Subgroups — #97×#3624 #17950266

Order 4352.

606 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544498	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3624	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763650	C₂	1	2	yes	maximal
4	C₂×C₈×#3624	C₂	1	2	yes	maximal
5	#97×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#3624	C₄	1	4	yes	—
7	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3624	C₂²	1	2	yes	—
10	#11763675	C₄	1	4	yes	—
11	#11763650	C₄	1	8	yes	complemented
12	C₈×#3624	C₄	1	8	yes	complemented
13	C₂×#10587474	C₄	1	1	yes	characteristic
14	C₂×#10587486	C₄	1	2	yes	—
15	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
16	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
18	C₂²×#3624	C₂×C₄	1	2	yes	—
19	C₂²×#3624	C₂×C₄	1	4	yes	—
20	C₂²×#3624	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	2	yes	—
28	C₄×#3624	—	2	8	no	—
29	C₄×#3624	C₂×C₄	1	4	yes	—
30	C₄×#3624	C₂×C₄	1	4	yes	—
31	C₄×#3624	C₂×C₄	1	4	yes	—
32	C₄×#3624	C₂×C₄	1	4	yes	—
33	C₁₇×#97	C₈	1	4	yes	complemented
34	#10587486	C₈	1	8	yes	complemented
35	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
36	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
37	#10587474	C₈	1	4	yes	complemented
38	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₄²	1	2	yes	—
41	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×#3624	—	2	4	no	—
44	C₂×#3624	—	2	8	no	—
45	C₂×#3624	C₄²	1	4	yes	—
46	C₂×#3624	C₄²	1	8	yes	—
47	C₂×#3624	C₄²	1	4	yes	—
48	C₂×#3624	M₁₆	1	8	yes	—
49	C₂×C₈×C₁₇	C₄²	1	4	yes	—
50	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
52	C₄²×C₁₇	C₂×C₈	1	2	yes	—
53	C₄²×C₁₇	C₂×C₈	1	2	yes	—
54	C₄×Dic₁₇	—	2	4	no	—
55	C₄×Dic₁₇	—	2	4	no	—
56	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
57	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
58	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
59	C₂×#3622	C₄²	1	4	yes	—
60	C₂×#3622	C₂×C₈	1	4	yes	—
61	C₂×#3622	C₂×C₈	1	4	yes	—
62	C₈×#97	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
67	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×Dic₁₇	—	2	2	no	—
72	C₂×Dic₁₇	—	2	2	no	—
73	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
74	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×Dic₁₇	C₂×C₄²	1	2	yes	—
77	C₂×Dic₁₇	C₂×M₁₆	1	4	yes	—
78	C₂×Dic₁₇	#131	1	4	yes	—
79	#3624	—	2	4	no	—
80	#3624	—	2	8	no	—
81	#3624	#97	1	8	yes	direct product factor
82	C₈×C₁₇	C₄×C₈	1	8	yes	complemented
83	#3622	C₄×C₈	1	8	yes	complemented
84	#1744	—	17	1	no	—
85	C₄×#97	—	17	1	no	—
86	C₄²×C₈	—	17	1	no	—
87	C₂×#322	—	17	2	no	—
88	C₂×C₈²	—	17	2	no	—
89	C₂²×C₁₇	C₄³	1	1	yes	characteristic
90	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
92	C₂²×C₁₇	C₄×M₁₆	1	2	yes	—
93	C₂²×C₁₇	#405	1	2	yes	—
94	C₄×C₁₇	—	2	4	no	—
95	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
96	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
97	Dic₁₇	—	2	4	no	—
98	Dic₁₇	C₂×#97	1	2	yes	—
99	Dic₁₇	C₄×M₁₆	1	2	yes	—
100	Dic₁₇	#405	1	4	yes	—
101	C₂×#97	—	17	2	no	—
102	C₂×#97	—	17	1	no	—
103	C₂×#97	#595	1	1	yes	characteristic
104	C₂×C₄×C₈	—	17	1	no	—
105	C₂×C₄×C₈	—	17	2	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	17	2	no	—
108	C₂×C₄×C₈	—	17	4	no	—
109	C₄³	—	17	1	no	—
