Subgroups — #17981085

Order 4352.

352 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545928	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3633	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×#595	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3633	—	2	1	no	—
5	C₂×C₄⋊#3633	—	2	1	no	—
6	C₂×C₄⋊#3633	—	2	1	no	—
7	C₂×C₄⋊#3633	C₄	1	2	yes	—
8	C₂×C₄×#3633	—	2	1	no	—
9	(C₂×#3633)⋊C₄	C₄	1	2	yes	—
10	#11763810	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	—	2	1	no	—
12	C₄²×#595	C₂²	1	1	yes	characteristic
13	#11763674	—	2	1	no	—
14	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
15	C₂²×#3633	—	2	2	no	—
16	C₂²×#3633	—	2	2	no	—
17	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×#595	D₄	1	1	yes	characteristic
20	C₂×C₄×#595	D₄	1	1	yes	characteristic
21	C₂×C₄×#595	D₄	1	1	yes	characteristic
22	C₂×C₄×#595	Q₈	1	1	yes	characteristic
23	C₂×#63242	—	2	1	no	—
24	C₄⋊#3633	—	2	2	no	—
25	C₄⋊#3633	—	2	2	no	—
26	C₄⋊#3633	—	4	2	no	—
27	C₄⋊#3633	—	4	1	no	—
28	C₄⋊#3633	—	4	1	no	—
29	C₄⋊#3633	—	4	1	no	—
30	C₄⋊#3633	—	4	1	no	—
31	C₄×#3633	—	4	1	no	—
32	C₄×#3633	—	4	1	no	—
33	C₂×C₈×D₁₇	—	2	1	no	—
34	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
35	#10587544	C₈	1	2	yes	complemented
36	C₄⋊#3624	—	2	1	no	—
37	C₈×#595	—	4	1	no	—
38	C₈×#595	—	4	1	no	—
39	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic, complemented
40	#10587485	—	2	1	no	—
41	C₄⋊#3622	C₈	1	1	yes	characteristic, complemented
42	C₂²×#595	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×#595	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×#595	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×#595	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#3633	—	2	4	no	—
47	C₂×#3633	—	4	2	no	—
48	C₂×#3633	—	4	2	no	—
49	C₂×#3633	—	4	2	no	—
50	C₂×#3633	—	4	2	no	—
51	C₂×#3633	—	4	2	no	—
52	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
53	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×#3624	—	2	2	no	—
56	C₄×#595	—	2	1	no	—
57	C₄×#595	—	2	1	no	—
58	C₄×#595	—	2	1	no	—
59	C₄×#595	—	2	1	no	—
60	C₄×#595	—	2	1	no	—
61	C₄×#595	—	2	1	no	—
62	C₄×#595	—	4	1	no	—
63	C₄×#595	D₈	1	1	yes	characteristic
64	C₄×#595	D₈	1	1	yes	characteristic
65	C₄×#595	SD₁₆	1	1	yes	characteristic
66	C₄×#595	SD₁₆	1	1	yes	characteristic
67	C₄×#595	SD₁₆	1	1	yes	characteristic
68	C₄×#595	SD₁₆	1	1	yes	characteristic
69	C₄×#595	Q₁₆	1	1	yes	characteristic
70	C₄×#595	Q₁₆	1	1	yes	characteristic
71	#63242	—	4	1	no	—
72	#63242	—	4	1	no	—
73	C₂×C₈×C₁₇	—	2	1	no	—
74	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
75	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
76	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
77	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
78	C₂×#3622	—	2	1	no	—
79	C₈×D₁₇	—	4	1	no	—
80	C₈×D₁₇	—	4	1	no	—
81	#7031	—	17	1	no	maximal, Sylow
82	C₂²×D₁₇	#95	1	1	yes	characteristic
83	C₂×#595	—	2	1	no	—
84	C₂×#595	—	2	1	no	—
85	C₂×#595	—	2	1	no	—
86	C₂×#595	—	2	1	no	—
87	C₂×#595	—	2	1	no	—
88	C₂×#595	—	2	1	no	—
89	C₂×#595	—	2	1	no	—
90	C₂×#595	#100	1	1	yes	characteristic
91	C₂×#595	#101	1	1	yes	characteristic
92	C₂×#595	#102	1	1	yes	characteristic
