Subgroups — #17978773

Order 4352.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	6	yes	maximal, complemented
2	Q₈×#63216	C₂	1	2	yes	maximal, complemented
3	#14545145	C₂	1	3	yes	maximal
4	#14545141	C₂	1	3	yes	maximal
5	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#63216	C₂²	1	6	yes	—
7	C₄⋊#63216	C₂²	1	6	yes	—
8	C₂²⋊#63216	C₂²	1	1	yes	characteristic
9	C₂²⋊#63216	C₂²	1	3	yes	—
10	#11763697	C₂²	1	6	yes	—
11	C₄×#63216	C₂²	1	6	yes	—
12	Q₈×#3622	C₄	1	3	yes	—
13	Q₈×#3622	C₄	1	1	yes	characteristic
14	Q₈×#3622	C₂²	1	1	yes	characteristic
15	Q₈×#3622	C₂²	1	3	yes	—
16	C₂×C₄⋊#3622	C₄	1	3	yes	—
17	C₂×C₄×#3622	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
19	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₈	1	3	yes	—
21	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
22	C₄×Q₈×C₁₇	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	C₈	1	3	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
26	C₂²×#3622	C₂×C₄	1	3	yes	—
27	C₂²×#3622	C₂³	1	1	yes	characteristic
28	C₂×#63216	C₂³	1	2	yes	—
29	C₂×#63216	C₂³	1	6	yes	—
30	C₄⋊#3622	C₂×C₄	1	3	yes	—
31	C₄⋊#3622	C₂×C₄	1	6	yes	—
32	C₄⋊#3622	C₂×C₄	1	3	yes	—
33	C₄×#3622	C₂×C₄	1	3	yes	—
34	C₄×#3622	C₂×C₄	1	3	yes	—
35	C₄×#3622	C₂³	1	3	yes	—
36	C₄×#3622	C₂³	1	3	yes	—
37	C₂×Q₈×C₁₇	—	2	4	no	—
38	C₂×Q₈×C₁₇	C₂×C₈	1	3	yes	—
39	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₄⋊C₄×C₁₇	C₂×C₈	1	6	yes	—
44	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
45	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
46	C₄²×C₁₇	C₂×C₈	1	3	yes	—
47	C₄²×C₁₇	C₂×C₈	1	3	yes	—
48	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
49	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	—	4	3	no	—
52	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
53	C₂×#3622	C₂²×C₄	1	3	yes	—
54	C₂×#3622	C₂²×C₄	1	3	yes	—
55	C₂×#3622	C₂⁴	1	1	yes	characteristic
56	#63216	—	2	2	no	—
57	#63216	—	2	6	no	—
58	#33498	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	Q₈×C₁₇	—	2	6	no	—
61	Q₈×C₁₇	M₃₂	1	4	yes	—
62	C₂×C₄×C₁₇	—	2	6	no	—
63	C₂×C₄×C₁₇	—	2	1	no	—
64	C₂×C₄×C₁₇	—	2	1	no	—
65	C₂×C₄×C₁₇	—	2	6	no	—
66	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
72	#3622	—	2	1	no	—
73	#3622	—	2	3	no	—
74	#3622	—	2	3	no	—
75	#3622	2₋¹⁺⁴	1	2	yes	—
76	C₂×C₈×Q₈	—	17	1	no	—
77	#2183	—	17	6	no	—
78	Q₈×C₁₆	—	17	2	no	—
79	#2163	—	17	3	no	—
80	#2162	—	17	3	no	—
81	C₂²×C₁₇	—	2	1	no	—
82	C₂²×C₁₇	—	2	1	no	—
83	C₂²×C₁₇	#504	1	2	yes	—
84	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
85	C₄×C₁₇	—	2	3	no	—
86	C₄×C₁₇	—	2	3	no	—
87	C₄×C₁₇	—	2	3	no	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	C₂×M₃₂	1	6	yes	—
90	C₄×C₁₇	#519	1	1	yes	characteristic
91	C₄×C₁₇	#568	1	1	yes	characteristic