110	#336	—	17	4	no	—
111	#322	—	17	8	no	—
112	C₈²	—	17	8	no	—
113	C₂×C₁₇	C₈×M₁₆	1	2	yes	—
114	C₂×C₁₇	C₈×M₁₆	1	2	yes	—
115	C₂×C₁₇	C₄²×C₈	1	1	yes	characteristic
116	C₂×C₁₇	C₄×#97	1	1	yes	characteristic
117	C₂×C₁₇	#1726	1	1	yes	characteristic
118	C₂²×C₈	—	17	2	no	—
119	C₂²×C₈	—	17	2	no	—
120	C₂²×C₈	—	17	4	no	—
121	C₂²×C₈	—	17	2	no	—
122	C₂²×C₈	C₂×#595	1	2	yes	—
123	C₂×C₄²	—	17	1	no	—
124	C₂×C₄²	—	17	1	no	—
125	C₂×C₄²	—	17	2	no	—
126	C₂×C₄²	—	17	2	no	—
127	C₂×C₄²	C₂×#595	1	1	yes	characteristic
128	#97	—	17	8	no	—
129	#97	—	17	4	no	—
130	#97	#3624	1	4	yes	direct product factor
131	C₄×C₈	—	17	4	no	—
132	C₄×C₈	—	17	4	no	—
133	C₄×C₈	—	17	4	no	—
134	C₄×C₈	—	17	4	no	—
135	C₄×C₈	—	17	4	no	—
136	C₄×C₈	—	17	4	no	—
137	C₄×C₈	—	34	8	no	—
138	C₁₇	C₈×#97	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	—	17	1	no	—
140	C₂²×C₄	—	17	1	no	—
141	C₂²×C₄	—	17	2	no	—
142	C₂²×C₄	C₄×#595	1	2	yes	—
143	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
144	C₂×C₈	—	17	4	no	—
145	C₂×C₈	—	17	8	no	—
146	C₂×C₈	—	17	4	no	—
147	C₂×C₈	—	17	4	no	—
148	C₂×C₈	—	17	4	no	—
149	C₂×C₈	—	17	8	no	—
150	C₂×C₈	—	17	4	no	—
151	C₂×C₈	—	34	4	no	—
152	C₂×C₈	—	34	8	no	—
153	C₂×C₈	#63242	1	4	yes	—
154	C₂×C₈	C₄×#595	1	4	yes	—
155	C₂×C₈	C₂×#3624	1	4	yes	—
156	C₄²	—	17	4	no	—
157	C₄²	—	17	2	no	—
158	C₄²	—	17	2	no	—
159	C₄²	—	34	4	no	—
160	C₄²	—	34	4	no	—
161	C₄²	#63242	1	2	yes	—
162	C₄²	C₂×#3624	1	2	yes	—
163	C₂³	C₂×C₄×#595	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	17	1	no	—
167	C₂×C₄	—	17	1	no	—
168	C₂×C₄	—	17	2	no	—
169	C₂×C₄	—	17	4	no	—
170	C₂×C₄	—	17	4	no	—
171	C₂×C₄	—	17	4	no	—
172	C₂×C₄	—	34	2	no	—
173	C₂×C₄	—	34	2	no	—
174	C₂×C₄	C₄×#3624	1	4	yes	—
175	C₂×C₄	C₂×#63242	1	2	yes	—
176	C₂×C₄	C₂×#63242	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×#595	1	2	yes	—
178	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
179	C₈	—	17	8	no	—
180	C₈	—	17	8	no	—
181	C₈	—	34	4	no	—
182	C₈	—	34	8	no	—
183	C₈	C₄×#3624	1	8	yes	complemented
184	C₂²	C₄²×#595	1	1	yes	characteristic
185	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
186	C₂²	C₄×#63242	1	1	yes	characteristic
187	C₂²	M₁₆×#595	1	2	yes	—
188	C₂²	#11763775	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	17	2	no	—
191	C₄	—	17	2	no	—
192	C₄	—	17	4	no	—
193	C₄	—	34	4	no	—
194	C₄	C₂×C₄×#3624	1	2	yes	—
195	C₄	C₄×#63242	1	2	yes	—
196	C₂	M₁₆×#3624	1	2	yes	—
197	C₂	#14543982	1	2	yes	—
198	C₂	C₄²×#3624	1	1	yes	characteristic
199	C₂	#97×#595	1	1	yes	characteristic
200	C₂	#14544442	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂×C₁₇	C₄²×C₈
Fitting	C₂×C₁₇×#97	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#97	#595
hypocenter	C₁₇	C₈×#97
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
11. #11763650	C₄	semidirect
12. C₈×#3624	C₄	semidirect
33. C₁₇×#97	C₈	semidirect
34. #10587486	C₈	semidirect
37. #10587474	C₈	semidirect
81. #3624	#97	direct (N × H)
82. C₈×C₁₇	C₄×C₈	semidirect
83. #3622	C₄×C₈	semidirect
130. #97	#3624	direct (N × H)
138. C₁₇	C₈×#97	semidirect
183. C₈	C₄×#3624	semidirect

Maximal subgroups

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

subgroup	index
#14544498	2
C₄²×#3624	2
C₂×#11763650	2
C₂×C₈×#3624	2
#97×Dic₁₇	2
C₈×#97	17