Subgroups — #16925295

Order 4352.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528637	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762558	C₂	1	2	yes	maximal
4	#14526667	C₂	1	2	yes	maximal
5	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	2	yes	—
8	C₂×#10587474	—	2	1	no	—
9	C₂×C₄×#3622	—	2	1	no	—
10	#11762564	—	2	1	no	—
11	#11762564	—	2	1	no	—
12	#11762558	C₄	1	8	yes	complemented
13	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₇	C₄	1	2	yes	—
15	C₁₇×#391	C₄	1	2	yes	—
16	C₁₇×#389	—	2	1	no	—
17	C₁₇×#389	—	2	1	no	—
18	C₁₇×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
21	C₄³×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₄×Q₈×C₁₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₇	—	2	2	no	—
33	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₂²×#3622	—	2	2	no	—
41	C₂²×#3622	—	2	2	no	—
42	C₁₇×#95	—	2	2	no	—
43	C₁₇×#95	—	2	2	no	—
44	C₄⋊#3622	—	2	2	no	—
45	C₄⋊#3622	—	2	2	no	—
46	C₄⋊#3622	C₂×C₄	1	4	yes	—
47	C₄⋊#3622	C₂×C₄	1	4	yes	—
48	#10587474	—	4	2	no	—
49	C₄×#3622	—	4	2	no	—
50	C₂×Q₈×C₁₇	—	2	2	no	—
51	C₂×Q₈×C₁₇	—	4	2	no	—
52	C₂×Q₈×C₁₇	—	4	2	no	—
53	C₂×Q₈×C₁₇	C₄²	1	4	yes	—
54	C₂²×C₄×C₁₇	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₇	—	2	4	no	—
64	C₄⋊C₄×C₁₇	—	2	4	no	—
65	C₄⋊C₄×C₁₇	—	4	2	no	—
66	C₄⋊C₄×C₁₇	—	4	4	no	—
67	C₄⋊C₄×C₁₇	—	4	2	no	—
68	C₄⋊C₄×C₁₇	—	4	1	no	—
69	C₄⋊C₄×C₁₇	—	4	1	no	—
70	C₄²×C₁₇	—	2	2	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	—	2	1	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	—	4	1	no	—
76	C₄²×C₁₇	—	4	4	no	—
77	C₄²×C₁₇	SD₁₆	1	2	yes	—
78	C₄²×C₁₇	Q₁₆	1	2	yes	—
79	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₇	#41	1	2	yes	—
81	C₂×#3622	—	2	4	no	—
82	C₂×#3622	—	2	4	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	C₂×#3622	—	4	2	no	—
86	C₂×#3622	—	4	2	no	—
87	#8235	—	17	1	no	maximal, Sylow
88	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₁₇	—	4	4	no	—
90	Q₈×C₁₇	—	4	4	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	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	4	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	4	2	no	—
106	C₂×C₄×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	—	4	2	no	—
108	C₂×C₄×C₁₇	#100	1	2	yes	—
109	C₂×C₄×C₁₇	#103	1	2	yes	—
110	C₂×C₄×C₁₇	#103	1	2	yes	—
111	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₇	#135	1	2	yes	—
118	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₇	#137	1	2	yes	—
120	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
121	#3622	—	4	4	no	—
122	#3622	—	4	4	no	—
123	C₄×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
124	#1767	—	17	1	no	—
125	C₄×C₄⋊C₈	—	17	1	no	—
126	C₂×#333	—	17	2	no	—
127	#1296	—	17	2	no	—
128	C₂²×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
130	C₂²×C₁₇	#413	1	1	yes	characteristic
131	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
132	C₂²×C₁₇	#416	1	1	yes	characteristic
133	C₂²×C₁₇	#417	1	1	yes	characteristic
134	C₂²×C₁₇	#419	1	1	yes	characteristic
135	C₄×C₁₇	—	2	2	no	—
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	—	4	4	no	—
142	C₄×C₁₇	—	8	2	no	—
143	C₄×C₁₇	C₄×SD₁₆	1	2	yes	—
144	C₄×C₁₇	C₄×Q₁₆	1	2	yes	—
145	C₄×C₁₇	#440	1	2	yes	—
146	C₄×C₁₇	#441	1	2	yes	—
147	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
