Subgroups — #16934310

Order 4352.

528 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	#14526625	C₂	1	4	yes	maximal
4	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	—	2	1	no	—
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄×#3622	—	2	1	no	—
8	C₂×C₄×#3622	C₂²	1	2	yes	—
9	#11762564	—	2	1	no	—
10	#11762564	—	2	1	no	—
11	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₇	—	2	1	no	—
13	C₁₇×#391	—	2	1	no	—
14	C₁₇×#389	—	2	1	no	—
15	C₁₇×#389	C₄	1	2	yes	—
16	C₁₇×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
19	C₄³×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	2	1	no	—
21	C₁₇×C₄⋊Q₈	—	2	2	no	—
22	C₁₇×C₄⋊Q₈	C₈	1	4	yes	complemented
23	C₄×Q₈×C₁₇	—	4	2	no	—
24	C₄×Q₈×C₁₇	—	4	2	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₇	—	2	2	no	—
32	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₇	D₄	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	C₂³	1	1	yes	characteristic
39	C₂²×#3622	—	2	2	no	—
40	C₂²×#3622	—	2	2	no	—
41	C₁₇×#95	—	2	2	no	—
42	C₁₇×#95	—	2	2	no	—
43	C₄⋊#3622	—	2	4	no	—
44	C₄⋊#3622	—	2	2	no	—
45	C₄⋊#3622	Q₈	1	4	yes	complemented
46	C₄×#3622	—	2	4	no	—
47	C₄×#3622	—	4	2	no	—
48	C₂×Q₈×C₁₇	—	2	4	no	—
49	C₂×Q₈×C₁₇	—	4	2	no	—
50	C₂×Q₈×C₁₇	—	4	2	no	—
51	C₂²×C₄×C₁₇	—	2	2	no	—
52	C₂²×C₄×C₁₇	—	2	2	no	—
53	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₇	—	2	4	no	—
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊C₄×C₁₇	—	2	4	no	—
63	C₄⋊C₄×C₁₇	—	4	2	no	—
64	C₄⋊C₄×C₁₇	—	4	4	no	—
65	C₄⋊C₄×C₁₇	—	4	2	no	—
66	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
67	C₄²×C₁₇	—	2	2	no	—
68	C₄²×C₁₇	—	2	2	no	—
69	C₄²×C₁₇	—	2	2	no	—
70	C₄²×C₁₇	—	4	1	no	—
71	C₄²×C₁₇	—	4	2	no	—
72	C₄²×C₁₇	—	4	2	no	—
73	C₄²×C₁₇	C₂×C₈	1	2	yes	—
74	C₄²×C₁₇	M₁₆	1	2	yes	—
75	C₄²×C₁₇	D₈	1	2	yes	—
76	C₄²×C₁₇	SD₁₆	1	2	yes	—
77	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
78	C₄²×C₁₇	#41	1	2	yes	—
79	C₂×#3622	—	2	4	no	—
80	C₂×#3622	—	2	4	no	—
81	C₂×#3622	—	4	2	no	—
82	C₂×#3622	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	#9865	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₁₇	—	4	8	no	—
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	4	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	—	4	2	no	—
99	C₂×C₄×C₁₇	—	4	2	no	—
100	C₂×C₄×C₁₇	—	4	2	no	—
101	C₂×C₄×C₁₇	—	4	2	no	—
102	C₂×C₄×C₁₇	—	4	2	no	—
103	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₁₇	#102	1	2	yes	—
106	C₂×C₄×C₁₇	#102	1	2	yes	—
107	C₂×C₄×C₁₇	#104	1	2	yes	—
108	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₇	#131	1	2	yes	—
115	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₇	#137	1	2	yes	—
118	#3622	—	4	4	no	—
119	#3622	—	4	4	no	—
120	C₄×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
121	C₄×C₄⋊C₈	—	17	1	no	—
122	C₄×C₄⋊C₈	—	17	1	no	—
123	#1279	—	17	4	no	—
124	C₂²×C₁₇	#391	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₂²×C₁₇	#408	1	1	yes	characteristic
127	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
128	C₂²×C₁₇	#419	1	1	yes	characteristic
129	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
130	C₂²×C₁₇	#421	1	1	yes	characteristic
131	C₄×C₁₇	—	2	1	no	—
132	C₄×C₁₇	—	2	1	no	—
133	C₄×C₁₇	—	2	2	no	—
134	C₄×C₁₇	—	2	2	no	—
135	C₄×C₁₇	—	4	4	no	—
136	C₄×C₁₇	—	4	1	no	—
137	C₄×C₁₇	—	8	2	no	—
138	C₄×C₁₇	D₄⋊C₈	1	4	yes	—
139	C₄×C₁₇	C₈×Q₈	1	2	yes	—
140	C₄×C₁₇	#446	1	2	yes	—
141	C₄×C₁₇	#474	1	2	yes	—
142	C₄×C₁₇	#475	1	2	yes	—
143	C₂×C₄⋊Q₈	Dic₁₇	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×C₄⋊C₈	—	17	2	no	—
