Subgroups — #17961007

Order 4352.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763730	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763695	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763695	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
5	#14543823	C₂	1	1	yes	maximal, characteristic
6	#14543822	C₂	1	1	yes	maximal, characteristic
7	#14528645	C₂	1	1	yes	maximal, characteristic
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	C₂²×#63216	C₂²	1	1	yes	characteristic
10	C₂²×#63216	C₂²	1	1	yes	characteristic
11	C₂²×#63216	C₂²	1	1	yes	characteristic
12	#11763730	C₄	1	4	yes	complemented
13	#11763695	—	2	2	no	—
14	#11763695	C₄	1	4	yes	complemented
15	C₄×#63216	—	2	2	no	—
16	C₂×#10587485	C₄	1	1	yes	characteristic
17	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₄×#63225	C₄	1	1	yes	characteristic
21	#11762563	C₄	1	1	yes	characteristic
22	C₁₇×#336	C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×#63225	C₈	1	2	yes	—
27	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
30	C₂²×#3622	C₂³	1	1	yes	characteristic
31	C₂×#63216	—	2	1	no	—
32	C₂×#63216	—	2	1	no	—
33	C₂×#63216	—	2	1	no	—
34	C₂×#63216	—	2	1	no	—
35	C₂×#63216	—	2	1	no	—
36	C₂×#63216	—	2	1	no	—
37	C₂×#63216	C₂×C₄	1	2	yes	—
38	C₂×#63216	C₂×C₄	1	2	yes	—
39	C₂×#63216	C₂×C₄	1	2	yes	—
40	C₂×#63216	C₂×C₄	1	2	yes	—
41	C₂×#63216	D₄	1	2	yes	—
42	C₂×#63216	Q₈	1	2	yes	—
43	#10587502	C₈	1	2	yes	—
44	#10587485	C₈	1	4	yes	—
45	C₈×Dic₁₇	—	2	1	no	—
46	C₈×Dic₁₇	—	2	1	no	—
47	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
49	C₄×#3622	—	2	1	no	—
50	C₄×#3622	—	2	1	no	—
51	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
55	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₁₇	—	2	1	no	—
57	C₂×C₈×C₁₇	—	2	1	no	—
58	C₂×C₈×C₁₇	—	2	1	no	—
59	C₂×C₈×C₁₇	—	2	1	no	—
60	C₂×C₈×C₁₇	C₄²	1	2	yes	—
61	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
62	C₄²×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	—	2	1	no	—
64	#63225	C₁₆	1	8	yes	complemented
65	C₄×Dic₁₇	—	2	1	no	—
66	C₄×Dic₁₇	—	2	1	no	—
67	C₄×Dic₁₇	C₄²	1	2	yes	—
68	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
69	C₂×#3622	—	2	1	no	—
70	C₂×#3622	—	2	1	no	—
71	C₂×#3622	C₂×C₈	1	2	yes	—
72	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
73	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
74	C₂×#3622	C₂×D₄	1	1	yes	characteristic
75	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
76	C₂×#3622	#41	1	1	yes	characteristic
77	C₂×#3622	#41	1	1	yes	characteristic
78	#63216	—	2	2	no	—
79	#63216	—	2	2	no	—
80	#63216	—	2	2	no	—
81	#63216	C₄⋊C₄	1	4	yes	complemented
82	C₁₆×C₄⋊C₄	—	17	1	no	maximal, Sylow
83	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
87	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
93	C₂×Dic₁₇	—	2	2	no	—
94	C₂×Dic₁₇	—	2	2	no	—
95	C₂×Dic₁₇	—	2	2	no	—
96	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
97	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
98	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
99	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
100	C₈×C₁₇	—	2	2	no	—
101	C₈×C₁₇	—	2	2	no	—
102	#3622	—	2	2	no	—
103	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
104	#3622	#117	1	1	yes	characteristic
105	#3622	C₄×D₄	1	1	yes	characteristic
106	#3622	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄⋊C₁₆	—	17	1	no	—
108	C₂×C₄×C₁₆	—	17	1	no	—
109	C₂×C₄×C₁₆	—	17	1	no	—
110	C₂×C₄×C₁₆	—	17	1	no	—
111	C₈×C₄⋊C₄	—	17	1	no	—
112	#1374	—	17	1	no	—
113	#1374	—	17	1	no	—
114	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
116	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
117	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
118	C₂²×C₁₇	#504	1	2	yes	—
119	C₄×C₁₇	—	2	4	no	—
120	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₁₇	#405	1	1	yes	characteristic
