Subgroups — #17948272

Order 4352.

352 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544716	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3624	—	2	1	no	—
5	C₂×C₄⋊#3624	—	2	1	no	—
6	C₂×C₄⋊#3624	—	2	1	no	—
7	C₂×C₄⋊#3624	C₄	1	2	yes	—
8	C₂×C₄×#3624	—	2	1	no	—
9	#11763675	C₄	1	2	yes	—
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	—	2	2	no	—
28	C₄⋊#3624	—	4	1	no	—
29	C₄⋊#3624	—	4	1	no	—
30	C₄⋊#3624	—	4	1	no	—
31	C₄⋊#3624	—	4	1	no	—
32	C₄⋊#3624	—	4	2	no	—
33	C₄×#3624	—	4	1	no	—
34	C₄×#3624	—	4	1	no	—
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	—	4	2	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×#3624	—	4	2	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	1	no	—
64	C₄×Dic₁₇	—	2	1	no	—
65	C₄×Dic₁₇	—	2	1	no	—
66	C₄×Dic₁₇	—	2	1	no	—
67	C₄×Dic₁₇	—	4	1	no	—
68	C₄×Dic₁₇	D₈	1	1	yes	characteristic
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	SD₁₆	1	1	yes	characteristic
73	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
74	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
75	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
76	C₂×#3622	—	2	2	no	—
77	C₂×#3622	—	4	1	no	—
78	C₂×#3622	—	4	1	no	—
79	#7031	—	17	1	no	maximal, Sylow
80	C₂³×C₁₇	#95	1	1	yes	characteristic
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
97	C₂×Dic₁₇	—	2	1	no	—
98	C₂×Dic₁₇	—	2	1	no	—
99	C₂×Dic₁₇	—	2	1	no	—
100	C₂×Dic₁₇	—	2	1	no	—
101	C₂×Dic₁₇	—	2	1	no	—
102	C₂×Dic₁₇	—	2	1	no	—
103	C₂×Dic₁₇	—	2	1	no	—
104	C₂×Dic₁₇	#100	1	1	yes	characteristic
105	C₂×Dic₁₇	#101	1	1	yes	characteristic
106	C₂×Dic₁₇	#102	1	1	yes	characteristic
107	C₂×Dic₁₇	#102	1	1	yes	characteristic
108	C₂×Dic₁₇	#102	1	1	yes	characteristic
109	C₂×Dic₁₇	#103	1	1	yes	characteristic
110	C₂×Dic₁₇	#103	1	1	yes	characteristic
111	C₂×Dic₁₇	#103	1	1	yes	characteristic
112	C₂×Dic₁₇	#106	1	1	yes	characteristic
113	C₂×Dic₁₇	#107	1	1	yes	characteristic
114	#3624	—	4	4	no	—
115	#3624	—	8	2	no	—
116	C₈×C₁₇	—	4	2	no	—
117	#3622	—	4	2	no	—
118	#1842	—	17	1	no	—
119	C₄×C₄⋊C₈	—	17	1	no	—
120	C₄×C₄⋊C₈	—	17	1	no	—
121	C₂²×C₁₇	#336	1	1	yes	characteristic
122	C₂²×C₁₇	#339	1	1	yes	characteristic
123	C₂²×C₁₇	#340	1	1	yes	characteristic
124	C₂²×C₁₇	#340	1	1	yes	characteristic
125	C₂²×C₁₇	#341	1	1	yes	characteristic
126	C₂²×C₁₇	#343	1	1	yes	characteristic
127	C₂²×C₁₇	#344	1	1	yes	characteristic
128	C₄×C₁₇	—	2	1	no	—
129	C₄×C₁₇	—	2	1	no	—
130	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
131	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
132	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
133	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
