Subgroups — #16925539

Order 4352.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
2	#14528530	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762561	C₂	1	2	yes	maximal
4	#14526618	C₂	1	2	yes	maximal
5	C₁₇×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄×#3622	—	2	1	no	—
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	#11762564	—	2	1	no	—
11	#11762564	C₄	1	4	yes	—
12	#11762561	C₄	1	8	yes	complemented
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₇×#97	—	2	1	no	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#336	—	2	1	no	—
17	C₂²×C₈×C₁₇	—	2	2	no	—
18	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×#3622	—	2	2	no	—
24	C₂²×#3622	C₂×C₄	1	2	yes	—
25	C₂²×#3622	C₂×C₄	1	2	yes	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₁₇×#97	—	4	2	no	—
29	C₄⋊#3622	—	2	2	no	—
30	C₄⋊#3622	C₂×C₄	1	4	yes	—
31	C₄×#3622	—	2	2	no	—
32	C₄×#3622	—	4	2	no	—
33	C₄×#3622	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₇	—	2	4	no	—
40	C₂×C₈×C₁₇	—	4	2	no	—
41	C₂×C₈×C₁₇	—	4	2	no	—
42	C₄²×C₁₇	—	2	4	no	—
43	C₄²×C₁₇	—	2	2	no	—
44	C₄²×C₁₇	—	4	2	no	—
45	C₄²×C₁₇	D₈	1	2	yes	—
46	C₄²×C₁₇	Q₁₆	1	2	yes	—
47	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₁₇	#41	1	2	yes	—
49	C₂×#3622	—	2	4	no	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	—	2	4	no	—
52	C₂×#3622	—	2	4	no	—
53	C₂×#3622	—	4	2	no	—
54	C₂×#3622	—	4	2	no	—
55	C₂×#3622	C₄²	1	4	yes	—
56	#8278	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	#107	1	4	yes	—
65	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₁₇	#131	1	4	yes	—
70	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#135	1	2	yes	—
73	C₈×C₁₇	—	4	4	no	—
74	#3622	—	2	4	no	—
75	#3622	—	4	2	no	—
76	#3622	—	4	4	no	—
77	#1767	D₁₇	1	1	yes	characteristic
78	C₄×C₄⋊C₈	—	17	1	no	—
79	#1744	—	17	1	no	—
80	C₂×#335	—	17	2	no	—
81	#1277	—	17	2	no	—
82	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₁₇	#424	1	2	yes	—
84	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
85	C₂²×C₁₇	#427	1	1	yes	characteristic
86	C₂²×C₁₇	#430	1	2	yes	—
87	C₄×C₁₇	—	2	4	no	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	4	2	no	—
90	C₄×C₁₇	#405	1	2	yes	—
91	C₄×C₁₇	#433	1	2	yes	—
92	C₄×C₁₇	C₄×D₈	1	2	yes	—
93	C₄×C₁₇	C₄×Q₁₆	1	2	yes	—
94	C₂×C₄⋊C₈	—	17	2	no	—
95	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
96	C₂×#97	—	2	1	no	—
97	C₂×C₄×C₈	—	17	1	no	—
98	C₂×C₄×C₈	—	17	1	no	—
99	C₂×C₄×C₈	—	34	1	no	—
100	C₄³	C₂×D₁₇	1	1	yes	characteristic
101	#336	—	2	1	no	—
102	#336	—	17	4	no	—
103	#336	—	34	1	no	—
104	#335	—	17	8	no	—
105	C₂×C₁₇	#1571	1	2	yes	—
106	C₂×C₁₇	#1572	1	2	yes	—
107	C₂×C₁₇	#1768	1	1	yes	characteristic
108	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
109	C₂×C₁₇	#1778	1	1	yes	characteristic
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	17	2	no	—
112	C₂²×C₈	—	17	2	no	—
113	C₂²×C₈	—	34	2	no	—
114	C₂×C₄²	Dic₃₄	1	2	yes	—
115	C₂×C₄²	C₄×D₁₇	1	2	yes	—
116	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	17	4	no	—
121	C₄⋊C₈	—	34	2	no	—
122	C₄⋊C₈	C₄×D₁₇	1	4	yes	—
123	#97	—	4	2	no	—
124	C₄×C₈	—	17	4	no	—
125	C₄×C₈	—	34	2	no	—
126	C₄×C₈	—	68	2	no	—
127	C₁₇	#8278	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	#63225	1	2	yes	—
129	C₂²×C₄	#63225	1	2	yes	—
130	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	17	4	no	—
137	C₂×C₈	—	34	4	no	—
138	C₂×C₈	—	34	2	no	—
139	C₂×C₈	—	34	4	no	—
140	C₂×C₈	—	34	4	no	—
141	C₂×C₈	—	68	2	no	—
142	C₂×C₈	—	68	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#63228	1	2	yes	—
147	C₄²	#63231	1	2	yes	—
148	C₄²	C₂×C₄×D₁₇	1	2	yes	—
149	C₄²	#63252	1	2	yes	—
150	C₂³	C₂×#63225	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#10587478	1	4	yes	—
158	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
159	C₂×C₄	C₄²×D₁₇	1	1	yes	characteristic
160	C₂×C₄	#10587558	1	1	yes	characteristic
161	C₂×C₄	#10587593	1	4	yes	—
162	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
163	C₂×C₄	C₂×#63228	1	1	yes	characteristic
164	C₂×C₄	C₂×#63231	1	1	yes	characteristic
165	C₂×C₄	#10587636	1	2	yes	—
166	C₈	—	4	4	no	—
167	C₈	—	34	4	no	—
168	C₈	—	68	2	no	—
169	C₈	—	68	4	no	—
170	C₂²	C₄×#63225	1	1	yes	characteristic
171	C₂²	#11762837	1	2	yes	—
172	C₂²	C₂×#10587478	1	1	yes	characteristic
173	C₂²	#11762845	1	1	yes	characteristic
174	C₂²	#11762857	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	#11762766	1	2	yes	—
179	C₄	#11762871	1	2	yes	—
180	C₄	C₄×#63228	1	2	yes	—
181	C₄	C₄×#63231	1	2	yes	—
182	C₂	#14527560	1	2	yes	—
183	C₂	#14527565	1	2	yes	—
184	C₂	#14528644	1	1	yes	characteristic
185	C₂	C₄×#10587478	1	1	yes	characteristic
186	C₂	#14528691	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1767	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1767	D₁₇
hypocenter	C₁₇	#8278
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
12. #11762561	C₄	semidirect
127. C₁₇	#8278	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3622	2
#14528530	2
C₂×#11762561	2
#14526618	2
C₁₇×#1767	2
#8278	17