆	—	68	1	no	—
199	C₂×M₁₆	—	68	2	no	—
200	C₂²×C₈	—	17	1	no	—
201	C₂²×C₈	—	17	1	no	—
202	C₂²×C₈	—	17	1	no	—
203	C₂²×C₈	—	17	1	no	—
204	C₂²×C₈	—	17	1	no	—
205	C₂²×C₈	—	34	1	no	—
206	C₂²×C₈	—	34	1	no	—
207	C₂²×C₈	—	34	1	no	—
208	C₂²×C₈	—	34	1	no	—
209	C₂²×C₈	—	68	1	no	—
210	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	17	2	no	—
214	C₂×C₁₆	—	34	1	no	—
215	C₂×C₁₆	—	34	1	no	—
216	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	17	2	no	—
221	#108	—	34	4	no	—
222	#108	—	34	1	no	—
223	#108	—	34	1	no	—
224	#108	—	34	1	no	—
225	#108	D₆₈	1	2	yes	—
226	C₂²⋊C₈	—	68	1	no	—
227	C₂²⋊C₈	—	68	1	no	—
228	C₁₇	#12922	1	1	yes	characteristic, complemented, Sylow
229	C₂⁴	—	17	1	no	—
230	C₂²×C₄	—	17	1	no	—
231	C₂²×C₄	—	17	1	no	—
232	C₂²×C₄	—	17	1	no	—
233	C₂²×C₄	—	17	1	no	—
234	C₂²×C₄	—	17	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	34	2	no	—
245	M₁₆	—	68	4	no	—
246	M₁₆	—	68	1	no	—
247	M₁₆	—	68	1	no	—
248	M₁₆	—	68	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	17	1	no	—
252	C₂×C₈	—	17	1	no	—
253	C₂×C₈	—	34	1	no	—
254	C₂×C₈	—	34	1	no	—
255	C₂×C₈	—	34	1	no	—
256	C₂×C₈	—	34	1	no	—
257	C₂×C₈	—	34	1	no	—
258	C₂×C₈	—	34	1	no	—
259	C₂×C₈	—	34	1	no	—
260	C₂×C₈	—	34	1	no	—
261	C₂×C₈	—	34	1	no	—
262	C₂×C₈	—	34	1	no	—
263	C₂×C₈	—	68	4	no	—
264	C₂×C₈	—	68	1	no	—
265	C₂×C₈	—	68	1	no	—
266	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
267	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
268	C₂×C₈	#63252	1	1	yes	characteristic
269	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
270	C₂×C₈	#63254	1	1	yes	characteristic
271	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	34	2	no	—
274	C₂³	—	17	1	no	—
275	C₂³	—	17	1	no	—
276	C₂³	—	34	2	no	—
277	C₂³	—	34	1	no	—
278	C₂³	—	34	1	no	—
279	C₂³	—	68	1	no	—
280	C₂³	Q₁₆×D₁₇	1	1	yes	characteristic
281	C₂×C₄	—	17	1	no	—
282	C₂×C₄	—	17	1	no	—
283	C₂×C₄	—	34	1	no	—
284	C₂×C₄	—	34	1	no	—
285	C₂×C₄	—	34	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	#10587568	1	1	yes	characteristic
294	C₂×C₄	#10587569	1	1	yes	characteristic
295	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
296	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
297	C₂×C₄	#10587600	1	1	yes	characteristic
298	C₂×C₄	#10587601	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	34	1	no	—
302	C₈	—	34	1	no	—
303	C₈	—	68	2	no	—
304	C₈	—	136	1	no	—
305	C₈	C₄×D₆₈	1	1	yes	characteristic
306	C₈	#10587567	1	1	yes	characteristic
307	C₈	#10587582	1	1	yes	characteristic
308	C₈	#10587583	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	17	2	no	—
312	C₂²	—	34	1	no	—
313	C₂²	—	34	1	no	—
314	C₂²	—	34	1	no	—
315	C₂²	—	34	1	no	—
316	C₂²	—	34	1	no	—
317	C₂²	—	68	2	no	—
318	C₂²	—	68	2	no	—
319	C₂²	#107×D₁₇	1	1	yes	characteristic
320	C₂²	#11762966	1	1	yes	characteristic
321	C₂²	#11763143	1	1	yes	characteristic
322	C₄	—	34	1	no	—
323	C₄	—	34	1	no	—
324	C₄	#11762683	1	1	yes	characteristic
325	C₄	#11762848	1	1	yes	characteristic
326	C₄	#11762944	1	1	yes	characteristic
327	C₄	#11763121	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	34	2	no	—
331	C₂	—	68	1	no	—
332	C₂	#14528973	1	1	yes	characteristic
333	C₂	D₁₇×#368	1	1	yes	characteristic
334	C₂	#14530577	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1381	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×D₁₇	Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1381	D₁₇
hypocenter	C₁₇	#12922
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

Complements

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

N	H ≅ G/N	type
2. C₂×#11762639	C₂	semidirect
3. C₂×#11762638	C₂	semidirect
6. #14526877	C₂	semidirect
7. C₁₇×#1381	C₂	semidirect
148. #1381	D₁₇	semidirect
161. C₂×D₁₇	#368	semidirect
228. C₁₇	#12922	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₁₇	2
C₂×#11762639	2
C₂×#11762638	2
C₂×#11762588	2
#14528819	2
#14526877	2
C₁₇×#1381	2
#12922	17