Subgroups — #17948231

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3624	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3624	C₄	1	2	yes	—
5	C₂×C₄×#3624	—	2	1	no	—
6	C₂×C₄×#3624	—	2	1	no	—
7	C₂×C₄×#3624	—	2	1	no	—
8	C₂×C₄×#3624	C₄	1	2	yes	—
9	#11763675	—	2	1	no	—
10	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	—	2	1	no	—
12	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
13	#11762563	—	2	1	no	—
14	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₂²×#3624	—	2	2	no	—
16	C₂²×#3624	—	2	2	no	—
17	C₂²×C₈×C₁₇	—	2	1	no	—
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
25	C₂²×#3622	—	2	1	no	—
26	C₄⋊#3624	—	2	2	no	—
27	C₄⋊#3624	C₈	1	4	yes	complemented
28	C₄×#3624	—	2	2	no	—
29	C₄×#3624	—	2	2	no	—
30	C₄×#3624	—	2	4	no	—
31	C₄×#3624	—	2	2	no	—
32	C₄×#3624	—	4	1	no	—
33	C₄×#3624	—	4	1	no	—
34	C₄×#3624	C₈	1	4	yes	complemented
35	C₁₇×C₄⋊C₈	—	2	1	no	—
36	C₁₇×C₄⋊C₈	C₈	1	2	yes	complemented
37	C₈×Dic₁₇	—	4	1	no	—
38	C₈×Dic₁₇	—	4	1	no	—
39	C₄⋊#3622	—	2	1	no	—
40	C₄⋊#3622	C₈	1	2	yes	complemented
41	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#3624	—	2	4	no	—
49	C₂×#3624	—	2	4	no	—
50	C₂×#3624	—	2	4	no	—
51	C₂×#3624	—	2	4	no	—
52	C₂×#3624	—	4	2	no	—
53	C₂×#3624	—	4	2	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	4	1	no	—
56	C₂×C₈×C₁₇	—	4	1	no	—
57	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
60	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
61	C₄×Dic₁₇	—	2	1	no	—
62	C₄×Dic₁₇	—	2	1	no	—
63	C₄×Dic₁₇	—	2	2	no	—
64	C₄×Dic₁₇	—	2	2	no	—
65	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
67	C₄×Dic₁₇	M₁₆	1	2	yes	—
68	C₄×Dic₁₇	M₁₆	1	2	yes	—
69	C₄×Dic₁₇	D₈	1	1	yes	characteristic
70	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
71	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
72	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
73	C₂×#3622	—	2	2	no	—
74	C₂×#3622	—	4	1	no	—
75	C₂×#3622	—	4	1	no	—
76	#7015	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	#95	1	1	yes	characteristic
78	C₂×C₄×C₁₇	—	2	1	no	—
79	C₂×C₄×C₁₇	—	2	1	no	—
80	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
94	C₂×Dic₁₇	—	2	1	no	—
95	C₂×Dic₁₇	—	2	1	no	—
96	C₂×Dic₁₇	—	2	1	no	—
97	C₂×Dic₁₇	—	2	1	no	—
98	C₂×Dic₁₇	C₂²⋊C₈	1	2	yes	—
99	C₂×Dic₁₇	C₂²⋊C₈	1	2	yes	—
100	C₂×Dic₁₇	#102	1	1	yes	characteristic
101	C₂×Dic₁₇	#103	1	1	yes	characteristic
102	C₂×Dic₁₇	#104	1	2	yes	—
103	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
104	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
105	C₂×Dic₁₇	#106	1	1	yes	characteristic
106	C₂×Dic₁₇	#107	1	1	yes	characteristic
107	C₂×Dic₁₇	#108	1	2	yes	—
108	#3624	—	2	8	no	—
109	#3624	—	4	4	no	—
110	C₈×C₁₇	—	4	2	no	—
111	#3622	—	4	2	no	—
112	C₄×C₄⋊C₈	—	17	1	no	—
113	C₄×C₄⋊C₈	—	17	1	no	—
114	C₄²×C₈	—	17	1	no	—
