Subgroups — #17978844

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545208	C₂	1	8	yes	maximal, complemented
2	#14545205	C₂	1	4	yes	maximal, complemented
3	C₂×#11763741	C₂	1	2	yes	maximal, complemented
4	C₂×#11763227	C₂	1	1	yes	maximal, characteristic
5	#11763742	C₂²	1	8	yes	complemented
6	#11763741	C₂²	1	4	yes	—
7	#11763741	C₂²	1	4	yes	complemented
8	#11763740	C₂²	1	8	yes	complemented
9	C₂×#10587467	C₂²	1	4	yes	complemented
10	C₂×#10587587	C₄	1	1	yes	characteristic
11	#11763227	C₄	1	4	yes	—
12	#11763227	C₂²	1	4	yes	—
13	C₂×#10587469	C₄	1	2	yes	—
14	C₂²×#63214	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
16	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
17	C₂×#63252	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#63223	C₈	1	1	yes	characteristic
20	#10587587	C₈	1	4	yes	—
21	#10587587	C₂×C₄	1	4	yes	—
22	C₂×#63218	C₈	1	2	yes	—
23	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
24	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
26	C₂×#63214	C₂×C₄	1	1	yes	characteristic
27	C₂×#63214	C₂×C₄	1	2	yes	—
28	C₂×#63214	C₂³	1	1	yes	characteristic
29	C₂×#63214	C₂³	1	2	yes	—
30	#10587469	C₂×C₄	1	8	yes	—
31	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
32	C₁₆×D₁₇	C₂³	1	4	yes	—
33	#10587467	C₂³	1	8	yes	—
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₇	C₁₆	1	2	yes	—
36	#63252	C₁₆	1	4	yes	—
37	#63252	C₂×C₈	1	4	yes	—
38	C₂×D₆₈	C₁₆	1	1	yes	characteristic
39	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
40	C₂×Dic₃₄	C₁₆	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
45	#63223	C₂×C₈	1	4	yes	—
46	C₂×#3622	C₂×C₈	1	2	yes	—
47	#63218	C₂×C₈	1	8	yes	—
48	C₈×D₁₇	C₂×C₈	1	4	yes	—
49	C₈×D₁₇	C₂²×C₄	1	4	yes	—
50	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
51	#63214	C₂²×C₄	1	1	yes	characteristic
52	#63214	C₂²×C₄	1	2	yes	—
53	#63214	C₂⁴	1	1	yes	characteristic
54	#33974	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂²×D₁₇	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
61	C₂²⋊D₁₇	C₂×C₁₆	1	8	yes	—
62	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
63	D₆₈	C₂×C₁₆	1	4	yes	—
64	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
65	C₄×D₁₇	C₂²×C₈	1	4	yes	—
66	Dic₃₄	C₂×C₁₆	1	4	yes	—
67	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₈×C₁₇	C₂²×C₈	1	2	yes	—
69	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
70	#3622	C₂²×C₈	1	4	yes	—
71	C₂×#504	—	17	1	no	—
72	#2258	—	17	8	no	—
73	C₂×M₆₄	—	17	2	no	—
74	C₂×M₆₄	—	17	4	no	—
75	C₂²×C₁₇	—	4	1	no	—
76	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
77	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
78	C₂×D₁₇	—	4	2	no	—
79	C₂×D₁₇	C₂²×C₁₆	1	4	yes	—
80	C₄×C₁₇	C₂²×C₁₆	1	2	yes	—
81	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
82	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
83	Dic₁₇	C₂²×C₁₆	1	4	yes	—
