Subgroups — #16927136

Order 4352.

496 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528936	C₂	1	1	yes	maximal, characteristic
2	#14528901	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	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762564	—	2	1	no	—
11	#11762564	—	2	1	no	—
12	#11762558	—	2	4	no	—
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	1	no	—
24	C₁₇×C₄⋊Q₈	—	2	1	no	—
25	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
26	C₁₇×C₄⋊Q₈	Q₈	1	2	yes	—
27	C₄×Q₈×C₁₇	—	4	4	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₇	—	2	2	no	—
35	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	C₂×C₄	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	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×#3622	—	2	2	no	—
43	C₂²×#3622	—	2	2	no	—
44	C₁₇×#95	—	2	2	no	—
45	C₁₇×#95	—	2	2	no	—
46	C₄⋊#3622	—	2	2	no	—
47	C₄⋊#3622	—	2	2	no	—
48	C₄⋊#3622	—	2	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄⋊#3622	—	4	2	no	—
51	C₄⋊#3622	—	4	2	no	—
52	C₂×Q₈×C₁₇	—	2	2	no	—
53	C₂×Q₈×C₁₇	—	4	2	no	—
54	C₂×Q₈×C₁₇	—	4	2	no	—
55	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₇	—	2	2	no	—
57	C₂²×C₄×C₁₇	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₇	—	2	2	no	—
66	C₄⋊C₄×C₁₇	—	2	2	no	—
67	C₄⋊C₄×C₁₇	—	2	2	no	—
68	C₄⋊C₄×C₁₇	—	2	2	no	—
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₄⋊C₄×C₁₇	—	4	4	no	—
71	C₄⋊C₄×C₁₇	—	4	2	no	—
72	C₄⋊C₄×C₁₇	—	4	1	no	—
73	C₄⋊C₄×C₁₇	—	4	1	no	—
74	C₄²×C₁₇	—	2	2	no	—
75	C₄²×C₁₇	—	2	2	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	—	2	2	no	—
78	C₄²×C₁₇	—	2	1	no	—
79	C₄²×C₁₇	—	4	1	no	—
80	C₄²×C₁₇	—	4	4	no	—
81	C₄²×C₁₇	SD₁₆	1	2	yes	—
82	C₄²×C₁₇	Q₁₆	1	2	yes	—
83	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₁₇	#41	1	1	yes	characteristic
86	C₄²×C₁₇	#41	1	1	yes	characteristic
87	C₂×#3622	—	4	2	no	—
88	C₂×#3622	—	4	2	no	—
89	C₂×#3622	—	4	2	no	—
90	C₂×#3622	—	4	2	no	—
91	C₂×#3622	—	4	2	no	—
92	C₂×#3622	—	4	2	no	—
93	#8590	—	17	1	no	maximal, Sylow
94	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₁₇	—	4	4	no	—
96	Q₈×C₁₇	—	4	4	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	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	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	4	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	—	4	2	no	—
111	C₂×C₄×C₁₇	—	4	2	no	—
112	C₂×C₄×C₁₇	—	4	2	no	—
113	C₂×C₄×C₁₇	—	4	2	no	—
114	C₂×C₄×C₁₇	#100	1	2	yes	—
115	C₂×C₄×C₁₇	#103	1	2	yes	—
116	C₂×C₄×C₁₇	#103	1	2	yes	—
117	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#135	1	2	yes	—
126	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₇	#137	1	2	yes	—
128	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
129	#3622	—	8	2	no	—
130	#3622	—	8	2	no	—
131	C₄×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
132	#1842	—	17	1	no	—
133	#1831	—	17	1	no	—
134	C₂×#333	—	17	2	no	—
135	#1296	—	17	2	no	—
136	C₂²×C₁₇	#380	1	1	yes	characteristic
137	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
138	C₂²×C₁₇	#413	1	1	yes	characteristic
139	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
140	C₂²×C₁₇	#416	1	1	yes	characteristic
141	C₂²×C₁₇	#417	1	1	yes	characteristic
142	C₂²×C₁₇	#419	1	1	yes	characteristic
143	C₄×C₁₇	—	2	2	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	2	no	—
147	C₄×C₁₇	—	2	2	no	—
148	C₄×C₁₇	—	4	4	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	8	2	no	—
151	C₄×C₁₇	#460	1	1	yes	characteristic
152	C₄×C₁₇	#462	1	1	yes	characteristic
153	C₄×C₁₇	#464	1	2	yes	—
154	C₄×C₁₇	#475	1	1	yes	characteristic
155	C₄×C₁₇	#477	1	1	yes	characteristic
156	C₄×C₁₇	#479	1	2	yes	—
