Subgroups — #17948248

Order 4352.

320 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544699	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
3	#14528639	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3624	—	2	1	no	—
5	C₂×C₄⋊#3624	—	2	1	no	—
6	C₂×C₄×#3624	C₄	1	2	yes	—
7	#11763675	—	2	1	no	—
8	#11763675	—	2	1	no	—
9	#11763675	C₄	1	2	yes	—
10	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
11	C₂×#10587486	—	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	2	no	—
31	C₁₇×C₄⋊C₈	—	2	1	no	—
32	C₁₇×C₄⋊C₈	C₈	1	2	yes	complemented
33	#10587486	—	4	1	no	—
34	#10587486	—	4	1	no	—
35	C₄⋊#3622	—	2	1	no	—
36	C₄⋊#3622	C₈	1	2	yes	complemented
37	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#3624	—	2	4	no	—
45	C₂×#3624	—	4	2	no	—
46	C₂×#3624	—	4	2	no	—
47	C₂×#3624	—	4	2	no	—
48	C₂×#3624	—	4	2	no	—
49	C₂×#3624	—	4	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	4	1	no	—
52	C₂×C₈×C₁₇	—	4	1	no	—
53	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
56	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
57	C₄×Dic₁₇	—	2	1	no	—
58	C₄×Dic₁₇	—	2	1	no	—
59	C₄×Dic₁₇	—	2	1	no	—
60	C₄×Dic₁₇	—	2	1	no	—
61	C₄×Dic₁₇	—	4	1	no	—
62	C₄×Dic₁₇	—	4	1	no	—
63	C₄×Dic₁₇	—	4	1	no	—
64	C₄×Dic₁₇	D₈	1	1	yes	characteristic
65	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
66	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
67	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
68	C₂×#3622	—	2	2	no	—
69	C₂×#3622	—	4	1	no	—
70	C₂×#3622	—	4	1	no	—
71	#7022	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	#95	1	1	yes	characteristic
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
87	C₂×Dic₁₇	—	2	1	no	—
88	C₂×Dic₁₇	—	2	1	no	—
89	C₂×Dic₁₇	—	2	1	no	—
90	C₂×Dic₁₇	—	2	1	no	—
91	C₂×Dic₁₇	—	2	1	no	—
92	C₂×Dic₁₇	—	2	1	no	—
93	C₂×Dic₁₇	—	2	1	no	—
94	C₂×Dic₁₇	—	2	1	no	—
95	C₂×Dic₁₇	—	2	1	no	—
96	C₂×Dic₁₇	#100	1	1	yes	characteristic
97	C₂×Dic₁₇	#101	1	1	yes	characteristic
98	C₂×Dic₁₇	#102	1	1	yes	characteristic
99	C₂×Dic₁₇	#103	1	1	yes	characteristic
100	C₂×Dic₁₇	#106	1	1	yes	characteristic
101	C₂×Dic₁₇	#107	1	1	yes	characteristic
102	#3624	—	4	4	no	—
103	#3624	—	8	2	no	—
104	C₈×C₁₇	—	4	2	no	—
105	#3622	—	4	2	no	—
106	#1831	—	17	1	no	—
107	#1767	—	17	1	no	—
108	C₄×C₄⋊C₈	—	17	1	no	—
109	C₂²×C₁₇	#336	1	1	yes	characteristic
110	C₂²×C₁₇	H(3,ℤ/4)	1	1	yes	characteristic
111	C₂²×C₁₇	#338	1	1	yes	characteristic
112	C₂²×C₁₇	#339	1	1	yes	characteristic
113	C₂²×C₁₇	#340	1	1	yes	characteristic
114	C₂²×C₁₇	#343	1	1	yes	characteristic
115	C₂²×C₁₇	#344	1	1	yes	characteristic
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	2	1	no	—
118	C₄×C₁₇	—	2	1	no	—
119	C₄×C₁₇	—	2	1	no	—
120	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
121	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
122	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
123	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
124	Dic₁₇	—	2	1	no	—
125	Dic₁₇	—	2	1	no	—
126	Dic₁₇	—	2	1	no	—
127	Dic₁₇	—	2	1	no	—
128	Dic₁₇	—	2	1	no	—
129	Dic₁₇	—	2	1	no	—
130	Dic₁₇	—	4	1	no	—
131	C₂×C₄⋊C₈	—	17	1	no	—
132	C₂×C₄⋊C₈	—	34	1	no	—
133	C₂×C₄⋊C₈	—	34	1	no	—
134	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
135	C₂×#97	—	34	1	no	—
136	C₂×C₄×C₈	—	17	2	no	—
137	C₄³	—	17	1	no	—
138	#336	—	17	2	no	—
139	#336	—	34	1	no	—
140	#336	—	34	1	no	—
141	#336	—	34	1	no	—
142	C₂×C₁₇	#1275	1	1	yes	characteristic
143	C₂×C₁₇	#1279	1	1	yes	characteristic
144	C₂×C₁₇	#1283	1	1	yes	characteristic
145	C₂×C₁₇	#1285	1	1	yes	characteristic
146	C₂×C₁₇	#1286	1	1	yes	characteristic
147	C₂×C₁₇	#1291	1	1	yes	characteristic
148	C₂×C₁₇	#1292	1	1	yes	characteristic
