Subgroups — #17978769

Order 4352.

486 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	8	yes	maximal, complemented
2	#14545145	C₂	1	4	yes	maximal
3	#14545141	C₂	1	2	yes	maximal
4	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#63216	C₂²	1	4	yes	—
6	C₄⋊#63216	C₂²	1	8	yes	—
7	C₂²⋊#63216	C₂²	1	4	yes	—
8	#11763697	C₂²	1	8	yes	—
9	C₄×#63216	C₂²	1	4	yes	—
10	Q₈×#3622	C₄	1	4	yes	—
11	Q₈×#3622	C₂²	1	4	yes	—
12	C₂×C₄⋊#3622	C₄	1	2	yes	—
13	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	C₂×C₄×#3622	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
17	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	C₈	1	4	yes	—
19	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂²×#3622	C₂×C₄	1	2	yes	—
26	C₂²×#3622	C₂³	1	1	yes	characteristic
27	C₂×#63216	C₂³	1	8	yes	—
28	C₄⋊#3622	C₂×C₄	1	8	yes	—
29	C₄⋊#3622	C₂×C₄	1	4	yes	—
30	C₄×#3622	C₂×C₄	1	2	yes	—
31	C₄×#3622	C₂×C₄	1	4	yes	—
32	C₄×#3622	C₂³	1	2	yes	—
33	C₄×#3622	C₂³	1	4	yes	—
34	C₂×Q₈×C₁₇	—	2	4	no	—
35	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
42	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
43	C₄²×C₁₇	C₂×C₈	1	4	yes	—
44	C₄²×C₁₇	C₂×C₈	1	2	yes	—
45	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
46	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
47	C₂×#3622	—	2	2	no	—
48	C₂×#3622	—	4	1	no	—
49	C₂×#3622	—	4	2	no	—
50	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
51	C₂×#3622	C₂²×C₄	1	2	yes	—
52	C₂×#3622	C₂²×C₄	1	4	yes	—
53	C₂×#3622	C₂⁴	1	1	yes	characteristic
54	#63216	—	2	8	no	—
55	#33497	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
57	Q₈×C₁₇	—	2	8	no	—
58	C₂×C₄×C₁₇	—	2	4	no	—
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	4	2	no	—
62	C₂×C₄×C₁₇	—	4	1	no	—
63	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
64	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
66	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
68	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
70	#3622	—	2	1	no	—
71	#3622	—	2	2	no	—
72	#3622	—	2	4	no	—
73	#3622	2₋¹⁺⁴	1	2	yes	—
74	C₂×C₈×Q₈	—	17	1	no	—
75	#2183	—	17	8	no	—
76	#2163	—	17	4	no	—
77	#2162	—	17	2	no	—
78	C₂²×C₁₇	—	2	1	no	—
79	C₂²×C₁₇	—	2	1	no	—
80	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
81	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
82	C₄×C₁₇	—	2	4	no	—
83	C₄×C₁₇	—	2	4	no	—
84	C₄×C₁₇	—	2	2	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	C₂×M₃₂	1	4	yes	—
87	C₄×C₁₇	#519	1	1	yes	characteristic
88	C₄×C₁₇	#568	1	1	yes	characteristic
89	C₂×C₄×Q₈	#595	1	1	yes	characteristic
90	C₈×Q₈	—	17	4	no	—
91	C₈×Q₈	—	17	4	no	—
92	C₂×C₄⋊C₈	—	17	2	no	—
93	C₂×C₄⋊C₈	—	17	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	C₂×C₄×C₈	—	17	2	no	—
96	C₄⋊C₁₆	—	17	4	no	—
97	C₄⋊C₁₆	—	17	8	no	—
98	C₂²⋊C₁₆	—	17	4	no	—
99	#346	—	17	8	no	—
100	C₄×C₁₆	—	17	4	no	—
101	C₂×C₁₇	—	2	2	no	—
102	C₂×C₁₇	#2982	1	1	yes	characteristic
103	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
104	C₂×C₁₇	#3407	1	1	yes	characteristic
105	C₂²×Q₈	#3624	1	1	yes	characteristic
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	17	2	no	—
109	C₄×Q₈	#3624	1	4	yes	—
110	C₄×Q₈	C₂×#595	1	4	yes	—
111	C₂×C₄⋊C₄	#3624	1	2	yes	—
112	C₂×C₄⋊C₄	#3624	1	1	yes	characteristic
113	C₂×C₄²	C₂×#595	1	2	yes	—
114	C₂×C₄²	C₂×#595	1	1	yes	characteristic
115	C₂×C₁₆	—	17	8	no	—
116	C₄⋊C₈	—	17	8	no	—
117	C₄⋊C₈	—	17	4	no	—
118	C₄×C₈	—	17	2	no	—
119	C₄×C₈	—	17	4	no	—
120	C₄×C₈	—	17	2	no	—
121	C₄×C₈	—	17	4	no	—
122	C₁₇	#33497	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂×#3624	1	4	yes	—
125	C₂²×C₄	C₂×#3624	1	2	yes	—
126	C₂²×C₄	C₂×#3624	1	2	yes	—
127	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
128	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
129	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	—	17	4	no	—
134	C₂×C₈	—	34	2	no	—
135	C₂×C₈	—	68	1	no	—
136	C₂×C₈	—	68	2	no	—
137	C₄⋊C₄	C₂×#3624	1	8	yes	—
138	C₄⋊C₄	C₂×#3624	1	4	yes	—
139	C₄²	C₂×#3624	1	4	yes	—
140	C₄²	C₂×#3624	1	2	yes	—
141	C₄²	C₂²×#595	1	4	yes	—
142	C₄²	C₂²×#595	1	2	yes	—
143	C₁₆	—	34	8	no	—
144	C₂³	C₂²×#3624	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	#10587538	1	4	yes	—
152	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
153	C₂×C₄	C₂²×#3624	1	4	yes	—
154	C₂×C₄	C₂²×#3624	1	2	yes	—
155	C₂×C₄	C₂²×#3624	1	4	yes	—
156	C₂×C₄	C₂²×#3624	1	2	yes	—
157	C₂×C₄	C₂³×#595	1	1	yes	characteristic
158	C₈	—	17	2	no	—
159	C₈	—	34	1	no	—
160	C₈	—	34	2	no	—
161	C₈	—	34	4	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#10587538	1	2	yes	—
165	C₂²	C₂³×#3624	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	C₂×#10587538	1	4	yes	—
171	C₄	#11763877	1	1	yes	characteristic
172	C₄	#11763889	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#14545677	1	1	yes	characteristic
175	C₂	C₂²×#10587538	1	1	yes	characteristic
176	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₇	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×C₄×Q₈	#595
hypocenter	C₁₇	#33497
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
122. C₁₇	#33497	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545145	2
#14545141	2
C₂×Q₈×#3622	2
#33497	17