92	C₂×C₄×Q₈	#595	1	1	yes	characteristic
93	C₈×Q₈	—	17	1	no	—
94	C₈×Q₈	—	17	3	no	—
95	C₈×Q₈	—	17	3	no	—
96	C₈×Q₈	—	17	1	no	—
97	C₂×C₄⋊C₈	—	17	3	no	—
98	C₂×C₄×C₈	—	17	3	no	—
99	C₄⋊C₁₆	—	17	6	no	—
100	C₄⋊C₁₆	—	17	6	no	—
101	C₂²⋊C₁₆	—	17	1	no	—
102	C₂²⋊C₁₆	—	17	3	no	—
103	#346	—	17	6	no	—
104	C₄×C₁₆	—	17	6	no	—
105	C₂×C₁₇	—	2	2	no	—
106	C₂×C₁₇	#2982	1	1	yes	characteristic
107	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
108	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
109	C₂²×Q₈	#3624	1	1	yes	characteristic
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	—	17	3	no	—
112	C₄×Q₈	#3624	1	3	yes	—
113	C₄×Q₈	#3624	1	1	yes	characteristic
114	C₄×Q₈	C₂×#595	1	3	yes	—
115	C₄×Q₈	C₂×#595	1	1	yes	characteristic
116	C₂×C₄⋊C₄	#3624	1	3	yes	—
117	C₂×C₄²	C₂×#595	1	3	yes	—
118	C₂×C₁₆	—	17	2	no	—
119	C₂×C₁₆	—	17	6	no	—
120	C₄⋊C₈	—	17	3	no	—
121	C₄⋊C₈	—	17	6	no	—
122	C₄⋊C₈	—	17	3	no	—
123	C₄×C₈	—	17	3	no	—
124	C₄×C₈	—	17	3	no	—
125	C₄×C₈	—	17	3	no	—
126	C₄×C₈	—	17	3	no	—
127	C₁₇	#33498	1	1	yes	characteristic, complemented, Sylow
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	C₂×#3624	1	3	yes	—
130	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
131	C₂²×C₄	C₂×#3624	1	3	yes	—
132	C₂²×C₄	C₂×#3624	1	3	yes	—
133	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	17	1	no	—
136	C₂×C₈	—	17	3	no	—
137	C₂×C₈	—	17	3	no	—
138	C₂×C₈	—	34	2	no	—
139	C₂×C₈	—	68	3	no	—
140	C₄⋊C₄	C₂×#3624	1	6	yes	—
141	C₄⋊C₄	C₂×#3624	1	3	yes	—
142	C₄⋊C₄	C₂×#3624	1	3	yes	—
143	C₄²	C₂×#3624	1	3	yes	—
144	C₄²	C₂×#3624	1	3	yes	—
145	C₄²	C₂²×#595	1	3	yes	—
146	C₄²	C₂²×#595	1	3	yes	—
147	C₁₆	—	34	2	no	—
148	C₁₆	—	34	6	no	—
149	C₂³	C₂²×#3624	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	#10587538	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
157	C₂×C₄	C₂²×#3624	1	3	yes	—
158	C₂×C₄	C₂²×#3624	1	3	yes	—
159	C₂×C₄	C₂²×#3624	1	3	yes	—
160	C₂×C₄	C₂²×#3624	1	3	yes	—
161	C₂×C₄	C₂³×#595	1	1	yes	characteristic
162	C₈	—	17	2	no	—
163	C₈	—	34	1	no	—
164	C₈	—	34	3	no	—
165	C₈	—	34	3	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#11763867	1	2	yes	—
169	C₂²	C₂³×#3624	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂×#10587538	1	6	yes	—
175	C₄	#11763877	1	1	yes	characteristic
176	C₄	#11763889	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	#14545677	1	1	yes	characteristic
179	C₂	C₂²×#10587538	1	1	yes	characteristic
180	C₂	C₂×#11763867	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	#33498
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
2. Q₈×#63216	C₂	semidirect
127. C₁₇	#33498	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
Q₈×#63216	2
#14545145	2
#14545141	2
C₂×Q₈×#3622	2
#33498	17