122	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
123	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
124	Dic₁₇	—	2	2	no	—
125	Dic₁₇	—	2	4	no	—
126	Dic₁₇	C₄×C₁₆	1	4	yes	complemented
127	C₂²×C₁₆	—	17	1	no	—
128	C₂²×C₁₆	—	17	1	no	—
129	C₂²×C₁₆	—	17	1	no	—
130	C₂²×C₁₆	—	17	1	no	—
131	C₂×C₄⋊C₈	—	17	1	no	—
132	C₂×C₄×C₈	—	17	1	no	—
133	C₂×C₄×C₈	—	17	1	no	—
134	C₂×C₄×C₈	—	17	1	no	—
135	C₄×C₄⋊C₄	—	17	1	no	—
136	C₄⋊C₁₆	—	17	4	no	—
137	C₄×C₁₆	—	17	4	no	—
138	C₄×C₁₆	—	34	2	no	—
139	C₄×C₁₆	—	34	2	no	—
140	#336	—	17	1	no	—
141	#336	#595	1	1	yes	characteristic
142	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
144	C₂×C₁₇	#2108	1	1	yes	characteristic
145	C₂×C₁₇	D₄×C₁₆	1	2	yes	—
146	C₂×C₁₇	Q₈×C₁₆	1	2	yes	—
147	C₂²×C₈	—	17	1	no	—
148	C₂²×C₈	—	17	1	no	—
149	C₂²×C₈	C₂×#595	1	1	yes	characteristic
150	C₂²×C₈	C₂×#595	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	17	2	no	—
152	C₂×C₄²	—	17	1	no	—
153	C₂×C₄²	—	17	1	no	—
154	C₂×C₄²	C₂×#595	1	1	yes	characteristic
155	C₂×C₁₆	—	17	2	no	—
156	C₂×C₁₆	—	17	2	no	—
157	C₂×C₁₆	—	17	2	no	—
158	C₂×C₁₆	—	17	2	no	—
159	C₂×C₁₆	—	17	2	no	—
160	C₂×C₁₆	—	17	2	no	—
161	C₂×C₁₆	—	34	1	no	—
162	C₂×C₁₆	—	34	1	no	—
163	C₂×C₁₆	—	34	1	no	—
164	C₂×C₁₆	—	34	1	no	—
165	C₂×C₁₆	—	34	1	no	—
166	C₂×C₁₆	—	34	1	no	—
167	C₄⋊C₈	—	17	4	no	—
168	C₄×C₈	—	17	2	no	—
169	C₄×C₈	—	17	2	no	—
170	C₄×C₈	—	34	1	no	—
171	C₄×C₈	—	34	1	no	—
172	C₄×C₈	—	34	1	no	—
173	C₄×C₈	—	34	1	no	—
174	#95	—	17	2	no	—
175	C₁₇	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
176	C₂²×C₄	—	17	1	no	—
177	C₂²×C₄	—	17	1	no	—
178	C₂²×C₄	—	17	2	no	—
179	C₂²×C₄	#63242	1	2	yes	—
180	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	17	1	no	—
186	C₂×C₈	—	17	1	no	—
187	C₂×C₈	—	17	1	no	—
188	C₂×C₈	—	17	1	no	—
189	C₂×C₈	—	17	1	no	—
190	C₂×C₈	—	17	1	no	—
191	C₂×C₈	—	17	2	no	—
192	C₂×C₈	—	34	1	no	—
193	C₂×C₈	—	34	1	no	—
194	C₂×C₈	#63242	1	2	yes	—
195	C₂×C₈	C₄×#595	1	2	yes	—
196	C₄⋊C₄	—	17	8	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	17	2	no	—
200	C₄²	—	17	2	no	—
201	C₄²	—	34	1	no	—
202	C₄²	—	34	1	no	—
203	C₁₆	—	17	4	no	—
204	C₁₆	—	34	2	no	—
205	C₁₆	—	34	2	no	—
206	C₁₆	—	34	2	no	—
207	C₂³	C₂×#63242	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	17	2	no	—
211	C₂×C₄	—	17	4	no	—
212	C₂×C₄	—	17	2	no	—
213	C₂×C₄	—	17	4	no	—
214	C₂×C₄	—	34	2	no	—
215	C₂×C₄	—	34	2	no	—
216	C₂×C₄	—	34	2	no	—
217	C₂×C₄	#10587529	1	4	yes	—
218	C₂×C₄	C₂×#63242	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
220	C₂×C₄	#10587678	1	1	yes	characteristic
221	C₂×C₄	#10587678	1	1	yes	characteristic
222	C₂×C₄	D₄×#595	1	1	yes	characteristic
223	C₂×C₄	Q₈×#595	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	17	1	no	—
227	C₈	—	17	1	no	—
228	C₈	—	17	1	no	—
229	C₈	—	17	1	no	—
230	C₈	—	34	2	no	—
231	C₂²	C₄×#63242	1	1	yes	characteristic
232	C₂²	#11763836	1	1	yes	characteristic
233	C₂²	#11763858	1	1	yes	characteristic
234	C₂²	C₂×#10587529	1	2	yes	—
235	C₂²	#11763868	1	2	yes	—
236	C₄	—	2	4	no	—
237	C₄	—	17	4	no	—
238	C₄	—	34	4	no	—
239	C₄	—	34	2	no	—
240	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
241	C₄	#11763773	1	1	yes	characteristic
242	C₄	#11763836	1	1	yes	characteristic
243	C₄	#11763858	1	1	yes	characteristic
244	C₂	#14544576	1	1	yes	characteristic
245	C₂	C₄×#10587529	1	1	yes	characteristic
246	C₂	#14544994	1	1	yes	characteristic
247	C₂	#14545162	1	2	yes	—
248	C₂	#14545198	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₁₇×#336	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	#336	#595
hypocenter	C₁₇	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
12. #11763730	C₄	semidirect
14. #11763695	C₄	semidirect
64. #63225	C₁₆	semidirect
81. #63216	C₄⋊C₄	semidirect
126. Dic₁₇	C₄×C₁₆	semidirect
175. C₁₇	C₁₆×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763730	2
C₂×#11763695	2
C₂×C₄×#63216	2
#14543823	2
#14543822	2
#14528645	2
C₁₆×C₄⋊C₄	17