Subgroups — #108×#3624 #17952391

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544331	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#3624	C₂	1	2	yes	maximal
3	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
4	#14543617	C₂	1	2	yes	maximal
5	#108×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763825	C₂²	1	2	yes	—
7	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
9	#11763672	C₄	1	2	yes	—
10	#11763670	C₄	1	2	yes	—
11	#11763650	—	2	2	no	—
12	C₈×#3624	—	2	2	no	—
13	C₈×#3624	C₄	1	2	yes	—
14	C₈×#3624	C₄	1	2	yes	—
15	C₂×#10587490	C₄	1	1	yes	characteristic
16	M₁₆×Dic₁₇	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
18	#11762573	C₄	1	2	yes	—
19	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
20	C₂²×#3624	—	2	1	no	—
21	C₂²×#3624	Q₈	1	2	yes	—
22	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
25	C₂×#63223	C₂×C₄	1	2	yes	—
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²⋊#3624	—	2	2	no	—
28	C₄⋊#3624	C₂×C₄	1	4	yes	—
29	C₄×#3624	—	2	2	no	—
30	C₄×#3624	—	2	1	no	—
31	C₄×#3624	C₂×C₄	1	2	yes	—
32	C₄×#3624	C₂×C₄	1	2	yes	—
33	C₄×#3624	D₄	1	1	yes	characteristic
34	C₄×#3624	D₄	1	1	yes	characteristic
35	C₁₇×#108	C₈	1	4	yes	complemented
36	#10587490	C₈	1	4	yes	complemented
37	#10587486	—	2	2	no	—
38	C₈×Dic₁₇	—	2	2	no	—
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₁₇	#41	1	1	yes	characteristic
46	C₂×#3624	—	2	4	no	—
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	—	2	4	no	—
49	C₂×#3624	—	2	4	no	—
50	C₂×#3624	C₄⋊C₄	1	2	yes	—
51	C₂×#3624	C₄⋊C₄	1	2	yes	—
52	M₁₆×C₁₇	—	2	2	no	—
53	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
59	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄×Dic₁₇	#41	1	1	yes	characteristic
63	#63223	—	2	2	no	—
64	#63223	C₂×C₈	1	4	yes	—
65	C₂×#3622	—	2	2	no	—
66	C₂×#3622	—	2	1	no	—
67	C₂×#3622	—	2	1	no	—
68	C₂×#3622	C₄²	1	1	yes	characteristic
69	C₂×#3622	C₄²	1	1	yes	characteristic
70	C₈×#108	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
78	C₂×Dic₁₇	—	2	2	no	—
79	C₂×Dic₁₇	—	2	2	no	—
80	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×Dic₁₇	#117	1	1	yes	characteristic
82	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
83	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
84	#3624	—	2	4	no	—
85	#3624	—	2	2	no	—
86	#3624	—	4	4	no	—
87	#3624	#108	1	4	yes	direct product factor
88	C₈×C₁₇	—	2	4	no	—
89	C₈×C₁₇	C₄×C₈	1	2	yes	—
90	C₈×C₁₇	C₄×C₈	1	2	yes	—
91	#3622	—	2	1	no	—
92	#3622	—	2	1	no	—
93	#3622	—	2	4	no	—
94	C₄×#108	—	17	1	no	—
95	#1564	—	17	1	no	—
96	C₈×M₁₆	—	17	2	no	—
97	C₂×C₈²	—	17	1	no	—
98	#1278	—	17	2	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
103	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
104	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
105	C₄×C₁₇	#405	1	1	yes	characteristic
106	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
107	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
108	Dic₁₇	—	2	1	no	—
109	Dic₁₇	—	2	1	no	—
110	Dic₁₇	C₂×#108	1	2	yes	—
111	Dic₁₇	#443	1	2	yes	—
112	#431	—	17	2	no	—
113	C₂×#108	—	17	1	no	—
114	C₂×#108	#595	1	1	yes	characteristic
115	C₄×M₁₆	—	17	2	no	—
116	C₂×C₄×C₈	—	17	1	no	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	#341	—	17	2	no	—
120	#335	—	17	2	no	—