148	C₂×C₄×Q₈	Dic₁₇	1	2	yes	—
149	C₂×C₄⋊C₈	—	17	2	no	—
150	C₂×C₄⋊C₈	—	17	2	no	—
151	C₂×#97	—	34	1	no	—
152	C₂×C₄×C₈	—	34	1	no	—
153	#391	Dic₁₇	1	2	yes	—
154	#389	—	2	1	no	—
155	#389	—	2	1	no	—
156	#384	C₂×D₁₇	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	2	1	no	—
158	C₄×C₄⋊C₄	—	2	1	no	—
159	C₄³	C₂×D₁₇	1	1	yes	characteristic
160	#336	—	34	1	no	—
161	#336	—	34	1	no	—
162	#333	—	17	8	no	—
163	C₂×C₁₇	#1548	1	1	yes	characteristic
164	C₂×C₁₇	#1550	1	1	yes	characteristic
165	C₂×C₁₇	#1551	1	1	yes	characteristic
166	C₂×C₁₇	#1554	1	1	yes	characteristic
167	C₂×C₁₇	C₄×#100	1	1	yes	characteristic
168	C₂×C₁₇	C₄×#103	1	1	yes	characteristic
169	C₂×C₁₇	#1763	1	1	yes	characteristic
170	C₂²×Q₈	C₂×Dic₁₇	1	2	yes	—
171	C₂²×C₈	—	34	2	no	—
172	C₂²×C₈	—	34	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₄×D₁₇	1	4	yes	—
175	C₄×Q₈	—	4	4	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
190	C₄⋊C₈	—	17	4	no	—
191	C₄⋊C₈	—	17	4	no	—
192	C₄⋊C₈	—	34	2	no	—
193	C₄⋊C₈	—	34	2	no	—
194	#97	—	68	2	no	—
195	C₄×C₈	—	68	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	C₁₇	#8235	1	1	yes	characteristic, complemented, Sylow
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	C₄×Dic₁₇	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
212	C₂×C₈	—	34	4	no	—
213	C₂×C₈	—	34	4	no	—
214	C₂×C₈	—	68	2	no	—
215	C₂×C₈	—	68	2	no	—
216	C₂×C₈	—	68	2	no	—
217	C₂×C₈	—	68	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	4	no	—
232	C₄²	#63230	1	2	yes	—
233	C₄²	#63231	1	2	yes	—
234	C₄²	C₂×C₄×D₁₇	1	2	yes	—
235	C₄²	#63252	1	2	yes	—
236	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	4	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	#10587504	1	2	yes	—
257	C₂×C₄	#10587506	1	2	yes	—
258	C₂×C₄	#10587506	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
260	C₂×C₄	#10587612	1	1	yes	characteristic
261	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
262	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
263	C₂×C₄	C₂×#63230	1	1	yes	characteristic
264	C₂×C₄	#10587628	1	2	yes	—
265	C₂×C₄	C₂×#63231	1	1	yes	characteristic
266	C₂×C₄	#10587635	1	1	yes	characteristic
267	C₂×C₄	#10587636	1	2	yes	—
268	C₂×C₄	#10587637	1	1	yes	characteristic
269	C₈	—	68	4	no	—
270	C₈	—	68	4	no	—
271	C₂²	C₄×C₂²⋊Dic₁₇	1	1	yes	characteristic
272	C₂²	C₂×#10587504	1	1	yes	characteristic
273	C₂²	#11762793	1	1	yes	characteristic
274	C₂²	C₂×#10587506	1	1	yes	characteristic
275	C₂²	#11762805	1	1	yes	characteristic
276	C₂²	#11762810	1	1	yes	characteristic
277	C₂²	#11762818	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄×#63230	1	2	yes	—
287	C₄	C₄×#63231	1	2	yes	—
288	C₄	#11762907	1	2	yes	—
289	C₄	#11762914	1	2	yes	—
290	C₂	#14527465	1	1	yes	characteristic
291	C₂	#14527474	1	1	yes	characteristic
292	C₂	#14527479	1	1	yes	characteristic
293	C₂	#14527494	1	1	yes	characteristic
294	C₂	C₄×#10587504	1	1	yes	characteristic
295	C₂	C₄×#10587506	1	1	yes	characteristic
296	C₂	#14528614	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#8235
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
12. #11762558	C₄	semidirect
198. C₁₇	#8235	semidirect

Maximal subgroups

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

subgroup	index
#14528637	2
C₄×C₄⋊#3622	2
C₂×#11762558	2
#14526667	2
C₄×C₁₇×C₄⋊Q₈	2
#8235	17