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	34	2	no	—
151	C₂²×C₈	—	34	1	no	—
152	C₂²×C₈	—	34	2	no	—
153	C₂×C₄²	—	17	1	no	—
154	C₂×C₄²	—	17	1	no	—
155	C₂×C₄²	—	17	1	no	—
156	C₂×C₄²	—	17	1	no	—
157	C₂×C₄²	—	17	1	no	—
158	C₂×C₄²	—	17	1	no	—
159	C₂×C₄²	C₂×#595	1	1	yes	characteristic
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	17	2	no	—
162	C₄⋊C₈	—	34	2	no	—
163	C₄⋊C₈	—	34	2	no	—
164	C₄⋊C₈	—	34	1	no	—
165	C₄⋊C₈	—	68	1	no	—
166	C₄⋊C₈	—	68	1	no	—
167	C₄⋊C₈	#3624	1	2	yes	complemented
168	#97	—	68	1	no	—
169	#97	—	68	1	no	—
170	C₄×C₈	—	68	2	no	—
171	C₁₇	#7022	1	1	yes	characteristic, complemented, Sylow
172	C₂²×C₄	—	17	1	no	—
173	C₂²×C₄	—	17	1	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂²×C₄	—	17	1	no	—
176	C₂²×C₄	C₄×#595	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	34	4	no	—
183	C₂×C₈	—	34	2	no	—
184	C₂×C₈	—	68	2	no	—
185	C₂×C₈	—	68	2	no	—
186	C₂×C₈	—	68	1	no	—
187	C₂×C₈	—	68	1	no	—
188	C₂×C₈	—	68	2	no	—
189	C₂×C₈	—	68	2	no	—
190	C₂×C₈	—	68	2	no	—
191	C₄²	—	17	1	no	—
192	C₄²	—	17	1	no	—
193	C₄²	—	17	1	no	—
194	C₄²	—	17	1	no	—
195	C₄²	—	34	1	no	—
196	C₄²	—	34	1	no	—
197	C₄²	—	34	1	no	—
198	C₄²	—	34	1	no	—
199	C₄²	—	68	1	no	—
200	C₄²	—	68	1	no	—
201	C₄²	—	68	1	no	—
202	C₄²	#63242	1	1	yes	characteristic
203	C₄²	#63243	1	1	yes	characteristic
204	C₄²	C₂×#3624	1	1	yes	characteristic
205	C₄²	#63247	1	1	yes	characteristic
206	C₂³	(C₂×#595)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	17	1	no	—
212	C₂×C₄	—	17	1	no	—
213	C₂×C₄	—	17	1	no	—
214	C₂×C₄	—	17	1	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	34	1	no	—
218	C₂×C₄	—	34	1	no	—
219	C₂×C₄	—	34	1	no	—
220	C₂×C₄	—	34	1	no	—
221	C₂×C₄	—	34	1	no	—
222	C₂×C₄	—	34	1	no	—
223	C₂×C₄	—	34	1	no	—
224	C₂×C₄	—	34	1	no	—
225	C₂×C₄	—	34	1	no	—
226	C₂×C₄	C₄×#3624	1	1	yes	characteristic
227	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
228	C₂×C₄	#10587542	1	1	yes	characteristic
229	C₂×C₄	#10587543	1	1	yes	characteristic
230	C₂×C₄	#10587544	1	1	yes	characteristic
231	C₂×C₄	#10587547	1	1	yes	characteristic
232	C₂×C₄	#10587548	1	1	yes	characteristic
233	C₂×C₄	#10587549	1	1	yes	characteristic
234	C₂×C₄	#10587550	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
236	C₈	—	4	2	no	—
237	C₈	—	68	4	no	—
238	C₈	—	68	2	no	—
239	C₈	—	136	2	no	—
240	C₂²	#11763675	1	1	yes	characteristic
241	C₂²	#11763677	1	1	yes	characteristic
242	C₂²	#11763678	1	1	yes	characteristic
243	C₂²	#11763679	1	1	yes	characteristic
244	C₂²	#11763682	1	1	yes	characteristic
245	C₂²	#11763688	1	1	yes	characteristic
246	C₂²	#11763692	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	34	1	no	—
252	C₄	—	34	1	no	—
253	C₄	—	34	1	no	—
254	C₄	—	34	1	no	—
255	C₄	—	34	1	no	—
256	C₄	—	34	1	no	—
257	C₄	—	68	1	no	—
258	C₄	D₄⋊#3624	1	1	yes	characteristic
259	C₄	#11763657	1	1	yes	characteristic
260	C₄	Q₈⋊#3624	1	1	yes	characteristic
261	C₄	#11763659	1	1	yes	characteristic
262	C₂	#14543610	1	1	yes	characteristic
263	C₂	#14543620	1	1	yes	characteristic
264	C₂	#14543630	1	1	yes	characteristic
265	C₂	#14543636	1	1	yes	characteristic
266	C₂	#14543639	1	1	yes	characteristic
267	C₂	#14543650	1	1	yes	characteristic
268	C₂	#14543653	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	#7022
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×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
32. C₁₇×C₄⋊C₈	C₈	semidirect
36. C₄⋊#3622	C₈	semidirect
167. C₄⋊C₈	#3624	semidirect
171. C₁₇	#7022	semidirect

Maximal subgroups

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

subgroup	index
#14544699	2
C₄×C₄⋊#3624	2
#14528639	2
#7022	17