121	#334	—	17	2	no	—
122	#322	—	34	2	no	—
123	C₈²	—	17	2	no	—
124	C₈²	—	17	2	no	—
125	C₈²	—	34	2	no	—
126	C₂×C₁₇	—	2	1	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₁₇	#1772	1	1	yes	characteristic
130	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
131	C₂×M₁₆	—	17	2	no	—
132	C₂×M₁₆	C₂×#595	1	2	yes	—
133	C₂²×C₈	—	17	2	no	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	34	1	no	—
136	C₂²×C₈	C₂×#595	1	1	yes	characteristic
137	C₂×C₄²	—	17	1	no	—
138	#108	—	17	4	no	—
139	#108	#3624	1	4	yes	direct product factor
140	C₄⋊C₈	—	17	4	no	—
141	C₂²⋊C₈	—	34	2	no	—
142	#97	—	34	2	no	—
143	C₄×C₈	—	17	2	no	—
144	C₄×C₈	—	17	1	no	—
145	C₄×C₈	—	17	1	no	—
146	C₄×C₈	—	17	1	no	—
147	C₄×C₈	—	17	1	no	—
148	C₄×C₈	—	17	1	no	—
149	C₄×C₈	—	17	1	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	34	2	no	—
152	C₄×C₈	—	34	2	no	—
153	C₄×C₈	—	34	1	no	—
154	C₁₇	C₈×#108	1	1	yes	characteristic, complemented, Sylow
155	C₂²×C₄	—	17	1	no	—
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	17	4	no	—
160	M₁₆	—	34	2	no	—
161	M₁₆	C₂×#3624	1	4	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	2	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	1	no	—
167	C₂×C₈	—	34	4	no	—
168	C₂×C₈	—	34	2	no	—
169	C₂×C₈	—	34	4	no	—
170	C₂×C₈	—	34	1	no	—
171	C₂×C₈	—	34	2	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	4	no	—
174	C₂×C₈	#63242	1	2	yes	—
175	C₂×C₈	C₄×#595	1	1	yes	characteristic
176	C₂×C₈	C₄×#595	1	1	yes	characteristic
177	C₂×C₈	C₂×#3624	1	2	yes	—
178	C₄²	—	17	1	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₂³	Q₈×#595	1	1	yes	characteristic
183	C₂×C₄	—	17	1	no	—
184	C₂×C₄	—	17	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	34	2	no	—
188	C₂×C₄	—	34	2	no	—
189	C₂×C₄	C₂×#63242	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
191	C₂×C₄	#10587678	1	1	yes	characteristic
192	C₂×C₄	D₄×#595	1	1	yes	characteristic
193	C₂×C₄	#10587680	1	1	yes	characteristic
194	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₈	—	17	4	no	—
197	C₈	—	34	2	no	—
198	C₈	—	34	4	no	—
199	C₈	—	34	1	no	—
200	C₈	—	34	1	no	—
201	C₈	—	34	4	no	—
202	C₈	—	68	4	no	—
203	C₈	C₄×#3624	1	2	yes	—
204	C₈	C₄×#3624	1	2	yes	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
208	C₂²	Q₈×#3624	1	1	yes	characteristic
209	C₂²	#11763858	1	1	yes	characteristic
210	C₄	—	17	2	no	—
211	C₄	—	17	2	no	—
212	C₄	—	34	1	no	—
213	C₄	—	34	1	no	—
214	C₄	C₂×C₄×#3624	1	1	yes	characteristic
215	C₄	#11763774	1	1	yes	characteristic
216	C₄	D₄×#3624	1	1	yes	characteristic
217	C₄	#11763837	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	C₄⋊C₄×#3624	1	1	yes	characteristic
221	C₂	#14544589	1	1	yes	characteristic
222	C₂	#108×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#108	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄×Q₈
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#108	#595
hypocenter	C₁₇	C₈×#108
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
35. C₁₇×#108	C₈	semidirect
36. #10587490	C₈	semidirect
87. #3624	#108	direct (N × H)
139. #108	#3624	direct (N × H)
154. C₁₇	C₈×#108	semidirect

Maximal subgroups

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

subgroup	index
#14544331	2
M₁₆×#3624	2
C₂×C₈×#3624	2
#14543617	2
#108×Dic₁₇	2
C₈×#108	17