Subgroups — #17948259

Order 4352.

338 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#14544498	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	—	2	1	no	—
7	#11763675	—	2	1	no	—
8	#11763675	C₄	1	2	yes	—
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	4	no	—
27	C₄×#3624	—	2	2	no	—
28	C₄×#3624	—	2	2	no	—
29	C₄×#3624	—	4	1	no	—
30	C₄×#3624	—	4	1	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	—	2	4	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	2	no	—
60	C₄×Dic₁₇	—	2	2	no	—
61	C₄×Dic₁₇	—	4	1	no	—
62	C₄×Dic₁₇	—	4	1	no	—
63	C₄×Dic₁₇	D₈	1	1	yes	characteristic
64	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
65	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
66	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
67	C₂×#3622	—	2	2	no	—
68	C₂×#3622	—	4	1	no	—
69	C₂×#3622	—	4	1	no	—
70	#7027	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	#95	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	1	no	—
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	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
86	C₂×Dic₁₇	—	2	1	no	—
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	#102	1	1	yes	characteristic
95	C₂×Dic₁₇	#103	1	1	yes	characteristic
96	C₂×Dic₁₇	#104	1	2	yes	—
97	C₂×Dic₁₇	#106	1	1	yes	characteristic
98	C₂×Dic₁₇	#107	1	1	yes	characteristic
99	C₂×Dic₁₇	#108	1	2	yes	—
100	#3624	—	4	4	no	—
101	#3624	—	4	4	no	—
102	C₈×C₁₇	—	4	2	no	—
103	#3622	—	4	2	no	—
104	#1767	—	17	1	no	—
105	C₄×C₄⋊C₈	—	17	1	no	—
106	#1744	—	17	1	no	—
107	C₂²×C₁₇	#336	1	1	yes	characteristic
108	C₂²×C₁₇	#339	1	1	yes	characteristic
109	C₂²×C₁₇	#339	1	1	yes	characteristic
110	C₂²×C₁₇	#340	1	1	yes	characteristic
111	C₂²×C₁₇	#341	1	1	yes	characteristic
112	C₂²×C₁₇	#344	1	1	yes	characteristic
113	C₂²×C₁₇	#344	1	1	yes	characteristic
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	2	1	no	—
118	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
119	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
120	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
121	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
122	Dic₁₇	—	2	2	no	—
123	Dic₁₇	—	2	1	no	—
124	Dic₁₇	—	2	1	no	—
125	Dic₁₇	—	2	2	no	—
126	Dic₁₇	—	2	1	no	—
127	Dic₁₇	—	2	1	no	—
128	C₂×C₄⋊C₈	—	17	1	no	—
129	C₂×C₄⋊C₈	—	34	1	no	—
130	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
131	C₂×#97	—	34	1	no	—
132	C₂×C₄×C₈	—	34	1	no	—
133	C₂×C₄×C₈	—	34	1	no	—
134	C₄³	—	17	1	no	—
135	#336	—	17	2	no	—
136	#336	—	17	2	no	—
137	#336	—	34	1	no	—
138	#336	—	34	1	no	—
139	C₂×C₁₇	#1278	1	1	yes	characteristic
140	C₂×C₁₇	#1279	1	1	yes	characteristic
141	C₂×C₁₇	#1281	1	1	yes	characteristic
142	C₂×C₁₇	#1300	1	1	yes	characteristic
143	C₂×C₁₇	#1301	1	1	yes	characteristic
144	C₂×C₁₇	#1302	1	1	yes	characteristic
145	C₂×C₁₇	#1303	1	1	yes	characteristic
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	34	2	no	—
148	C₂²×C₈	—	34	2	no	—
149	C₂²×C₈	—	34	1	no	—
150	C₂×C₄²	—	17	1	no	—
151	C₂×C₄²	—	17	1	no	—
152	C₂×C₄²	—	17	1	no	—
153	C₂×C₄²	—	17	1	no	—
154	C₂×C₄²	—	17	1	no	—
155	C₂×C₄²	—	17	1	no	—
156	C₂×C₄²	C₂×#595	1	1	yes	characteristic
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	17	2	no	—
