Subgroups — #16949382

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530395	C₂	1	2	yes	maximal
2	C₂×#11762587	C₂	1	2	yes	maximal, complemented
3	#14526864	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₂²	1	4	yes	—
6	C₂²⋊#63214	C₄	1	8	yes	complemented
7	#11762587	—	2	4	no	—
8	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
11	C₂×C₁₇×#97	—	2	4	no	—
12	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
13	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
14	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₇	—	2	1	no	—
17	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	—	2	2	no	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	—	2	1	no	—
29	C₂×C₄²×C₁₇	—	2	2	no	—
30	C₂×C₄²×C₁₇	—	2	2	no	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₁₇×#97	—	2	2	no	—
34	C₁₇×#97	—	2	2	no	—
35	C₁₇×#97	—	2	2	no	—
36	C₁₇×#97	D₄	1	4	yes	—
37	#10587483	—	2	4	no	—
38	#10587483	—	2	4	no	—
39	C₂×#63214	C₂×C₄	1	8	yes	complemented
40	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	—	2	1	no	—
42	C₂²×C₄×C₁₇	—	2	1	no	—
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	2	4	no	—
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	—	2	1	no	—
48	C₂²×C₄×C₁₇	—	4	4	no	—
49	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	—	2	8	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	—	2	1	no	—
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	—	2	8	no	—
61	C₂×C₈×C₁₇	C₄²	1	2	yes	—
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₇	#41	1	2	yes	—
67	C₄²×C₁₇	—	2	4	no	—
68	C₄²×C₁₇	—	2	1	no	—
69	C₄²×C₁₇	—	2	1	no	—
70	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₇	M₁₆	1	2	yes	—
72	#63214	—	2	8	no	—
73	#12605	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	—	2	4	no	—
75	C₂³×C₁₇	—	2	1	no	—
76	C₂³×C₁₇	—	2	1	no	—
77	C₂³×C₁₇	#97	1	2	yes	—
78	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	2	1	no	—
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	4	8	no	—
88	C₂×C₄×C₁₇	—	4	2	no	—
89	C₂×C₄×C₁₇	—	4	2	no	—
90	C₂×C₄×C₁₇	#97	1	2	yes	—
91	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	2	no	—
99	C₈×C₁₇	—	2	2	no	—
100	C₈×C₁₇	C₄×D₄	1	4	yes	—
101	C₂²×#97	D₁₇	1	1	yes	characteristic
102	#2112	—	17	2	no	—
103	C₂×#347	—	17	2	no	—
104	#1375	—	17	2	no	—
105	C₂²×C₁₇	—	2	4	no	—
106	C₂²×C₁₇	—	2	4	no	—
107	C₂²×C₁₇	—	2	2	no	—
108	C₂²×C₁₇	—	2	2	no	—
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	#347	1	4	yes	complemented
111	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
112	C₂²×C₁₇	#407	1	1	yes	characteristic
113	C₂²×C₁₇	#432	1	1	yes	characteristic
114	C₄×C₁₇	—	2	2	no	—
115	C₄×C₁₇	—	4	2	no	—
116	C₄×C₁₇	—	4	2	no	—
117	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
119	C₄×C₁₇	#407	1	1	yes	characteristic
120	C₄×C₁₇	#432	1	1	yes	characteristic
121	C₂³×C₈	Dic₁₇	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
123	C₂²×C₄²	Dic₁₇	1	1	yes	characteristic
124	C₂×M₃₂	—	17	4	no	—
125	C₂×#97	—	2	4	no	—
126	C₂×#97	Dic₁₇	1	1	yes	characteristic
127	C₂×#97	Dic₁₇	1	1	yes	characteristic
128	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
129	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
130	C₂²⋊C₁₆	—	17	8	no	—
131	#347	—	34	4	no	—
132	C₂×C₁₇	—	2	1	no	—
133	C₂×C₁₇	—	2	4	no	—
134	C₂×C₁₇	—	2	1	no	—
135	C₂×C₁₇	#1749	1	1	yes	characteristic
136	C₂×C₁₇	C₂×#347	1	2	yes	—
137	C₂³×C₄	—	2	1	no	—
138	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	C₄×D₁₇	1	4	yes	—
145	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
154	M₃₂	—	34	4	no	—
155	M₃₂	—	34	4	no	—
156	C₂×C₁₆	—	17	8	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	#97	—	2	2	no	—
160	#97	C₂²⋊D₁₇	1	4	yes	—
161	C₁₇	#12605	1	1	yes	characteristic, complemented, Sylow
162	C₂⁴	#63223	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	#63223	1	1	yes	characteristic
172	C₂²×C₄	#63223	1	2	yes	—
173	C₂²×C₄	C₄×Dic₁₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	#63223	1	2	yes	—
184	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
185	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
187	C₂×C₈	#63252	1	2	yes	—
188	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	#63223	1	2	yes	—
193	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
194	C₁₆	—	34	8	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	1	no	—
198	C₂³	#10587474	1	2	yes	—
199	C₂³	C₂×#63223	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	#10587474	1	2	yes	—
212	C₂×C₄	C₂×#63223	1	1	yes	characteristic
213	C₂×C₄	C₂×#63223	1	1	yes	characteristic
214	C₂×C₄	C₂×#63223	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
216	C₂×C₄	#10587612	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	C₄×C₂²⋊D₁₇	1	4	yes	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	#11762587	1	4	yes	complemented
228	C₂²	C₂×#10587474	1	1	yes	characteristic
229	C₂²	#11762773	1	1	yes	characteristic
230	C₂²	#11762866	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	C₄×C₂²⋊Dic₁₇	1	1	yes	characteristic
235	C₄	C₄×#63223	1	1	yes	characteristic
236	C₄	#11762773	1	1	yes	characteristic
237	C₄	#11762866	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	4	no	—
240	C₂	—	2	1	no	—
241	C₂	#14528546	1	1	yes	characteristic
242	C₂	C₂×#11762587	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂²×C₁₇×#97	C₂
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂²×#97	D₁₇
hypocenter	C₁₇	#12605
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#11762587	C₂	semidirect
3. #14526864	C₂	semidirect
6. C₂²⋊#63214	C₄	semidirect
39. C₂×#63214	C₂×C₄	semidirect
110. C₂²×C₁₇	#347	semidirect
161. C₁₇	#12605	semidirect
227. C₂²	#11762587	semidirect

Maximal subgroups

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

subgroup	index
#14530395	2
C₂×#11762587	2
#14526864	2
C₂²×C₁₇×#97	2
#12605	17