Subgroups — #17948488

Order 4352.

306 conjugacy classes of subgroups.

The subgroup lattice has 160 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	C₄	1	4	yes	—
4	C₄×#63216	—	2	2	no	—
5	C₄×#63216	—	2	2	no	—
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	C₈	1	2	yes	—
13	C₂²×#3622	C₂×C₄	1	2	yes	—
14	C₂×#63216	—	2	4	no	—
15	C₂×#63216	—	2	4	no	—
16	C₂×#63216	C₈	1	8	yes	complemented
17	C₄×C₈×C₁₇	—	2	1	no	—
18	C₄×C₈×C₁₇	—	2	1	no	—
19	C₄×#3622	—	2	1	no	—
20	C₄×#3622	—	2	1	no	—
21	C₄×#3622	D₄	1	2	yes	—
22	C₄×#3622	D₄	1	1	yes	characteristic
23	C₄×#3622	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₂×C₈	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	C₂²⋊C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
33	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
34	C₂×#3622	—	2	2	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	C₁₆	1	4	yes	complemented
37	C₂×#3622	C₂²⋊C₄	1	2	yes	—
38	C₂×#3622	C₄⋊C₄	1	2	yes	—
39	C₂×#3622	C₂×C₈	1	4	yes	—
40	C₂×#3622	M₁₆	1	4	yes	—
41	#63216	—	2	8	no	—
42	#7136	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₁₇	#97	1	2	yes	—
47	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
54	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
55	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
56	C₈×C₁₇	—	2	4	no	—
57	#3622	—	2	4	no	—
58	#3622	C₂²⋊C₈	1	4	yes	—
59	#3622	C₄⋊C₈	1	4	yes	—
60	C₂×C₄×C₁₆	—	17	2	no	—
61	C₂×C₈²	—	17	1	no	—
62	C₂²×C₁₇	C₈²	1	1	yes	characteristic
63	C₂²×C₁₇	#322	1	1	yes	characteristic
64	C₂²×C₁₇	#336	1	1	yes	characteristic
65	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
66	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
67	C₂²×C₁₇	#346	1	1	yes	characteristic
68	C₂²×C₁₇	#346	1	1	yes	characteristic
69	C₄×C₁₇	#322	1	2	yes	—
70	C₄×C₁₇	#336	1	2	yes	—
71	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
72	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
73	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
74	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
75	C₂²×C₁₆	—	17	4	no	—
76	C₂×C₄×C₈	—	17	1	no	—
77	C₂×C₄×C₈	—	17	1	no	—
78	C₂×C₄×C₈	#595	1	1	yes	characteristic
79	C₄×C₁₆	—	34	2	no	—
80	C₄×C₁₆	—	34	2	no	—
81	C₈²	—	34	2	no	—
82	C₂×C₁₇	#1273	1	1	yes	characteristic
83	C₂×C₁₇	C₈×C₁₆	1	1	yes	characteristic
84	C₂×C₁₇	#1311	1	1	yes	characteristic
85	C₂×C₁₇	#1312	1	1	yes	characteristic
86	C₂×C₁₇	#1312	1	1	yes	characteristic
87	C₂×C₁₇	#1374	1	1	yes	characteristic
88	C₂×C₁₇	#1374	1	1	yes	characteristic
89	C₂²×C₈	—	17	2	no	—
90	C₂²×C₈	—	17	2	no	—
91	C₂²×C₈	#3624	1	2	yes	—
92	C₂×C₄²	C₂×#595	1	1	yes	characteristic
93	C₂×C₁₆	—	17	8	no	—
94	C₂×C₁₆	—	34	4	no	—
95	C₂×C₁₆	—	34	4	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	17	1	no	—
99	C₄×C₈	—	17	1	no	—
100	C₄×C₈	—	17	2	no	—
101	C₄×C₈	—	34	1	no	—
102	C₄×C₈	—	34	1	no	—
103	C₁₇	#7136	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	#63242	1	1	yes	characteristic
105	C₂²×C₄	C₄×#595	1	1	yes	characteristic
106	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	17	4	no	—
110	C₂×C₈	—	17	4	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	—	17	4	no	—
114	C₂×C₈	—	34	2	no	—
115	C₂×C₈	—	34	2	no	—
116	C₂×C₈	#63216	1	4	yes	complemented
117	C₄²	#63243	1	1	yes	characteristic
118	C₄²	C₄⋊#595	1	1	yes	characteristic
119	C₄²	#63247	1	1	yes	characteristic
120	C₄²	C₂²⋊#595	1	1	yes	characteristic
121	C₁₆	—	34	8	no	—
122	C₂³	C₄×#3624	1	1	yes	characteristic
123	C₂×C₄	#10587529	1	2	yes	—
124	C₂×C₄	#10587530	1	2	yes	—
125	C₂×C₄	C₈×#595	1	2	yes	—
126	C₂×C₄	#10587532	1	2	yes	—
127	C₂×C₄	C₂×#63216	1	2	yes	—
128	C₂×C₄	#10587538	1	2	yes	—
129	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
130	C₂×C₄	#10587542	1	1	yes	characteristic
131	C₂×C₄	#10587543	1	1	yes	characteristic
132	C₂×C₄	#10587544	1	1	yes	characteristic
133	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	17	4	no	—
137	C₈	—	17	4	no	—
138	C₈	—	34	4	no	—
139	C₂²	C₈×#3624	1	1	yes	characteristic
140	C₂²	#11763650	1	1	yes	characteristic
141	C₂²	#11763675	1	1	yes	characteristic
142	C₂²	C₄×#63216	1	1	yes	characteristic
143	C₂²	#11763695	1	1	yes	characteristic
144	C₂²	#11763697	1	1	yes	characteristic
145	C₂²	#11763698	1	1	yes	characteristic
146	C₄	#11763651	1	2	yes	—
147	C₄	#11763674	1	2	yes	—
148	C₄	C₂²⋊#63216	1	2	yes	—
149	C₄	#11763704	1	2	yes	—
150	C₄	C₄⋊#63216	1	2	yes	—
151	C₄	#11763730	1	2	yes	—
152	C₂	#14543606	1	1	yes	characteristic
153	C₂	C₈×#63216	1	1	yes	characteristic
154	C₂	#14543696	1	1	yes	characteristic
155	C₂	#14543697	1	1	yes	characteristic
156	C₂	#14543699	1	1	yes	characteristic
157	C₂	#14543822	1	1	yes	characteristic
158	C₂	#14543823	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂×#595
commutator	C₂×C₁₇	C₈×C₁₆
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	#7136
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
16. C₂×#63216	C₈	semidirect
29. C₂×C₈×C₁₇	C₁₆	semidirect
36. C₂×#3622	C₁₆	semidirect
103. C₁₇	#7136	semidirect
116. 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
#7136	17