159	C₄⋊C₈	—	34	4	no	—
160	C₄⋊C₈	—	34	1	no	—
161	C₄⋊C₈	#3624	1	2	yes	complemented
162	#97	—	68	1	no	—
163	#97	—	68	1	no	—
164	C₄×C₈	—	34	2	no	—
165	C₄×C₈	—	34	2	no	—
166	C₄×C₈	—	68	1	no	—
167	C₄×C₈	—	68	1	no	—
168	C₁₇	#7027	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₄	—	17	1	no	—
170	C₂²×C₄	—	17	1	no	—
171	C₂²×C₄	—	17	1	no	—
172	C₂²×C₄	—	17	1	no	—
173	C₂²×C₄	C₄×#595	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	34	4	no	—
180	C₂×C₈	—	34	4	no	—
181	C₂×C₈	—	34	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	—	68	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	C₄²	—	17	1	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	17	1	no	—
191	C₄²	—	17	1	no	—
192	C₄²	—	34	2	no	—
193	C₄²	—	34	2	no	—
194	C₄²	—	34	1	no	—
195	C₄²	—	34	1	no	—
196	C₄²	—	68	1	no	—
197	C₄²	—	68	1	no	—
198	C₄²	#63242	1	1	yes	characteristic
199	C₄²	#63243	1	1	yes	characteristic
200	C₄²	C₂×#3624	1	1	yes	characteristic
201	C₄²	#63247	1	1	yes	characteristic
202	C₂³	(C₂×#595)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	17	2	no	—
208	C₂×C₄	—	17	1	no	—
209	C₂×C₄	—	17	1	no	—
210	C₂×C₄	—	17	1	no	—
211	C₂×C₄	—	17	1	no	—
212	C₂×C₄	—	17	2	no	—
213	C₂×C₄	—	34	1	no	—
214	C₂×C₄	—	34	1	no	—
215	C₂×C₄	—	34	1	no	—
216	C₂×C₄	—	34	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	C₄×#3624	1	1	yes	characteristic
222	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
223	C₂×C₄	#10587542	1	1	yes	characteristic
224	C₂×C₄	#10587543	1	1	yes	characteristic
225	C₂×C₄	#10587544	1	1	yes	characteristic
226	C₂×C₄	#10587547	1	1	yes	characteristic
227	C₂×C₄	#10587548	1	1	yes	characteristic
228	C₂×C₄	#10587549	1	1	yes	characteristic
229	C₂×C₄	#10587550	1	1	yes	characteristic
230	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
231	C₈	—	4	2	no	—
232	C₈	—	68	4	no	—
233	C₈	—	68	4	no	—
234	C₈	—	68	2	no	—
235	C₂²	#11763675	1	1	yes	characteristic
236	C₂²	#11763679	1	1	yes	characteristic
237	C₂²	#11763681	1	1	yes	characteristic
238	C₂²	#11763682	1	1	yes	characteristic
239	C₂²	#11763684	1	1	yes	characteristic
240	C₂²	#11763691	1	1	yes	characteristic
241	C₂²	#11763691	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	34	2	no	—
247	C₄	—	34	1	no	—
248	C₄	—	34	1	no	—
249	C₄	—	34	2	no	—
250	C₄	—	34	1	no	—
251	C₄	—	34	1	no	—
252	C₄	D₄⋊#3624	1	1	yes	characteristic
253	C₄	#11763657	1	1	yes	characteristic
254	C₄	Q₈⋊#3624	1	1	yes	characteristic
255	C₄	#11763659	1	1	yes	characteristic
256	C₂	#14543617	1	1	yes	characteristic
257	C₂	#14543620	1	1	yes	characteristic
258	C₂	#14543624	1	1	yes	characteristic
259	C₂	#14543671	1	1	yes	characteristic
260	C₂	#14543674	1	1	yes	characteristic
261	C₂	#14543677	1	1	yes	characteristic
262	C₂	#14543680	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#595
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	#7027
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×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
32. C₁₇×C₄⋊C₈	C₈	semidirect
36. C₄⋊#3622	C₈	semidirect
161. C₄⋊C₈	#3624	semidirect
168. C₁₇	#7027	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
#14544498	2
#14528639	2
#7027	17