134	C₄×C₁₇	#334	1	1	yes	characteristic
135	C₄×C₁₇	#334	1	1	yes	characteristic
136	C₄×C₁₇	#335	1	1	yes	characteristic
137	C₄×C₁₇	#335	1	1	yes	characteristic
138	Dic₁₇	—	2	1	no	—
139	Dic₁₇	—	2	1	no	—
140	Dic₁₇	—	2	1	no	—
141	Dic₁₇	—	2	1	no	—
142	Dic₁₇	—	4	1	no	—
143	Dic₁₇	#331	1	1	yes	characteristic
144	Dic₁₇	#332	1	1	yes	characteristic
145	Dic₁₇	#332	1	1	yes	characteristic
146	Dic₁₇	#333	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	17	1	no	—
148	C₂×C₄⋊C₈	—	17	2	no	—
149	C₂×C₄⋊C₈	—	34	1	no	—
150	C₂×C₄⋊C₈	—	34	1	no	—
151	C₂×C₄⋊C₈	—	34	1	no	—
152	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
153	C₂×C₄×C₈	—	34	1	no	—
154	C₂×C₄×C₈	—	34	1	no	—
155	C₄³	—	17	1	no	—
156	#336	—	17	2	no	—
157	#336	—	34	1	no	—
158	C₂×C₁₇	#1277	1	1	yes	characteristic
159	C₂×C₁₇	#1279	1	1	yes	characteristic
160	C₂×C₁₇	#1283	1	1	yes	characteristic
161	C₂×C₁₇	#1296	1	1	yes	characteristic
162	C₂×C₁₇	#1299	1	1	yes	characteristic
163	C₂×C₁₇	#1301	1	1	yes	characteristic
164	C₂×C₁₇	#1303	1	1	yes	characteristic
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	34	2	no	—
167	C₂²×C₈	—	34	1	no	—
168	C₂²×C₈	—	34	2	no	—
169	C₂×C₄²	—	17	1	no	—
170	C₂×C₄²	—	17	1	no	—
171	C₂×C₄²	—	17	1	no	—
172	C₂×C₄²	—	17	1	no	—
173	C₂×C₄²	—	17	1	no	—
174	C₂×C₄²	—	17	1	no	—
175	C₂×C₄²	C₂×#595	1	1	yes	characteristic
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	17	2	no	—
178	C₄⋊C₈	—	34	2	no	—
179	C₄⋊C₈	—	34	2	no	—
180	C₄⋊C₈	—	34	1	no	—
181	C₄⋊C₈	—	68	1	no	—
182	C₄⋊C₈	—	68	1	no	—
183	C₄⋊C₈	—	68	1	no	—
184	C₄⋊C₈	—	68	1	no	—
185	C₄⋊C₈	—	68	2	no	—
186	C₄⋊C₈	#3624	1	2	yes	complemented
187	C₄×C₈	—	68	1	no	—
188	C₄×C₈	—	68	1	no	—
189	C₄×C₈	—	68	1	no	—
190	C₄×C₈	—	68	1	no	—
191	C₁₇	#7031	1	1	yes	characteristic, complemented, Sylow
192	C₂²×C₄	—	17	1	no	—
193	C₂²×C₄	—	17	1	no	—
194	C₂²×C₄	—	17	1	no	—
195	C₂²×C₄	—	17	1	no	—
196	C₂²×C₄	C₄×#595	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	34	4	no	—
203	C₂×C₈	—	34	2	no	—
204	C₂×C₈	—	68	2	no	—
205	C₂×C₈	—	68	2	no	—
206	C₂×C₈	—	68	1	no	—
207	C₂×C₈	—	68	1	no	—
208	C₂×C₈	—	68	2	no	—
209	C₂×C₈	—	68	2	no	—
210	C₂×C₈	—	68	2	no	—
211	C₄²	—	17	1	no	—
212	C₄²	—	17	1	no	—
213	C₄²	—	17	1	no	—
214	C₄²	—	17	1	no	—
215	C₄²	—	17	1	no	—
216	C₄²	—	17	1	no	—
217	C₄²	—	17	1	no	—
218	C₄²	—	17	1	no	—
219	C₄²	—	34	1	no	—
220	C₄²	—	34	1	no	—
221	C₄²	—	34	1	no	—
222	C₄²	—	34	1	no	—
223	C₄²	—	34	1	no	—
224	C₄²	—	34	1	no	—
225	C₄²	—	68	1	no	—
226	C₄²	#63242	1	1	yes	characteristic
227	C₄²	#63243	1	1	yes	characteristic
228	C₄²	C₂×#3624	1	1	yes	characteristic
229	C₄²	#63247	1	1	yes	characteristic