84	C₂×#131	—	17	1	no	—
85	#504	—	17	4	no	—
86	#504	—	17	4	no	—
87	C₂×M₃₂	—	17	2	no	—
88	C₂×M₃₂	#595	1	1	yes	characteristic
89	C₂²×C₁₆	—	17	1	no	—
90	C₂²×C₁₆	—	17	2	no	—
91	M₆₄	—	17	4	no	—
92	M₆₄	—	17	4	no	—
93	M₆₄	—	17	8	no	—
94	C₂×C₃₂	—	17	4	no	—
95	C₂×C₃₂	—	17	8	no	—
96	C₂×C₁₇	—	2	1	no	—
97	C₂×C₁₇	—	2	2	no	—
98	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
99	D₁₇	—	2	4	no	—
100	C₂×#41	—	17	1	no	—
101	#131	—	17	4	no	—
102	#131	—	17	4	no	—
103	C₂×M₁₆	—	17	1	no	—
104	C₂×M₁₆	—	17	2	no	—
105	C₂²×C₈	—	17	2	no	—
106	C₂²×C₈	C₂×#595	1	1	yes	characteristic
107	M₃₂	—	17	8	no	—
108	M₃₂	C₂×#595	1	4	yes	—
109	C₂×C₁₆	—	17	1	no	—
110	C₂×C₁₆	—	17	2	no	—
111	C₂×C₁₆	—	17	2	no	—
112	C₂×C₁₆	—	17	1	no	—
113	C₂×C₁₆	—	17	4	no	—
114	C₂×C₁₆	—	17	4	no	—
115	C₂×C₁₆	C₂×#595	1	2	yes	—
116	C₃₂	—	17	8	no	—
117	C₁₇	#33974	1	1	yes	characteristic, complemented, Sylow
118	#41	—	17	4	no	—
119	#41	—	17	4	no	—
120	C₂×Q₈	—	17	1	no	—
121	C₂×D₄	—	17	2	no	—
122	C₂×D₄	—	17	1	no	—
123	C₂²×C₄	—	17	2	no	—
124	C₂²×C₄	#63242	1	1	yes	characteristic
125	M₁₆	—	17	4	no	—
126	M₁₆	—	17	8	no	—
127	C₂×C₈	—	17	2	no	—
128	C₂×C₈	—	17	4	no	—
129	C₂×C₈	—	17	4	no	—
130	C₂×C₈	#63242	1	1	yes	characteristic
131	C₂×C₈	#63242	1	2	yes	—
132	C₂×C₈	C₂²×#595	1	1	yes	characteristic
133	C₂×C₈	C₂²×#595	1	2	yes	—
134	C₁₆	—	17	1	no	—
135	C₁₆	—	17	1	no	—
136	C₁₆	—	17	2	no	—
137	C₁₆	C₂²×#595	1	4	yes	—
138	C₂³	—	17	2	no	—
139	C₂³	#10587529	1	1	yes	characteristic
140	Q₈	—	17	4	no	—
141	D₄	—	17	8	no	—
142	D₄	—	17	4	no	—
143	C₂×C₄	—	17	2	no	—
144	C₂×C₄	—	17	4	no	—
145	C₂×C₄	—	17	4	no	—
146	C₂×C₄	#10587529	1	1	yes	characteristic
147	C₂×C₄	#10587529	1	2	yes	—
148	C₂×C₄	C₂×#63242	1	1	yes	characteristic
149	C₂×C₄	C₂×#63242	1	2	yes	—
150	C₈	—	17	4	no	—
151	C₈	C₂×#63242	1	1	yes	characteristic
152	C₈	C₂×#63242	1	2	yes	—
153	C₈	C₂³×#595	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	—	17	4	no	—
156	C₂²	—	68	2	no	—
157	C₂²	C₂×#10587529	1	1	yes	characteristic
158	C₂²	C₂×#10587529	1	2	yes	—
159	C₄	—	17	4	no	—
160	C₄	C₂×#10587529	1	1	yes	characteristic
161	C₄	C₂×#10587529	1	2	yes	—
162	C₄	C₂²×#63242	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	2	no	—
165	C₂	—	34	4	no	—
166	C₂	C₂²×#10587529	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#595
commutator, socle	C₂×C₁₇	C₂³×C₁₆
Fitting	C₂×C₁₇×M₃₂	C₄
prime radical	C₂×#63252	C₈
cosocle	#63214	C₂⁴
hypercenter	C₂×M₃₂	#595
hypocenter	C₁₇	#33974
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. #14545208	C₂	semidirect
2. #14545205	C₂	semidirect
3. C₂×#11763741	C₂	semidirect
5. #11763742	C₂²	semidirect
7. #11763741	C₂²	semidirect
8. #11763740	C₂²	semidirect
9. C₂×#10587467	C₂²	semidirect
117. C₁₇	#33974	semidirect

Maximal subgroups

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

subgroup	index
#14545208	2
#14545205	2
C₂×#11763741	2
C₂×#11763227	2
#33974	17