115	C₂²×C₁₇	#336	1	1	yes	characteristic
116	C₂²×C₁₇	#336	1	1	yes	characteristic
117	C₂²×C₁₇	#336	1	1	yes	characteristic
118	C₂²×C₁₇	#339	1	1	yes	characteristic
119	C₂²×C₁₇	#339	1	1	yes	characteristic
120	C₂²×C₁₇	#340	1	1	yes	characteristic
121	C₂²×C₁₇	#341	1	1	yes	characteristic
122	C₄×C₁₇	—	2	1	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	C₈²	1	1	yes	characteristic
125	C₄×C₁₇	#322	1	1	yes	characteristic
126	C₄×C₁₇	#322	1	1	yes	characteristic
127	C₄×C₁₇	#322	1	1	yes	characteristic
128	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
129	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
130	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
131	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
132	Dic₁₇	—	2	2	no	—
133	Dic₁₇	—	2	2	no	—
134	Dic₁₇	D₄⋊C₈	1	2	yes	—
135	Dic₁₇	Q₈⋊C₈	1	2	yes	—
136	Dic₁₇	#334	1	2	yes	—
137	Dic₁₇	#335	1	2	yes	—
138	C₂×C₄⋊C₈	—	17	2	no	—
139	C₂×C₄⋊C₈	—	17	1	no	—
140	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
141	C₂×C₄×C₈	—	17	2	no	—
142	C₂×C₄×C₈	—	34	1	no	—
143	C₂×C₄×C₈	—	34	1	no	—
144	C₂×C₄×C₈	—	34	1	no	—
145	C₂×C₄×C₈	—	34	1	no	—
146	C₄³	—	17	1	no	—
147	#336	—	34	1	no	—
148	#336	—	34	1	no	—
149	C₂×C₁₇	#1273	1	1	yes	characteristic
150	C₂×C₁₇	#1276	1	1	yes	characteristic
151	C₂×C₁₇	#1277	1	1	yes	characteristic
152	C₂×C₁₇	#1278	1	1	yes	characteristic
153	C₂×C₁₇	#1278	1	1	yes	characteristic
154	C₂×C₁₇	#1279	1	1	yes	characteristic
155	C₂×C₁₇	#1279	1	1	yes	characteristic
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	34	2	no	—
158	C₂²×C₈	—	34	2	no	—
159	C₂²×C₈	—	34	1	no	—
160	C₂×C₄²	—	17	1	no	—
161	C₂×C₄²	—	17	1	no	—
162	C₂×C₄²	—	17	1	no	—
163	C₂×C₄²	—	17	1	no	—
164	C₂×C₄²	—	17	1	no	—
165	C₂×C₄²	—	17	1	no	—
166	C₂×C₄²	C₂×#595	1	1	yes	characteristic
167	C₄⋊C₈	—	2	1	no	—
168	C₄⋊C₈	—	17	4	no	—
169	C₄⋊C₈	—	17	2	no	—
170	C₄⋊C₈	—	34	2	no	—
171	C₄⋊C₈	—	34	1	no	—
172	C₄⋊C₈	#3624	1	2	yes	complemented
173	C₄×C₈	—	17	4	no	—
174	C₄×C₈	—	34	2	no	—
175	C₄×C₈	—	34	4	no	—
176	C₄×C₈	—	34	2	no	—
177	C₄×C₈	—	34	2	no	—
178	C₄×C₈	—	68	1	no	—
179	C₄×C₈	—	68	1	no	—
180	C₄×C₈	—	68	1	no	—
181	C₄×C₈	—	68	1	no	—
182	C₁₇	#7015	1	1	yes	characteristic, complemented, Sylow
183	C₂²×C₄	—	17	1	no	—
184	C₂²×C₄	—	17	1	no	—
185	C₂²×C₄	—	17	1	no	—
186	C₂²×C₄	—	17	1	no	—
187	C₂²×C₄	C₄×#595	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	34	4	no	—
194	C₂×C₈	—	34	4	no	—
195	C₂×C₈	—	34	4	no	—
196	C₂×C₈	—	34	4	no	—
197	C₂×C₈	—	34	2	no	—
198	C₂×C₈	—	68	2	no	—
199	C₂×C₈	—	68	2	no	—
200	C₂×C₈	—	68	1	no	—
201	C₂×C₈	—	68	1	no	—
202	C₄²	—	17	2	no	—
203	C₄²	—	17	2	no	—
204	C₄²	—	17	2	no	—
205	C₄²	—	17	2	no	—
206	C₄²	—	17	1	no	—
207	C₄²	—	17	1	no	—
208	C₄²	—	17	1	no	—
209	C₄²	—	17	1	no	—
210	C₄²	—	34	2	no	—
211	C₄²	—	34	2	no	—
212	C₄²	—	34	1	no	—
213	C₄²	—	34	1	no	—
214	C₄²	#63242	1	1	yes	characteristic