230	C₂³	(C₂×#595)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	1	no	—
235	C₂×C₄	—	17	1	no	—
236	C₂×C₄	—	17	1	no	—
237	C₂×C₄	—	17	1	no	—
238	C₂×C₄	—	17	1	no	—
239	C₂×C₄	—	17	1	no	—
240	C₂×C₄	—	17	1	no	—
241	C₂×C₄	—	17	1	no	—
242	C₂×C₄	—	17	1	no	—
243	C₂×C₄	—	34	1	no	—
244	C₂×C₄	—	34	1	no	—
245	C₂×C₄	—	34	1	no	—
246	C₂×C₄	—	34	1	no	—
247	C₂×C₄	—	34	1	no	—
248	C₂×C₄	—	34	1	no	—
249	C₂×C₄	—	34	1	no	—
250	C₂×C₄	#10587533	1	1	yes	characteristic
251	C₂×C₄	#10587534	1	1	yes	characteristic
252	C₂×C₄	#10587535	1	1	yes	characteristic
253	C₂×C₄	#10587536	1	1	yes	characteristic
254	C₂×C₄	C₄×#3624	1	1	yes	characteristic
255	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
256	C₂×C₄	#10587542	1	1	yes	characteristic
257	C₂×C₄	#10587543	1	1	yes	characteristic
258	C₂×C₄	#10587544	1	1	yes	characteristic
259	C₂×C₄	#10587547	1	1	yes	characteristic
260	C₂×C₄	#10587548	1	1	yes	characteristic
261	C₂×C₄	#10587549	1	1	yes	characteristic
262	C₂×C₄	#10587550	1	1	yes	characteristic
263	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
264	C₈	—	4	2	no	—
265	C₈	—	68	4	no	—
266	C₈	—	68	2	no	—
267	C₈	—	136	2	no	—
268	C₂²	#11763675	1	1	yes	characteristic
269	C₂²	#11763679	1	1	yes	characteristic
270	C₂²	#11763682	1	1	yes	characteristic
271	C₂²	#11763683	1	1	yes	characteristic
272	C₂²	#11763685	1	1	yes	characteristic
273	C₂²	#11763688	1	1	yes	characteristic
274	C₂²	#11763692	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	17	1	no	—
278	C₄	—	17	1	no	—
279	C₄	—	17	1	no	—
280	C₄	—	17	1	no	—
281	C₄	—	34	1	no	—
282	C₄	—	34	1	no	—
283	C₄	—	34	1	no	—
284	C₄	—	34	1	no	—
285	C₄	—	68	1	no	—
286	C₄	D₄⋊#3624	1	1	yes	characteristic
287	C₄	#11763657	1	1	yes	characteristic
288	C₄	Q₈⋊#3624	1	1	yes	characteristic
289	C₄	#11763659	1	1	yes	characteristic
290	C₄	#11763670	1	1	yes	characteristic
291	C₄	#11763671	1	1	yes	characteristic
292	C₄	#11763672	1	1	yes	characteristic
293	C₄	#11763673	1	1	yes	characteristic
294	C₂	#14543614	1	1	yes	characteristic
295	C₂	#14543620	1	1	yes	characteristic
296	C₂	#14543630	1	1	yes	characteristic
297	C₂	#14543663	1	1	yes	characteristic
298	C₂	#14543670	1	1	yes	characteristic
299	C₂	#14543676	1	1	yes	characteristic
300	C₂	#14543681	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
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	#7031
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
36. C₁₇×C₄⋊C₈	C₈	semidirect
40. C₄⋊#3622	C₈	semidirect
186. C₄⋊C₈	#3624	semidirect
191. C₁₇	#7031	semidirect

Maximal subgroups

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

subgroup	index
#14544716	2
C₄×C₄⋊#3624	2
C₄⋊C₈×Dic₁₇	2
#7031	17