146	C₂×C₄⋊C₈	—	34	1	no	—
147	C₂×C₄×C₈	—	17	2	no	—
148	C₂×C₄×C₈	—	34	1	no	—
149	#391	—	2	1	no	—
150	#389	—	2	1	no	—
151	#389	Dic₁₇	1	2	yes	—
152	#384	Dic₁₇	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	2	1	no	—
154	C₄×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
155	C₄³	C₂×D₁₇	1	1	yes	characteristic
156	#336	—	34	1	no	—
157	#336	—	34	1	no	—
158	C₂×C₁₇	C₂×D₄⋊C₈	1	1	yes	characteristic
159	C₂×C₁₇	#1479	1	1	yes	characteristic
160	C₂×C₁₇	#1480	1	1	yes	characteristic
161	C₂×C₁₇	#1485	1	1	yes	characteristic
162	C₂×C₁₇	#1984	1	1	yes	characteristic
163	C₂×C₁₇	#1986	1	1	yes	characteristic
164	C₂×C₁₇	#1988	1	1	yes	characteristic
165	C₂²×Q₈	—	2	1	no	—
166	C₂²×C₈	—	34	2	no	—
167	C₂²×C₈	—	34	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	#3622	1	4	yes	complemented
170	C₄×Q₈	—	4	2	no	—
171	C₄×Q₈	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
186	C₄⋊C₈	—	17	4	no	—
187	C₄⋊C₈	—	34	4	no	—
188	C₄⋊C₈	—	34	2	no	—
189	C₄×C₈	—	34	4	no	—
190	C₄×C₈	—	68	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₁₇	#9865	1	1	yes	characteristic, complemented, Sylow
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
206	C₂×C₈	—	34	4	no	—
207	C₂×C₈	—	34	4	no	—
208	C₂×C₈	—	68	2	no	—
209	C₂×C₈	—	68	2	no	—
210	C₂×C₈	—	68	2	no	—
211	C₂×C₈	—	68	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	C₂×#3622	1	4	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×#3622	1	2	yes	—
226	C₄²	#63223	1	2	yes	—
227	C₄²	#63228	1	2	yes	—
228	C₄²	#63229	1	2	yes	—
229	C₄²	Q₈×D₁₇	1	2	yes	—
230	C₄²	#63256	1	2	yes	—
231	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
232	Q₈	—	4	8	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂²⋊#3622	1	2	yes	—
249	C₂×C₄	C₂²⋊#3622	1	2	yes	—
250	C₂×C₄	#10587503	1	2	yes	—
251	C₂×C₄	#10587503	1	2	yes	—
252	C₂×C₄	#10587508	1	2	yes	—
253	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
254	C₂×C₄	C₂×#63223	1	1	yes	characteristic
255	C₂×C₄	C₂×#63228	1	1	yes	characteristic
256	C₂×C₄	C₂×#63229	1	1	yes	characteristic
257	C₂×C₄	#10587628	1	2	yes	—
258	C₂×C₄	#10587630	1	1	yes	characteristic
259	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
260	C₂×C₄	#10587632	1	2	yes	—
261	C₂×C₄	#10587633	1	1	yes	characteristic
262	C₂×C₄	#10587634	1	2	yes	—
263	C₈	—	68	4	no	—
264	C₈	—	68	4	no	—
265	C₂²	#11762716	1	1	yes	characteristic
266	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
267	C₂²	#11762776	1	1	yes	characteristic
268	C₂²	C₂×#10587503	1	1	yes	characteristic
269	C₂²	#11762818	1	1	yes	characteristic
270	C₂²	C₂×#10587508	1	1	yes	characteristic
271	C₂²	#11762826	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	1	no	—
278	C₄	—	8	2	no	—
279	C₄	D₄⋊#3622	1	4	yes	—
280	C₄	Q₈×#3622	1	2	yes	—
281	C₄	#11762940	1	2	yes	—
282	C₄	#11763086	1	2	yes	—
283	C₄	#11763091	1	2	yes	—
284	C₂	C₂×D₄⋊#3622	1	1	yes	characteristic
285	C₂	#14527179	1	1	yes	characteristic
286	C₂	#14527183	1	1	yes	characteristic
287	C₂	#14527205	1	1	yes	characteristic
288	C₂	#14529721	1	1	yes	characteristic
289	C₂	#14529729	1	1	yes	characteristic
290	C₂	#14529737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
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	#9865
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. C₁₇×C₄⋊Q₈	C₈	semidirect
45. C₄⋊#3622	Q₈	semidirect
169. C₄⋊Q₈	#3622	semidirect
193. C₁₇	#9865	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3622	2
#14526625	2
C₄×C₁₇×C₄⋊Q₈	2
#9865	17