215	C₄²	#63243	1	1	yes	characteristic
216	C₄²	C₂×#3624	1	1	yes	characteristic
217	C₄²	#63247	1	1	yes	characteristic
218	C₂³	(C₂×#595)⋊C₄	1	1	yes	characteristic
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	17	2	no	—
222	C₂×C₄	—	17	2	no	—
223	C₂×C₄	—	17	2	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	1	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	1	no	—
228	C₂×C₄	—	17	2	no	—
229	C₂×C₄	—	17	2	no	—
230	C₂×C₄	—	17	2	no	—
231	C₂×C₄	—	34	1	no	—
232	C₂×C₄	—	34	1	no	—
233	C₂×C₄	—	34	1	no	—
234	C₂×C₄	—	34	1	no	—
235	C₂×C₄	C₈×#595	1	1	yes	characteristic
236	C₂×C₄	C₈×#595	1	1	yes	characteristic
237	C₂×C₄	#10587532	1	1	yes	characteristic
238	C₂×C₄	#10587532	1	1	yes	characteristic
239	C₂×C₄	C₄×#3624	1	1	yes	characteristic
240	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
241	C₂×C₄	#10587542	1	1	yes	characteristic
242	C₂×C₄	#10587543	1	1	yes	characteristic
243	C₂×C₄	#10587544	1	1	yes	characteristic
244	C₂×C₄	#10587547	1	1	yes	characteristic
245	C₂×C₄	#10587548	1	1	yes	characteristic
246	C₂×C₄	#10587549	1	1	yes	characteristic
247	C₂×C₄	#10587550	1	1	yes	characteristic
248	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
249	C₈	—	4	2	no	—
250	C₈	—	34	8	no	—
251	C₈	—	68	4	no	—
252	C₈	—	68	2	no	—
253	C₂²	#11763674	1	1	yes	characteristic
254	C₂²	#11763674	1	1	yes	characteristic
255	C₂²	#11763675	1	1	yes	characteristic
256	C₂²	#11763679	1	1	yes	characteristic
257	C₂²	#11763681	1	1	yes	characteristic
258	C₂²	#11763682	1	1	yes	characteristic
259	C₂²	#11763684	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	17	2	no	—
263	C₄	—	17	2	no	—
264	C₄	—	17	2	no	—
265	C₄	—	17	2	no	—
266	C₄	—	34	2	no	—
267	C₄	—	34	2	no	—
268	C₄	C₈×#3624	1	1	yes	characteristic
269	C₄	#11763650	1	1	yes	characteristic
270	C₄	#11763651	1	1	yes	characteristic
271	C₄	#11763651	1	1	yes	characteristic
272	C₄	D₄⋊#3624	1	1	yes	characteristic
273	C₄	#11763657	1	1	yes	characteristic
274	C₄	Q₈⋊#3624	1	1	yes	characteristic
275	C₄	#11763659	1	1	yes	characteristic
276	C₂	#14543606	1	1	yes	characteristic
277	C₂	#14543611	1	1	yes	characteristic
278	C₂	#14543613	1	1	yes	characteristic
279	C₂	#14543615	1	1	yes	characteristic
280	C₂	#14543617	1	1	yes	characteristic
281	C₂	#14543618	1	1	yes	characteristic
282	C₂	#14543620	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₈²
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₇	#95
cosocle	C₄²×Dic₁₇	C₂²
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	#7015
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₄⋊#3624	C₈	semidirect
34. C₄×#3624	C₈	semidirect
36. C₁₇×C₄⋊C₈	C₈	semidirect
40. C₄⋊#3622	C₈	semidirect
172. C₄⋊C₈	#3624	semidirect
182. C₁₇	#7015	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3624	2
C₄²×#3624	2
C₄⋊C₈×Dic₁₇	2
#7015	17