93	C₂×#595	#102	1	1	yes	characteristic
94	C₂×#595	#102	1	1	yes	characteristic
95	C₂×#595	#103	1	1	yes	characteristic
96	C₂×#595	#103	1	1	yes	characteristic
97	C₂×#595	#103	1	1	yes	characteristic
98	C₂×#595	#106	1	1	yes	characteristic
99	C₂×#595	#107	1	1	yes	characteristic
100	#3633	—	4	4	no	—
101	#3633	—	8	2	no	—
102	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
105	C₂×Dic₁₇	#97	1	1	yes	characteristic
106	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
108	C₄×D₁₇	—	2	1	no	—
109	C₄×D₁₇	—	2	1	no	—
110	C₄×D₁₇	#102	1	1	yes	characteristic
111	C₄×D₁₇	#103	1	1	yes	characteristic
112	C₄×D₁₇	#104	1	1	yes	characteristic
113	C₄×D₁₇	#104	1	1	yes	characteristic
114	C₄×D₁₇	#106	1	1	yes	characteristic
115	C₄×D₁₇	#107	1	1	yes	characteristic
116	C₄×D₁₇	#108	1	1	yes	characteristic
117	C₄×D₁₇	#108	1	1	yes	characteristic
118	#3624	—	4	2	no	—
119	C₈×C₁₇	—	4	1	no	—
120	#3622	—	4	1	no	—
121	#1842	—	17	1	no	—
122	C₄×C₄⋊C₈	—	17	1	no	—
123	C₄×C₄⋊C₈	—	17	1	no	—
124	C₂²×C₁₇	#336	1	1	yes	characteristic
125	C₂×D₁₇	#339	1	1	yes	characteristic
126	C₂×D₁₇	#340	1	1	yes	characteristic
127	C₂×D₁₇	#340	1	1	yes	characteristic
128	C₂×D₁₇	#341	1	1	yes	characteristic
129	C₂×D₁₇	#343	1	1	yes	characteristic
130	C₂×D₁₇	#344	1	1	yes	characteristic
131	#595	—	2	1	no	—
132	#595	—	2	1	no	—
133	#595	—	2	1	no	—
134	#595	—	2	1	no	—
135	#595	—	4	1	no	—
136	#595	#331	1	1	yes	characteristic
137	#595	#332	1	1	yes	characteristic
138	#595	#332	1	1	yes	characteristic
139	#595	#333	1	1	yes	characteristic
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
142	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
143	C₄×C₁₇	#334	1	1	yes	characteristic
144	C₄×C₁₇	#335	1	1	yes	characteristic
145	Dic₁₇	—	2	1	no	—
146	Dic₁₇	D₄⋊C₈	1	1	yes	characteristic
147	Dic₁₇	Q₈⋊C₈	1	1	yes	characteristic
148	Dic₁₇	#334	1	1	yes	characteristic
149	Dic₁₇	#335	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	17	1	no	—
151	C₂×C₄⋊C₈	—	17	1	no	—
152	C₂×C₄⋊C₈	—	17	2	no	—
153	C₂×C₄⋊C₈	—	34	1	no	—
154	C₂×C₄⋊C₈	—	34	1	no	—
155	C₂×C₄⋊C₈	—	34	1	no	—
156	C₂×C₄×C₈	—	34	1	no	—
157	C₂×C₄×C₈	—	34	1	no	—
158	C₄³	—	17	1	no	—
159	#336	—	17	2	no	—
160	#336	—	34	1	no	—
161	C₂×C₁₇	#1277	1	1	yes	characteristic
162	C₂×C₁₇	#1279	1	1	yes	characteristic
163	C₂×C₁₇	#1283	1	1	yes	characteristic
164	D₁₇	#1296	1	1	yes	characteristic
165	D₁₇	#1299	1	1	yes	characteristic
166	D₁₇	#1301	1	1	yes	characteristic
167	D₁₇	#1303	1	1	yes	characteristic
168	C₂²×C₈	—	34	1	no	—
169	C₂²×C₈	—	34	1	no	—
170	C₂²×C₈	—	34	2	no	—
171	C₂²×C₈	—	34	2	no	—
172	C₂×C₄²	—	17	1	no	—
173	C₂×C₄²	—	17	1	no	—
174	C₂×C₄²	—	17	1	no	—
175	C₂×C₄²	—	17	1	no	—
176	C₂×C₄²	—	17	1	no	—
177	C₂×C₄²	—	17	1	no	—
178	C₂×C₄²	—	17	1	no	—
179	C₄⋊C₈	—	17	2	no	—
180	C₄⋊C₈	—	17	1	no	—
181	C₄⋊C₈	—	34	1	no	—
182	C₄⋊C₈	—	34	1	no	—
183	C₄⋊C₈	—	34	2	no	—
184	C₄⋊C₈	—	34	2	no	—
185	C₄⋊C₈	—	68	1	no	—
186	C₄⋊C₈	—	68	1	no	—
187	C₄⋊C₈	—	68	1	no	—
188	C₄⋊C₈	—	68	1	no	—
189	C₄⋊C₈	—	68	2	no	—
190	C₄⋊C₈	#3633	1	1	yes	characteristic, complemented
191	C₄×C₈	—	68	1	no	—
192	C₄×C₈	—	68	1	no	—
193	C₄×C₈	—	68	1	no	—
194	C₄×C₈	—	68	1	no	—
195	C₁₇	#7031	1	1	yes	characteristic, complemented, Sylow
