Subgroups — #17948532

Order 4352.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63216	C₂	1	2	yes	maximal
2	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	—	2	2	no	—
4	C₄⋊#63216	—	2	2	no	—
5	C₄⋊#63216	C₄	1	4	yes	—
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	C₈×#3622	—	2	2	no	—
9	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₈	1	2	yes	—
11	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×#3622	—	2	1	no	—
13	C₂²×#3622	D₄	1	1	yes	characteristic
14	C₂²×#3622	Q₈	1	1	yes	characteristic
15	C₂×#63216	—	2	4	no	—
16	C₂×#63216	—	4	2	no	—
17	C₂×#63216	—	4	2	no	—
18	C₄×C₈×C₁₇	—	2	1	no	—
19	C₄×C₈×C₁₇	—	2	1	no	—
20	C₄×#3622	—	2	1	no	—
21	C₄×#3622	C₈	1	2	yes	—
22	C₄×#3622	C₂×C₄	1	2	yes	—
23	C₄×#3622	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	C₁₆	1	4	yes	complemented
30	C₄²×C₁₇	C₄²	1	1	yes	characteristic
31	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
34	C₂×#3622	—	2	4	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	—	2	1	no	—
37	C₂×#3622	—	2	1	no	—
38	C₂×#3622	C₂²⋊C₄	1	2	yes	—
39	C₂×#3622	C₄⋊C₄	1	2	yes	—
40	C₂×#3622	D₈	1	1	yes	characteristic
41	C₂×#3622	SD₁₆	1	2	yes	—
42	C₂×#3622	Q₁₆	1	1	yes	characteristic
43	#63216	—	4	4	no	—
44	#7151	—	17	1	no	maximal, Sylow
45	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	—	2	2	no	—
47	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
59	C₈×C₁₇	—	2	4	no	—
60	#3622	—	2	4	no	—
61	#3622	—	2	2	no	—
62	#3622	#102	1	2	yes	—
63	#3622	#103	1	2	yes	—
64	C₂×C₄⋊C₁₆	—	17	2	no	—
65	C₂×C₈²	—	17	1	no	—
66	C₂²×C₁₇	#334	1	1	yes	characteristic
67	C₂²×C₁₇	#335	1	1	yes	characteristic
68	C₂²×C₁₇	#336	1	1	yes	characteristic
69	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
70	C₂²×C₁₇	#364	1	2	yes	—
71	C₄×C₁₇	—	2	1	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	#334	1	1	yes	characteristic
74	C₄×C₁₇	#335	1	1	yes	characteristic
75	C₄×C₁₇	#340	1	1	yes	characteristic
76	C₄×C₁₇	#341	1	1	yes	characteristic
77	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
78	C₄×C₁₇	#346	1	1	yes	characteristic
79	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
80	C₂²×C₁₆	—	34	2	no	—
81	C₂×C₄×C₈	—	17	1	no	—
82	C₂×C₄×C₈	—	17	1	no	—
83	C₂×C₄×C₈	#595	1	1	yes	characteristic
84	C₄⋊C₁₆	—	17	4	no	—
85	C₄⋊C₁₆	—	34	2	no	—
86	C₈²	—	34	2	no	—
87	C₂×C₁₇	#1277	1	1	yes	characteristic
88	C₂×C₁₇	#1367	1	2	yes	—
89	C₂×C₁₇	#1370	1	2	yes	—
90	C₂×C₁₇	#1374	1	1	yes	characteristic
91	C₂×C₁₇	#1376	1	1	yes	characteristic
92	C₂²×C₈	—	17	1	no	—
93	C₂²×C₈	—	17	1	no	—
94	C₂²×C₈	—	34	1	no	—
95	C₂²×C₈	#3624	1	2	yes	—
96	C₂×C₄²	C₂×#595	1	1	yes	characteristic
97	C₂×C₁₆	—	34	4	no	—
98	C₂×C₁₆	—	68	2	no	—
99	C₂×C₁₆	—	68	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	17	2	no	—
103	C₄×C₈	—	17	2	no	—
104	C₄×C₈	—	17	2	no	—
105	C₄×C₈	—	34	1	no	—
106	C₁₇	#7151	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₄	#63243	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
109	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	—	17	1	no	—
114	C₂×C₈	—	17	1	no	—
115	C₂×C₈	—	17	2	no	—
116	C₂×C₈	—	17	2	no	—
117	C₂×C₈	—	34	4	no	—
118	C₂×C₈	—	34	1	no	—
119	C₂×C₈	—	34	1	no	—
120	C₂×C₈	—	34	2	no	—
121	C₂×C₈	#63216	1	4	yes	complemented
122	C₄²	#63242	1	1	yes	characteristic
123	C₄²	C₄×#595	1	1	yes	characteristic
124	C₄²	#63247	1	1	yes	characteristic
125	C₄²	C₂²⋊#595	1	1	yes	characteristic
126	C₁₆	—	68	4	no	—
127	C₂³	C₄⋊#3624	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#10587533	1	1	yes	characteristic
130	C₂×C₄	#10587534	1	1	yes	characteristic
131	C₂×C₄	#10587535	1	1	yes	characteristic
132	C₂×C₄	#10587536	1	1	yes	characteristic
133	C₂×C₄	C₂×#63216	1	2	yes	—
134	C₂×C₄	#10587538	1	2	yes	—
135	C₂×C₄	C₄×#3624	1	1	yes	characteristic
136	C₂×C₄	#10587542	1	1	yes	characteristic
137	C₂×C₄	#10587543	1	1	yes	characteristic
138	C₂×C₄	#10587544	1	1	yes	characteristic
139	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	17	2	no	—
143	C₈	—	17	2	no	—
144	C₈	—	34	4	no	—
145	C₈	—	34	2	no	—
146	C₂²	#11763670	1	1	yes	characteristic
147	C₂²	#11763672	1	1	yes	characteristic
148	C₂²	#11763675	1	1	yes	characteristic
149	C₂²	C₄⋊#63216	1	2	yes	—
150	C₂²	#11763731	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#11763671	1	1	yes	characteristic
154	C₄	#11763673	1	1	yes	characteristic
155	C₄	#11763683	1	1	yes	characteristic
156	C₄	#11763685	1	1	yes	characteristic
157	C₄	C₄×#63216	1	1	yes	characteristic
158	C₄	#11763697	1	1	yes	characteristic
159	C₄	C₂²⋊#63216	1	2	yes	—
160	C₂	#14543614	1	1	yes	characteristic
161	C₂	#14543806	1	2	yes	—
162	C₂	#14543813	1	2	yes	—
163	C₂	#14543822	1	1	yes	characteristic
164	C₂	#14543829	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#595
commutator	C₄×C₁₇	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7151
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
29. C₂×C₈×C₁₇	C₁₆	semidirect
106. C₁₇	#7151	semidirect
121. C₂×C₈	#63216	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63216	2
C₂×C₈×#3622	2
#7151	17