157	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
158	C₂×C₄×Q₈	Dic₁₇	1	2	yes	—
159	C₂×C₄⋊C₈	—	17	2	no	—
160	C₂×C₄⋊C₈	—	17	2	no	—
161	C₂×C₄⋊C₈	—	34	1	no	—
162	C₂×C₄⋊C₈	—	34	1	no	—
163	#391	Dic₁₇	1	2	yes	—
164	#389	—	2	1	no	—
165	#389	—	2	1	no	—
166	#384	C₂×D₁₇	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	1	no	—
168	C₄×C₄⋊C₄	—	2	1	no	—
169	C₄³	C₂×D₁₇	1	1	yes	characteristic
170	#336	—	34	1	no	—
171	#336	—	34	1	no	—
172	#333	—	34	4	no	—
173	C₂×C₁₇	#1542	1	1	yes	characteristic
174	C₂×C₁₇	#1548	1	1	yes	characteristic
175	C₂×C₁₇	#1551	1	1	yes	characteristic
176	C₂×C₁₇	#1559	1	1	yes	characteristic
177	C₂×C₁₇	#1800	1	1	yes	characteristic
178	C₂×C₁₇	#1803	1	1	yes	characteristic
179	C₂×C₁₇	#1805	1	1	yes	characteristic
180	C₂²×Q₈	C₂×Dic₁₇	1	2	yes	—
181	C₂²×C₈	—	34	2	no	—
182	C₂²×C₈	—	34	2	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	Dic₃₄	1	2	yes	—
186	C₄⋊Q₈	D₆₈	1	2	yes	—
187	C₄×Q₈	—	4	4	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
196	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
202	C₄⋊C₈	—	34	2	no	—
203	C₄⋊C₈	—	34	2	no	—
204	C₄⋊C₈	—	34	2	no	—
205	C₄⋊C₈	—	34	2	no	—
206	C₄⋊C₈	—	68	2	no	—
207	C₄⋊C₈	—	68	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₁₇	#8590	1	1	yes	characteristic, complemented, Sylow
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
224	C₂×C₈	—	68	2	no	—
225	C₂×C₈	—	68	2	no	—
226	C₂×C₈	—	68	2	no	—
227	C₂×C₈	—	68	2	no	—
228	C₂×C₈	—	68	2	no	—
229	C₂×C₈	—	68	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	4	no	—
246	C₄²	#63230	1	2	yes	—
247	C₄²	#63231	1	2	yes	—
248	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
249	C₄²	C₂×D₆₈	1	1	yes	characteristic
250	C₄²	#63252	1	1	yes	characteristic
251	C₄²	#63252	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	#10587504	1	2	yes	—
273	C₂×C₄	#10587506	1	2	yes	—
274	C₂×C₄	#10587506	1	2	yes	—
275	C₂×C₄	#10587610	1	1	yes	characteristic
276	C₂×C₄	#10587610	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
278	C₂×C₄	#10587612	1	1	yes	characteristic
279	C₂×C₄	#10587616	1	1	yes	characteristic
280	C₂×C₄	#10587616	1	1	yes	characteristic
281	C₂×C₄	C₂×#63230	1	1	yes	characteristic
282	C₂×C₄	#10587628	1	2	yes	—
283	C₂×C₄	C₂×#63231	1	1	yes	characteristic
284	C₂×C₄	#10587635	1	1	yes	characteristic
285	C₂×C₄	#10587636	1	2	yes	—
286	C₂×C₄	#10587637	1	1	yes	characteristic
287	C₈	—	136	2	no	—
288	C₈	—	136	2	no	—
289	C₂²	#11762669	1	1	yes	characteristic
290	C₂²	C₂×#10587504	1	1	yes	characteristic
291	C₂²	#11762793	1	1	yes	characteristic
292	C₂²	C₂×#10587506	1	1	yes	characteristic
293	C₂²	#11762805	1	1	yes	characteristic
294	C₂²	#11762810	1	1	yes	characteristic
295	C₂²	#11762818	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	#11763009	1	1	yes	characteristic
305	C₄	#11763019	1	1	yes	characteristic
306	C₄	#11763031	1	2	yes	—
307	C₄	#11763094	1	1	yes	characteristic
308	C₄	#11763104	1	1	yes	characteristic
309	C₄	#11763116	1	2	yes	—
310	C₂	#14527441	1	1	yes	characteristic
311	C₂	#14527465	1	1	yes	characteristic
312	C₂	#14527479	1	1	yes	characteristic
313	C₂	#14527514	1	1	yes	characteristic
314	C₂	#14528776	1	1	yes	characteristic
315	C₂	#14528789	1	1	yes	characteristic
316	C₂	#14528799	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
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	#8590
LCS	G ▷ C₂²×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
210. C₁₇	#8590	semidirect

Maximal subgroups

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

subgroup	index
#14528936	2
#14528901	2
C₂×#11762558	2
#14526667	2
C₄×C₁₇×C₄⋊Q₈	2
#8590	17