196	C₂²×C₄	—	17	1	no	—
197	C₂²×C₄	—	17	1	no	—
198	C₂²×C₄	—	17	1	no	—
199	C₂²×C₄	—	17	1	no	—
200	C₂²×C₄	—	17	1	no	—
201	C₂²×C₄	—	17	1	no	—
202	C₂²×C₄	—	17	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	34	2	no	—
205	C₂×C₈	—	34	4	no	—
206	C₂×C₈	—	34	1	no	—
207	C₂×C₈	—	68	1	no	—
208	C₂×C₈	—	68	1	no	—
209	C₂×C₈	—	68	1	no	—
210	C₂×C₈	—	68	1	no	—
211	C₂×C₈	—	68	2	no	—
212	C₂×C₈	—	68	2	no	—
213	C₂×C₈	—	68	2	no	—
214	C₂×C₈	—	68	2	no	—
215	C₂×C₈	—	68	2	no	—
216	C₄²	—	17	1	no	—
217	C₄²	—	17	1	no	—
218	C₄²	—	17	1	no	—
219	C₄²	—	17	1	no	—
220	C₄²	—	17	1	no	—
221	C₄²	—	17	1	no	—
222	C₄²	—	17	1	no	—
223	C₄²	—	17	1	no	—
224	C₄²	—	17	1	no	—
225	C₄²	—	17	1	no	—
226	C₄²	—	17	1	no	—
227	C₄²	—	34	1	no	—
228	C₄²	—	34	1	no	—
229	C₄²	—	34	1	no	—
230	C₄²	—	34	1	no	—
231	C₄²	—	34	1	no	—
232	C₄²	—	34	1	no	—
233	C₄²	—	68	1	no	—
234	C₄²	C₂×#3633	1	1	yes	characteristic
235	C₂³	—	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	—	17	1	no	—
244	C₂×C₄	—	17	1	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	1	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	1	no	—
250	C₂×C₄	—	17	1	no	—
251	C₂×C₄	—	17	1	no	—
252	C₂×C₄	—	17	1	no	—
253	C₂×C₄	—	17	1	no	—
254	C₂×C₄	—	17	1	no	—
255	C₂×C₄	—	17	1	no	—
256	C₂×C₄	—	17	1	no	—
257	C₂×C₄	—	34	1	no	—
258	C₂×C₄	—	34	1	no	—
259	C₂×C₄	—	34	1	no	—
260	C₂×C₄	—	34	1	no	—
261	C₂×C₄	—	34	1	no	—
262	C₂×C₄	—	34	1	no	—
263	C₂×C₄	—	34	1	no	—
264	C₂×C₄	—	34	1	no	—
265	C₂×C₄	—	34	1	no	—
266	C₂×C₄	C₄×#3633	1	1	yes	characteristic
267	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
268	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
269	C₈	—	4	1	no	—
270	C₈	—	68	2	no	—
271	C₈	—	68	4	no	—
272	C₈	—	68	1	no	—
273	C₈	—	136	2	no	—
274	C₂²	—	17	1	no	—
275	C₂²	—	17	1	no	—
276	C₂²	—	17	1	no	—
277	C₂²	—	17	1	no	—
278	C₂²	—	17	1	no	—
279	C₂²	—	17	1	no	—
280	C₂²	(C₂×#3633)⋊C₄	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	17	1	no	—
283	C₄	—	17	1	no	—
284	C₄	—	17	1	no	—
285	C₄	—	17	1	no	—
286	C₄	—	17	1	no	—
287	C₄	—	17	1	no	—
288	C₄	—	17	1	no	—
289	C₄	—	17	1	no	—
290	C₄	—	34	1	no	—
291	C₄	—	34	1	no	—
292	C₄	—	34	1	no	—
293	C₄	—	34	1	no	—
294	C₄	—	34	1	no	—
295	C₄	—	68	1	no	—
296	C₄	D₄⋊#3633	1	1	yes	characteristic
297	C₄	Q₈⋊#3633	1	1	yes	characteristic
298	C₄	#11763898	1	1	yes	characteristic
299	C₄	#11763899	1	1	yes	characteristic
300	C₂	—	17	1	no	—
301	C₂	—	17	1	no	—
302	C₂	—	17	1	no	—
303	C₂	—	17	1	no	—
304	C₂	#14545790	1	1	yes	characteristic
305	C₂	#14545792	1	1	yes	characteristic
306	C₂	#14545796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#3633)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₄²	C₂×#3633
Fitting	C₁₇×C₄⋊C₈	C₈
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	#95
cosocle	C₄²×#595	C₂²
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	#7031
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
35. #10587544	C₈	semidirect
39. C₁₇×C₄⋊C₈	C₈	semidirect
41. C₄⋊#3622	C₈	semidirect
190. C₄⋊C₈	#3633	semidirect
195. C₁₇	#7031	semidirect

Maximal subgroups

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

subgroup	index
#14545928	2
C₄×C₄⋊#3633	2
C₄⋊C₈×#595	2
#7031	17